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Shamos’s Catalog of the Real Numbers
by Michael Ian Shamos, Ph.D., J.D.
School of Computer Science Carnegie Mellon University
2011
Copyright © 2011 Michael Ian Shamos. All rights reserved.
PREFACE i
Preface
Have you ever looked at the first few digits the decimal expansion of a number and tried to recognize which number it was? You probably do it all the time without realizing it. When you see 3.14159... or 2.7182818... you have no trouble spotting and e, respectively. But are
you on familiar terms with 0.572467033... or 0.405465108...? These are 2 3
12
and
log log3 2 , both of which have interesting stories to tell. You will find them, along with those of more than 10,000 other numbers, listed in this book. But why would someone bother to compile a listing of such facts, which may seem on the surface to be a compendium of numerical trivia?
Unless you had spent some time with 2 3
12
, you might not realize that it is the sum of two
very different series, ( )cos
1 21
k
k
k
k and
1
)12()2(k
kkk . The proof that each sum
equals 2 3
12
is elementary; what connection there may be between the two series remains
elusive. The number log log3 2 turns up as the sum of several series and the value of certain
definite integrals: 1
31k
k k
, ( )
1
2
1
1
k
kk k
,
1 )2)(1( xx
dx and
0 12 xe
dx . If your research led to
either number, this information might well lead you to additional findings or help you explain what you had already discovered.
Mathematics is an adventure, and a very personal one. Its practitioners are all exploring the same territory, but without a roadmap they all follow different paths. Unfortunately, the mathematical literature dwells primarily on final results, the destinations, if you will, rather than the paths the explorers took in arriving at them. We are presented with an orderly sequence of definitions, lemmas, theorems and corollaries, but are often left mystified as to how they were discovered in the first place.
The real empirical process of mathematics, that is, the laborious trek through false conjectures and blind alleys and taking wrong turns, is a messy business, ill-suited to the crisp format of referred journals. Possibly the unattractiveness of the topic accounts for the scarcity of writing on the subject. How many unsuccessful models of computation did Turing invent before he hit on the Turing machine? How did he even generate the models? How did he realize that the Halting Problem was interesting to consider, let alone imagine that it was unsolvable? What did his scrap paper look like?
This book is a field guide to the real numbers, similar in many ways to a naturalist’s handbook. When a birdwatcher spots what he thinks may be a rare species, he notes some of its characteristics, such as color or shape of beak, and looks it up in a field guide to verify his identification. Likewise, when you see part of a number (its initial decimal digits), you should be able to look it up here and find out more about it. The book is just a lexicographic list of 10,000 numbers arranged by initial digits of their decimal expansions, along with expressions whose values share the same initial digits.
PREFACE ii
This book was inspired by Neil Sloane’s A Handbook of Integer Sequences. That work, first published in 1971, is an elaborately-compiled list of the initial terms of over 2000 integer sequences arranged in lexicographic order. When one is confronted with the first few term of an unfamiliar sequence, such as {1, 2, 5, 16, 65, 326, 1957,...}, a glance at Sloane reveals that this sequence, number 589, is the total number of permutations of all subsets of n objects, that
is,
n
k
n
k
n
k jn
kn
n
k
nk
111 !
1!
)!(
!! , which is the closest integer to n e! . Sloane’s book is
extremely useful for finding relationships among combinatorial problems. It leads to discoveries, which help in turn generate conjectures that hopefully lead to theorems.
What impelled me to compile this catalog was an investigation into Riemann’s -function that I began in 1990. It struck me as incredible that we have landed men on the moon but still
have not found a closed-form expression for ( )31
31
kk
. I say incredible because in the 18th
century Euler proved that ( )sk s
k
1
1
is a rational multiple of s for all even s, and gave a
formula for the coefficient in terms of Bernoulli numbers. No such formula is known for any odd s. It is not even known whether a closed form exists.
I began exploring such sums as 1
2 k ks s tk
, 1
1 k ks s tk
and ( )
1
2
k
s s tk k k
, whose values
can often be written as simple expressions involving various values of the -function. For
example, 1 5
43
5 32 k kk
( ), 2
coth1
6
1 2
124
k kk and the remarkable
1 1
124 22 k kk
. To keep track of these many results, I kept them in a list by numerical
value, conveniently obtained using Mathematica®. Very quickly, after experimenting with
sums of the form ( )ak bk
11
, I began to notice connections that led to the following
theorem. Theorem 1. For a 1 an integer, b an integer such that a b 2 and c a positive real
number, then 1
1 1ck k
aj b
ca b bk
jj
( ) .
A proof is given at the end of the chapter. The proof is elementary, but I never would have been motivated to write down the theorem in the first place without studying the emerging list of expressions sorted by numerical value.
After a time, I became somewhat obsessed with expanding the list of values and expressions and found that as it grew it became more and more useful. I began to add material not directly related to my research and found that it raised more questions than it answered. It became for me both a handbook and a research manifesto.
PREFACE iii
For example, when I observed that the sums 1
221
k
k
and ( )2
4 11
kk
k
shared the same 20-
digit prefix, this evoked considerable interest, since sums of the form 1
0 abk
k
are very difficult
to treat analytically. After verifying that the sums were equal to over 500 digits, I began to
look for generalizations, exploring such sums as 1
230
k
k ,
1
440
k
k ,
( )2
2 11
kk
k
and ( )3
4 11
kk
k
, to
give a few examples. The results suggested a conjecture, which soon yielded the following theorem:
Theorem 2. For c > 1 real and p prime, 1
11 1c
kp
cpi
kpk
i
( ), where ( )n is the
Moebius function. For a proof, see the end of the chapter. Alternatively, the theorem may be expressed in the
form
1 /11
1
1
)(
kppk
k k
aa
kp . This allows us to relate such strange expressions as
12
1
kk
and
13
3
1
kk
e to Moebius function sums. Unfortunately, the theorem seems to
shed no light on trickier sums like 12
1 kk
k
and 1
1 kkk
k
.
This book is what is known in computer science as a hash table, a structure for indexing in a small space a potentially huge number of objects. It is similar to the drawers in the card catalog of a library. Imagine that a library only has 26 card drawers and that it places catalog cards in each draw without sorting them. To find a book by Taylor, you must go to drawer “T” and look through all of the cards. If the library has only a hundred books and the authors’ names are reasonably distributed over the alphabet, you can expect to have to look at only two or three cards to find the book you want (if the library has it). If the library has a million books, this method would be impractical because you would have to leaf through more than 20,000 cards on the average. That’s because about 40,000 cards will “hash” to the same drawer. However, if the library had more card drawers, let’s say 17,576 of them labeled “AAA” through “ZZZ,” you would have a much easier time since each drawer would only have about 60 cards and you would expect to leaf through 30 of them to find a book the library holds (all 60 to verify that the book is not there).
Hashing was originally used to create small lookup tables for lengthy data elements. The term “hashing” in this context means cutting up into small pieces. Suppose that a company has 300 employees and used social security numbers (which have nine digits) to identify them. A table large enough to accommodate all possible social security numbers would need 109 entries, too large even to store on most computer hard disks. Such a table seems wasteful anyway, since only a few hundred locations are going to be occupied. One might try using just the first three digits of the number, which reduces the space required to just a thousand locations. A drawback is that the records for several employees would have to be stored at that location (if their social security numbers began with the same three digits). An attempt to store
PREFACE iv
a record in a location where one already exists is called a “collision” and must be resolved using other methods. There will not be many collisions in a sparse table unless the numbers tend to cluster strongly (as the initial digits of social security numbers do). Using this scheme in a small community where workers have lived all their lives might result in everyone hashing to the same location because of the geographical scheme used to assign social security numbers. A better method would be to use the three least-significant digits or to construct a “hash function,” one that is designed to spread the records evenly over the storage locations. A suitable hash function for social security numbers might be to square them and use the middle three digits. (I haven’t tried this.)
Getting back to the library, notice that any two cards in a drawer may be identical (for multiple copies of the same book) or different. The only thing they are guaranteed to have in common is the first three letters of the author’s last name. Now imagine that the library sets up 2620 drawers. Aside from the fact that you might have to travel for some time to get to a drawer, once you arrive there you will probably find very few cards in it, and the chances are excellent that if there is more than one card, you are dealing with the same author or two authors who have identical names. Of course, it’s still possible for two authors to have different names that share the same 20 first letters.
This book is a hash table for the real numbers. It has 1020 drawers, but to save paper only the nonempty drawers are shown. But it differs very significantly from the card catalog of a library, in which the indexing terms (the “keys”) have finite length. When you examine two catalog cards, you can tell immediately whether the authors have the same name, even if their names are very long. When you look up an entry in this book, you can’t tell immediately whether the expressions given are identically equal or not. That’s because their decimal expansions are of infinite length and all you know is that they share the same first 20 digits. With that warning, I can tell you there is no case in this book known to the author in which two listed quantities have the same initial 20 digits but are known not to be equal. There are many cases, however, in which the author has no idea whether two expressions for the same entry are equal, but therein lies the opportunity for much research by author and reader alike.
As a final example, seeing with some surprise that sin sink
k
k
kk k
1
2
21
1
2
led me to an
investigation of the conditions under which the sum of an infinite series equals the sum of the squares of its individual terms. (We may call such series “element-squaring.”) Element-squaring series rarely occur naturally but are in fact highly abundant:
Theorem 3. Every convergent series of reals B bkk
1
such that B bkk
( )2 2
1
can
be transformed into an element-squaring real series
B bkk 0
by prepending to B a single real
term b0 iff B B( )2 1
4 . Proof: see the end of the chapter.
I hope that these examples show that the list offers both information and challenges to the reader. It is a reference work, but also a storehouse of questions. Anyone who is fascinated by mathematics will be able to scan through the pages here and wonder if, how and why the entries for a given number are related.
Eventually it became clear that perusing the catalog in pursuit of gems by eyeball would not be effective. I then wrote software to search for relationships among the entries, seeking,
PREFACE v
for example, numbers p, q, r and s such that p = f (q, r, s) for interesting functions f. With 10,000 entries, such searches are fruitful. With a million entries more computing power would be needed. Some of the results of these investigations can be found in the author’s papers, e.g., Overcounting Functions1 and Property Enumerators and a Partial Sum Theorem2. I contend that this method of search yields a rich stream of conjectures, which in turn lead to interesting mathematical investigations.
The selection of entries in this book necessarily reflects the interests and prejudices of the author. This will explain the relatively large number of expressions involving the -function. For this I make no apology; there are more interesting real numbers than would fit in any book, even listing only their first 20 digits. The reader is encouraged to develop his own supplements to this catalog by using Mathematica to develop a list of numbers useful in his or her own field of study.
Professor George Hardy once related the now-famous story of his visit to an ailing Ramanujan in which Ramanujan asked him the number of the taxicab that brought him. Hardy thought the number, 1729, was a dull one, but Ramanujan instantly countered that it was the smallest integer that can be written as the sum of two cubes in two different ways, as 1 123 3 and 9 103 3 . Hardy marveled that Ramanujan seemed to be on intimate terms with the integers and regarded them as his personal friends3. It is hoped that this book will help make the real numbers the reader’s personal friends.
Proofs of Theorems 1-3
Theorem 1. For a 1 an integer, b an integer such that a b 2 and c a positive real
number, then ( )aj b
c ck kjj
a b bk
1 1
1 . A proof is given at the end of the chapter.
Proof: The left-hand sum may be written as the double sum 1
11 k caj b jkj
. Reversing the
order of summation,
1 11 11 1
11111
k jjab
k jjajb
j kjbaj
ckkckkck , which equals
1 1
1
1
1 1k ck ck kbk
a a b bk
. QED.
Theorem 2. For c > 1 real and p prime, 1
11 1c
kp
cpi
kpk
i
( ), where ( )n is the Moebius
function.
Proof: ( ) ( )kp
c
kp
ckpk
jkpjk
11 11
. Note that every term of 1
1 c pi
i
appears at least once in
the double sum, certainly in the case when k 1 and j pi . We will show that the sum of all 1 Available at http://euro.ecom.cmu.edu/people/faculty/mshamos/Overcounting.pdf 2 Available at http://euro.ecom.cmu.edu/people/faculty/mshamos/PST.pdf 3This incident is partially related in Kanigel, R. The Man Who Knew Infinity. New York: Washington Square Press (1991). ISBN 0-671-75061-5. The book is a fascinating biography of Srinivasa Ramanujan.
PREFACE vi
coefficients of 1
c pi in the double sum is zero except when k 1 and j is a power of p, in which
case ( )kp 1, and the result follows. Suppose that jk is a power of p. Then if k 1, ( )kp 0 since kp has a repeated factor.
Therefore, only the terms with k 1 contribute to the sum and ( )kp 1 each j a power of p. Now consider all pairs j, k with jk i not a power of p. Suppose that i has the factorization
p paqaq
11 , with each of the ai 0 . If p k or if k has any repeated prime factors, then
( )kp 0. Since k i, it suffices to consider only those k whose factors are products of subsets Pk of distinct primes taken from the set pppP q 1 , a set having cardinality q 1 or q,
depending on whether P contains p. If Pk has an odd number of elements, then ( )kp 1; otherwise, if Pk has an even number of elements, then ( )kp 1. ( )kp 0 cannot be zero since p does not divide k. But the number of subsets of Pk having an odd number of elements is the same as the number of subsets having an even number of elements. Therefore, the terms cancel each other and terms with jk not a power of p do not contribute to the double sum. QED.
Theorem 3. Every convergent series of reals B bkk
1
such that B bkk
( )2 2
1
can
be transformed into an element-squaring series
B bkk 0
by appending to B a single real term
b0 iff B B( )2 1
4 .
Proof: For B to be element-squaring, it is necessary that
b b b b B b Bkk
kk
20
2
0
20
2
1
20
( ) . Solving this quadratic equation for b0 yields
bB B
0
21 1 4
2
( )( )
, which is real iff B B( )2 1
4 . Since b0 is finite, B b B0
converges. It is also elementary to show that if B B( )2 1
4 , then there is no real element that
can be prepended to B to make it element-squaring. QED.
Note that B does not necessarily imply that B( )2 . For example, let bkk
k
( )1
.
Then
2
1,
2
1B , which is finite, but Bkk
( )2
1
1
, which diverges. Likewise, B( )2 does
not imply that B . If bkk 1
, then B( )22
6
but B diverges.
USING THIS BOOK vii
Using This Book
This book is a catalog of almost 10,000 real numbers. A typical entry looks like:
.20273255405408219099... log log3 2
215
arctanh AS 4.5.64, J941, K148
= 1
5 2 11
22 10
1
11
kk
k
kkk k
( )( )
=
12)12(
log
x
dxx
= dxx
xxx1
0
5sinsin)( GR 1.513.7
Each entry begins with a real number at the left, with its integer and fractional parts separated. Entries are aligned on the decimal point. Three dots are used to indicate either (1) a non-terminating expansion, as above, or (2) a rational number whose period is greater than 18. Three dots are always followed by the symbol , which denotes “approximately equal to.”
The center portion of an entry contains a list of expressions whose numerical values match the portion of the decimal expansion shown. The entries in the list are separated by the equal sign =; however, it is not to be inferred that the entries are identically equal, merely that their decimal expansions share the same prefix to the precision shown.
There is no precise ordering to the expressions in an entry. In general, closed forms are given first, followed by sums, products and then integrals. At the right margin next to expression may appear one or more citations to references relating the expression to the numerical value given or another expression in the entry. When more than one expression appears on a line containing a citation, it means that the citation refers to at least one expression on that line. See also, “Citations,” below.
Format of Numbers
Numbers are given to 19 decimal places in most cases. Where fewer than 15 digits are given, no more are known to the author. Rational numbers are specified by repeating the periodic portion two or more times, with the digits of the last repetition underlined. For example, 1/22 would be written as .04504545, which is an abbreviation for the non-terminating decimal .0454545454545045... . Since rational numbers can be specified exactly, the = symbol rather than is used to the right of the decimal. Where space does not permit a repetition of the periodic part, it is given once only. If the period is longer than 18 decimals, the sign is used to denote that the decimal has not been written exactly. No negative entries appear.
Ordering of Entries
Entries are in lexicographic order by decimal part. Entries having the same fractional part but different integer parts are listed in increasing magnitude by integer part. Rational numbers are treated as if their decimal expansions were fully elaborated. For example, these entries would appear in the following order:
USING THIS BOOK viii
.505002034457717739... .5050150501 = .505015060150701508... .5050 = .505069928817260012...
Arrangement of Expressions
In general, terms in a multiplicative expression are listed in decreasing order of magnitude except that constants, including such factors as ( )1 k appear at the left. For example, 4 2 3k kk! is used in preference to 4 23k kk !. Additive expressions are ordered to avoid initial minus signs, where possible. For example, 3 2 k is used instead of 2 3k . These principles are applied separately in the numerators and denominators of fractions.
Citations
Abbreviations flushed against the right margin next to an expression denote a reference to an equation, page or section of a reference work. Page numbers are prefixed by the letter p, as in “p. 434.” A list of cited works appears at the end of the book.
Other Catalogs
Simon Plouffe maintains a server at the University of Quebec at Montreal that allows searching among 215 million real numbers for equivalent symbolic expressions. See http://pi.lacim.uqam.ca/. This book, by contrast, contains only about 10,000 entries. One would expect, therefore, that almost all of the numbers in the book would also be found in Plouffe’s index. To test this hypothesis, I had a student write a script to look up each of the 10,000 entries on Plouffe’s server to see how many it would find. The result was approximately five percent. The reason for such a low degree of overlap, it appears, is that the expressions in Plouffe’s index were primarily generated by computer from templates, which the real numbers in this book were assembled for the most part from the mathematical literature.
WARNING
While great effort has been expended to ensure accuracy, the numerical values and expressions in this book have not been checked thoroughly and should not be relied upon unless verified independently.
FIRST-DIGIT PHENOMENON ix
The First-Digit Phenomenon
In 1881, Simon Newcomb observed that in books of logarithmic tables pages near the front tended to receive more wear than the later pages, suggesting that the earlier pages were used more frequently. That implies that the numbers whose logarithms were being looked up were not uniformly distributed. In particular, numbers whose leading digit was a 1 occurred more frequently than those beginning with 2 (or any other digit). Newcomb proposed that the frequency of first digit n in a random list of numbers (assuming meaning can be ascribed to such a concept), should be log10 (n+1) – log10 n. That is, the frequency of numbers beginning with 1 ought to be about 30%, while only 4.6% of numbers should begin with 9.
This phenomenon was rediscovered by Frank Benford in 1938 and it is now referred to as Benford’s law. Its generalization to numbers in base b is that the frequency of digit d is logb
(d+1) – logb d. This catalog (and Plouffe’s collection) provide experimental data concerning the first digits of the fractional parts of lists of numbers. In fact, both lists exhibit the first digit phenomenon very strongly. Because the fractional first digits range from 0 to 9 (rather than 1 to 9), the appropriate form of Benson’s law is to take b to be 11, not 10.
The bar chart below shows a plot of log10 (d+1) – log10 d (solid black bars) versus the actual percentage of entries in the catalog whose fractional part begins with the digit d (diagonal shading).
No explanation is offered to explain the deviation from the numerical predictions of Benford’s law in this instance, but the shape of the decay as d increases is similar. Various mathematical derivations of Benford’s law have been offered, but none are rigorous because of the futility of attempting to define a random sample of numbers form an unbounded set. It is somewhat easier to understand the law informally. When I explained Benford’s Law to the late John Gaschnig, an artificial intelligence graduate student at Carnegie Mellon in the late 1970’s, he pondered it for a few minutes and then announced, “Yes, it’s because 1 is first.”
NOTATION x
Notation
a n( ) greatest odd divisor of integer n. Ai x( ) Airy function, solution to y xy 0 . AS 10.4.1
b x( ) ( )
1
0
k
k x k.
Bk kth Bernoulli number, B Bk k ( )0 . AS 23.1.2
B xk ( ) kth Bernoulli polynomial, defined by te
eB x
t
k
xt
t k
k
k
1 0
( )!. AS 23.1.1
bp n( ) number of divisors of n that are primes or powers of primes.
ck coefficient of xk in the expansion of ( )x .
C x( )
0
2
142
0
2
)14(2)!2(
)1(cos
2
1)(
kk
kkkx
kk
xdttxC
. AS 7.3.1, 7.3.11
Ci x( ) cosine integral,
x
k
kk
kk
xxdt
t
txxCi
0 1
2
)2()!2(
)1(log
1coslog)( .
AS 5.2.2, AS 5.2.16 d n( ) number of divisors of n, d n n( ) ( ) 0
dn nth derangement number, d d d d1 2 3 40 1 2 9 , , , , nearest integer to en /! . Riordan 3.5
)(ne number of exponents in prime factorization of n.
E x( ) complete elliptic integral
dxxE 2/
0
2sin1)( .
En nth Euler number,
2
12 k
kk EE . AS 23.1.2
E xn( ) nth Euler number, defined by 2
1 0
e
eE x
t
k
xt
t k
k
k
( )!
. AS 23.1.1
Ei x( ) exponential integral
1 !
log)(k
k
x
t
kk
xx
t
dtexEi . AS 5.5.1
Erf x( ) error function
0
12
0
12
0 !)!12(
2)1(2
)12(!
)1(22 22
k
kkkx
k
kkxt
k
xe
kk
xdte
. AS 7.1.5
Erfc x( ) complimentary error function = 1 Erf x( ). AS 7.1.2 f n( ) number of representations of n as an ordered product of factors
Fn nth Fibonacci number, F F F F Fn n n1 2 1 21 ;
NOTATION xi
0 1F (; ; )a x hypergeometric function x
k a
k
kk ! ( )
0
, where a a a a kk( ) ( )...( ) 1 . Wolfram A.3.9
1 1F ( ; ; )a b x Kummer confluent hypergeometric function x a
k b
kk
kk
( )
( )!
0
,
where a a a a kk( ) ( )...( ) 1 . Wolfram A.3.9
2 1F ( ; ; ; )a b c x hypergeometric function x a b
k c
kk k
kk
( ) ( )
( )!
0
, where a a a a kk( ) ( )...( ) 1 . Wolfram A.3.9
G Catalan’s constant ( )
( ). ...
12 1
0 9159655941772190150520
k
k k
Gn
1 )13(
1
)13(
11
knn kk
gn
1 )13(
1
)13(
11
knn kk
gd z( ) Gudermannian, gd z ez( ) arctan 22
Hn harmonic number: Hkn
k
n
1
1
Hen even harmonic sum: H
ke
n
k
n
1
21
H on odd harmonic sum: H
ko
n
k
n
1
2 11
H jn
( ) generalized harmonic number: Hk
jn
jjj
n jk
n j j( )
( )
( )( ) ( )
( )!,
1 1 1
11
1
1 1
hn
0 )56(
)1(
)16(
)1(
kn
k
n
k
kk J314
hg x( ) Ramanujan’s generalization of the harmonic numbers, hg xx
k k xk
( )( )
1
I xn( ) modified Bessel function of the first kind, which satisfies the differential equation
z y zy z n y2 2 2 0 ( ) . Wolfram A.3.9
jn
1 )13(
1
)13(
11
knn kk
J311
J xn( ) Bessel function of the first kind ( )
!( )!
12
2
20
k n k
n kk
x
k n k LY 6.579
k 1 2 k
K x( ) complete elliptic integral of the first kind
dx
2/
02sin1
1
K xn( ) modified Bessel function of the second kind, which satisfies the differential equation
NOTATION xii
z y zy z n y2 2 2 0 ( ) . Wolfram A.3.9
l x( )
n
kk k
k
xk
xk
k
k
11
log)log(log , when x is an integer n.
nL “little” Fibonacci numbers: 1;2 1021 LLLLL nnn OR
nL Lucas numbers: 1;2; 1021 LLLLL nnn
li x( ) logarithmic integral x
t
dtxli
0 log)(
Li xn( ) polylog function Li xx
kn
k
nk
( )
1
log x natural logarithm loge x
p a prime number. p denotes a sum over all primes.
pd n( ) product of divisors pd n dd n
( )|
pf n( ) partial factorial pf nkk
n
( )!
1
0
pfac n( ) product of primes not exceeding n, pfac n pp n
( )
pq n( ) product of quadratfrei divisors of n, pq n pr n( ) ( ) 2 pr n( ) number of prime factors of n
)(nprime nth prime
r n( ) 4 ( )|
dd n HW 17.9
s n( ) smallest prime factor of n sd n( ) sum of the divisors of n, sd n n( ) ( ) 1
S x( ) 34
1
2
0
2
)34()!12(
)2/()1(
2sin)(
k
k
kkx
xkk
dtt
xS
. AS7.3.2, AS 7.3.13
S n m1( , ) Stirling number of the first kind, defined by the recurrence relation:
x x x n S n m xm
m
n
( )...( ) ( , ) 1 1 1
0
. AS 24.1.3
S n m2 ( , ) Stirling number of the first kind, nm
k
km kk
m
mmnS
02 )1(
!
1),( . AS 24.1.4
Si x( ) sine integral,
x
k
kk
kk
xdt
t
txSi
0 0
12
)12()!12(
)1(sin)( . AS 5.2.1, AS 5.2.14
NOTATION xiii
)(ntk
i
i
k
ek
1
1, 1)1( kt , where
i
ei
ipn
This is the number of ways of writing n as a product of k factors.
T s( ) 1ks
k , where k has an odd number of prime factors
)(nuk number of unordered k-tuples kaa ...,1 whose product is n.
w x( ) w x e erfc ixix
kx
k
k
( ) ( )( )
( / )
2
2 10
Wn
0 )56(
)1(
)16(
)1(
kn
k
n
k
kk J313
)(nz number of factorizations of n.
z x2 ( ) z x k xx
x xk
k2
2
12
11( ) ( ( ) ) ( ( ))
( )s ( )( )
( )s
k
k
sk
1
2 1
1
0
Euler's constant
n k
n
knlim log
1
1
0.57721566490153286 AS 6.1.3
f
n
kn
dxxfkf1 1
)()(lim
( )s Riemann zeta function 1
1 ksk
( ) ( )m s derivative of the Riemann zeta function ( log )
k
k
m
sk 1
)(sp Prime zeta function
1 )(
1
kskprime
( )s ( )( )
( ) ( )sk
sk
sk
s
1
1 21
1
1
( )s ( )( )
( ) ( )sk
ss
k
s
1
2 11 2
0
)(k Mangoldt lambda, kk log)( if k is a prime power; 0 otherwise
( )n Moebius function, ( )1 1 , ( ) ( )n k 1 if n has exactly k distinct prime factors, 0 otherwise. ( )n number of different prime factors of n.
NOTATION xiv
v1
2
23)31(
2
23)31(1
ii
iiv
)(s )()1(2
2/ sss s
k n( ) sum of kth powers of divisors of n, dk
d n|
golden ratio, 1 5
21 61803397498948482
. ...
( )n Euler totient function, the number of positive integers less than and relatively prime to n.
)(zn
0
12
)12()()(
2
1
kn
k
nn k
zzLizLi [Ramanujan] Berndt Ch. 9
( )n polygamma function ( )nkk
n
1
1
1
( )x kth derivative of the polygamma function ( ) ( )( )k
k
kx
d x
dx
( )n ( ) ( )n kk
n
1
( )s )()1(2
1)( 2/ sss
s s
( )n overcounting function ( )n -1 + the number of divisors of the g.c.d. of the exponents of the prime factorization of n.
REFERENCES xv
References
Citations given in the right margin of the number list refer to works in which more information about a number or equation may be found. The form of citation is generally a word or abbreviation followed by a reference giving a page, section or equation number. In the case of journals, the citation is by volume and page. Following is an alphabetical listing of reference abbreviations used in the text. Andrews Andrews, L. Special Functions for Engineers and Applied Mathematicians.
New York: Macmillan. AMM American Mathematical Monthly AS Abramowitz, M. and Stegun, I. A. Handbook of Mathematical Functions.
New York: Dover (1964). Berndt Berndt, B. C. Ramanujan’s Notebooks. New York: Springer-Verlag
(1985). ISBN 0-387-96110-0 CFG Croft, H. T., Falconer, K. J. and Guy, R. K. Unsolved Problems in
Geometry. New York: Springer-Verlag (1991). ISBN 0-387-97506-3. CRC Handbook of Mathematical Tables. Cleveland: Chemical Rubber Co. (2nd
ed., 1964) Dingle Dingle, R. B. Asymptotic Expansions: Their Derivation and Interpretation.
London: Academic Press (1973). ex exercise GKP Graham, R. E., Knuth, D. E. and Patashnik, O. Concrete Mathematics.
Reading, MA: Addison-Wesley (1989). ISBN 0-201-14236-8. GR Gradshteyn, I. S. and Ryzhik, I. M. Table of Integrals, Series and
Products. San Diego: Academic Press (1979). ISBN 1-12-294760-6. Edwards, H. M. Henrici
Riemann’s Zeta Function. New York: Academic Press (1974). ISBN 0-12-232750-0.
Henrici, P. Applied and Computational Complex Analysis. Hirschman Hirschman Jr., I. I. Infinite Series. New York: Holt (1962). ISBN 0-03-
11040-8. HW Hardy, G. H. and Wright, E. M. Theory of Numbers. Glasgow: Oxford
(4th ed., 1960) J Jolley, L. B. W. Summation of Series. New York: Dover (1961). K Knopp, K. Theory and Application of Infinite Series. New York: Dover
(1990). LY Lyusternik, L. A. and Yanpol’skii, A. R. Mathematical Analysis:
Functions, Limits, Series and Continued Fractions. Oxford: Pergamon (1965).
Marsden Marsden, J. E. Basic Complex Analysis. San Francisco: Freeman (1973). ISBN 0-7167-0451-X.
Melzak Melzak, Z. A. Companion to Concrete Mathematics. New York: Wiley (1973). ISBN 0-471-59338-9.
MI Mathematical Intelligencer
NOTATION xvi
Riordan Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley (1958).
Seaborn Seaborn, J. B. Hypergeometric Functions and Their Applications. New York: Springer-Verlag (1991). ISBN 0-387-97558-6.
Sloane Titchmarsh
Sloane, N. J. A. A Handbook of Integer Sequences. New York: Academic Press (1973). ISBN 0-12-648550-X.
The Theory of the Riemann Zeta Function. London: Oxford (1951). Vardi Vardi, I. Computational Recreations in Mathematica. Redwood City:
Addison-Wesley (1991). ISBN 0-201-52989-0 Wilf Wilf, H. S. generatingfunctionology. Boston: Academic Press (1989).
ISBN 0-12-751955-6. Wolfram Wolfram, S. Mathematica: A System for Doing Mathematics by Computer.
Reading, MA: Addison-Wesley (2nd ed., 1991). ISBN 0-201-51502-4.
ACKNOWLEDGMENTS xvii
Acknowledgments
No book like this one has previously appeared, probably because of the monumental difficulty of performing the required calculations even with computer assistance. However, with the development of the Mathematica® program, a product of Wolfram Research, Inc., it has been possible to obtain 20-digit expansions of most of the entries without undue difficulty. The symbolic summation capability of that program has been invaluable in developing closed-form expressions for many of the sums and products listed. Even though my involvement with computers extends back more than 40 years, I have no hesitation in declaring that Mathematica is the most important piece of mathematical software that has ever been written and my debt to its authors is, paradoxically, incalculable.
The tendency to be fascinated with numbers, as opposed to other, abstract structures of mathematics, is not uncommon among mathematicians, though to be afflicted to the same degree as this author may be unusual. If so, it because of the role played by a collection of people and books in my early life. My grandfather, Max Shamos, was a surveyor who, to the end of his 93-year life, performed elaborate calculations manually using tables of seven-place logarithms. The work never bored him — he took an infectious pride in obtaining precise results. When calculators were developed (and, later, computers), he never used them, not out of stubbornness or a refusal to embrace new technology, but because, like a cabinetmaker, he needed to use his hands.
I read later in Courant and Robbins’ What is Mathematics, of the accomplishments of the calculating prodigies like Johann Dase and of William Shanks, who devoted decades to a manual determination of to 707 decimal places (526 of them correct). At the time, I couldn’t imagine why someone would do that, but I later understood that he did it because he just had to know what lay out there. I expect that people will ask the same question about my compiling this book, and the answer is the same. The challenge set by Shanks is still alive and well — every year brings a story of a new calculation — is now known to several billion digits.
My father, Morris Shamos, was an experimental physicist with a healthy knowledge of mathematics. When I was in high school, I was puzzled by the formula for the moment of
inertia of a sphere, M mr2
52 . That is was proportional to the square of the radius made
sense enough, but I couldn’t understand the factor of 2/5. Dad set me straight on the road to advanced calculus by showing me how to calculate the moment by evaluating a triple integral.
As a teenager during the 1960s, I worked for Jacob T. (“Jake,” later “Jack”) Schwartz at the New York University Computer Center, programming the IBM 7094. (That machine, which cost millions, had a main memory equivalent to about 125,000 bytes, about one-tenth of the capacity of a single floppy disk that today can be purchased for a dollar.) NYU had a large filing cabinet with punched-cards containing FORTRAN subroutines for calculating the values of numerous functions, such as the double-precision (in the parlance of the day) arctangent. By poring over these cards, I learned something of the trouble even a computer must go through in its calculations of mathematical functions. It was this exposure that drew me to field that later came to be called computer science.
For the past 25 years I have been an enthusiastic user of Neil Sloane’s A Handbook of Integer Sequences (now co-authored with Simon Plouffe) in combinatorial research, so it is
ACKNOWLEDGMENTS xviii
particularly pleasing to note that this book itself, in a sense, consists entirely of such sequences.
My greatest debt, however, is to Stephen Wolfram for his creation of Mathematica, the single greatest piece of software ever written. In developing this catalog, I used Mathematica on and off over a 15-year period to explore symbolic sums and integrals and to compute most of the constants that appear in this book to 20 decimal places.
ABOUT THE AUTHOR xix
About the Author
Michael Ian Shamos was born in New York in 1947. He attended the Horace Mann School and the Columbia University Science Honors Program during high school. He learned computer programming at Columbia and the NYU Computer Center on the IBM 1401, 1620 and 7094 computers during the 1960s. He graduated from Princeton University in 1968 with an A.B. degree in Physics and a minor in Mathematics. His senior thesis, written under the guidance of Prof. John Wheeler, was devoted to the theory of gravitational radiation.
He joined IBM as an engineer in 1968, developing programming systems to operate automated circuit testing equipment and obtained an M.A. from Vassar College in 1970, working under Prof. Morton Tavel in the field of acoustics. In 1970, he joined the National Cancer Institute to write software in support of the Institute’s research in cancer chemotherapy.
In 1970, he began graduate study at the Yale University Department of Computer Science where he laid the foundations for the field of computational geometry and was awarded the M.S., M.Phil. and Ph.D. degrees, the latter in 1978. In 1975, he joined the faculty of the Computer Science Department at Carnegie Mellon University, doing research in combinatorics and the analysis of algorithms.
In 1979, Dr. Shamos founded Unilogic, Ltd., a computer software company in Pittsburgh. He obtained the J.D. degree from Duquesne University School of Law in 1981 and has engaged in the practice of intellectual property law since then, dividing his time with Unilogic until 1987. He practiced intellectual property law with the Webb Law Firm in Pittsburgh, concentrating in computer-related matters, from 1990-1998. From 1980-2000, he has acted as an examiner of computerized voting systems for the Secretary of the Commonwealth of Pennsylvania and the Attorney General of Texas.
Dr. Shamos is now Distinguished Career Professor in the School of Computer Science at Carnegie Mellon, where he directs a graduate program in eBusiness Technology.
Dr. Shamos is the author of four published books, Computational Geometry (1985), with F. P. Preparata, Pool (1990), The Illustrated Encyclopedia of Pool, Billiards and Snooker (1993), and Shooting Pool (1998). He serves as Curator of the Billiard Archive, a non-profit organization devoted the preservation of the history of cue games, as Contributing Editor of Billiards Digest magazine, and Chairman of the Statistics and Records Committee of the Billiard Congress of America.
Dr. Shamos lives in the Shadyside neighborhood of Pittsburgh with his wife Julie. They were married in 1973. Their son Alex is a student at the University of Arizona in Tucson. Their daughter Josselyn Crane lives nearby with her husband Dan and their infant daughter Harlow.
For more information, consult the Author’s web page: http://euro.ecom.cmu.edu/shamos.html
.00000000000000000000 = ( ) ( )
!
( )
!
1 1 1
0
1 2
1
k
k
k
k
k
k
k
k GR 1.212
= ( ) ( )
!
( )
!!( )
1 2 2 11
2
20 1
2
1
k k
k
k
k
kk
k
k
k
k
k
k
k
k
= ( )k
kk
1
Berndt Ch. 24 (14.3)
= ( )kk
k 2 11
= ( )sin
1 13
1
k
k
k
k
= log2
20 1
x
x
1 .000000000000000000000 = 1
11 k kk ( )
J396
= 1
2 21 kk
½
= k
kk ( )!
11
GR 0.245.4
= 2
20
k
k
k
k( )!
= 1
10 ( )! !k kk
=
0 !!!)!2((
1
)!3(
!!
k kkk
k
= 2 1 3 3 1
3 3
4 3
72
2
6 5 4 31
k k
k k
k k
k k k kk k
= 2 1
4 1
2 1
2 11 1
k
k k
k
k kkk k
( )
( )!!
( )! ( )
=
0224
2
)1(2
342
kk k
k
k
k
=
( ) logk k
kk 1
=
2 1
1
k kk
k
FF
F
= F k
kk
4 2
1 8
=
1 )2)(1(k
k
kk
H
=
22 22k
k
kk
H
=
1 3kkkL
= ( )kk
12
=
2
)(
k k
k
=
( ) logk k
kk 1
Berndt Ch. 24, Eq. 14.3
=
2 1
)(
)1(
)(
k k
k
kk
k
= 1
1
S
, where S is the set of all non-trivial integer powers
(Goldbach, 1729) JAMA 134, 129-140(1988)
=1
21 n
k
k
n
kn !
log
!
= ( )sin
121
1
k
k
k
k
= 11 1
1
( )k
k k J1065
= dx
x
xdx
x21
21
log
= dx
x( ) /2 3 20 1
= dx
x x( )1 1 20
1
= x x
edxx
log( )1
0
=
cossin
xx
x
dx
x0
Berndt I, p. 318
=
2/
02)cos(sin
xx
dx
= arctan x
xdx2
1
2 .000000000000000000000 =
111 !)!1()!2(
2!
2)0,1(½,
kk
k
kk k
k
k
kk
=
6
1
6
7
216
1
8
1
8
9
512
1 )2()2()2()2(
=
122 )4/1(k k
k Prud. 5.1.25.31
= ( )
( )
k H F
k
k
kk
kk
kk
kk
k
1
2 2
1
4 1
2
1 1 0
=
1
1)2(exp
k k
k
=
12
)(
k k
kz
=
1
122 12
)12(
12
)2(
kkk
kk
=
1
2
2
22
2
1 2
12
1)2(
11
kkk kk
kk
k
k
kk
= 11
220
k
k
Prud. 6.2.3.1
= e ek kk
k
k
k
( ) / /
1
1
1 2
1
1
= log
(cos )
2
21
2
20
2
1
x
xdx
x
edx
dx
o
= x x
xdx
sin
cos
/
10
2
GR 3.791.1
= logsin cos/
x xdx2
0
2
= e dxx
0
1
=
0
11
3xe
dxx
3 .000000000000000000000 =
k
kk
kk
12
60
=
1 3kk
kk LF
=
1 )3(
21
k kk
4 .000000000000000000000 = k k k
k
k
kkk
kk
kk2
1
2
1
4 1
2 21
1 2 0
( )
( )
= ( )!!
( )!! ( )
( )!!
( )!! ( )
2
2 1
2
2 122
121
k
k k k
k
k k kk k
=
12 4
31
k kk
6 .000000000000000000000 =
1
3
1
2
32)0,2(½,
kkkk
kk
FFk
= k
k kk
2
3 2 2( )( )
= x dx
e
xdx
xx
3
0
3
21
log
10 .000000000000000000000 = ( )k F k
kk
kk
k
4
2 20 1
12 .000000000000000000000 = ( )
log
k dx
x xxe dxk
k
x
1
2
2
0 2 0
15 .000000000000000000000 = ( / , , )1 3 4 03
4
1
kk
k
20 .000000000000000000000 = k
k kk
2
4 3 3( )( )
24 .000000000000000000000 = log4
20
xdx
x
26 .000000000000000000000 = (½, , )
3 02
3
1
kk
k
52 .000000000000000000000 = ( )k
kk
1
2
3
0
70 .000000000000000000000 = k
k kk
2
5 4 4( )( )
94 .000000000000000000000 = F kk
kk
2
1 2
150 .000000000000000000000 = (½, , )
4 02
4
1
kk
k
1082 .000000000000000000000 = (½, , )
5 02
5
1
kk
k
1330 .000000000000000000000 = F kk
kk
3
1 2
25102. 000000000000000000000 = F kk
kk
4
1 2
1 .00000027557319224027... 1
101 ( )!kk
1 .000002739583286472... 697
6613488
8
8
W J313
.00000484813681109536...
648000, the number of radians in one second
1 .000006974709035616233... 2coth
.00001067827922686153... 10
1 .0000115515612717522...
17
1
7
1
910
7
)16(
)1(
)56(
)1(
532
61547
k
kk
kk
J328
1 .000014085447167... W7 J313
1 .0000248015873015873...
13 )!(
1
k k
.00003354680357208869... 9 1 .00003380783857369614... sec1
.00004439438838973293... 33434arctan634arctan62arctan1272
1
13
32037log
3
5arctan233343log3log3
72
1
= 1
22528
11
121
23
12
1
4096 12 1 122
Fdx
x, , ,
.00004539992976248485... e10
1 .0000513451838437726... ( )
( )
( )9
511 9
512
1
2 1 90
kk
AS 23.2.20
1 .00005516079486946347... 1
358318080
1
12
5
12
7
12
11
125 5 5 5 ( ) ( ) ( ) ( )
= ( )
( )
( )
( )
1
6 5
1
6 1
1
6
1
61
k k
k k k
1 .0000733495124908981... W6
691
87480
J313
.00010539039165349367... 8
.00012340980408667955... e9
6 .00014580284304486564...
( ) ( )log3
3
21
2k
kk
1 .000155179025296119303... ( )
( )
( )8
17
161280
255 8
256
1
2 1
8
80
kk
AS 23.2.20
1 .0002024735580587153... 3
31
.0002480158734938207... 1
81 ( )!kk
1 .000249304876753261748...
1
8192
1
16
1
16 12
30
( )
( )kk
.0002499644008724365... 16 6 112 6
2 1k kk
k k
( )
1 .0002572430074464511...
15
1
5
1
67
7
)16(
)1(
)56(
)1(
32
615
k
kk
kk
J328
.00026041666666666666 = 1
3840
1
5 25
!
.0002908882086657216...
10800, the number of radians in one minute of arc.
.000291375291375291375 =1
34321
492
8
kk
.00029347092362294782...
( )
( )( )3
4096
1
81921
1
162
31
kk
.00033109368017756676... 7 .00033546262790251184... e8
1 . 00038992014989212800...
1
555 )56(
1
)16(
1
3888
)5(3751
k kkW
J313
1 .000443474605655004... 2655360
307 7
=
17
1
7
1
)14(
)1(
)34(
)1(
k
kk
kk J326
=
07
2/
)12(
)1(
k
k
k Prud. 5.1.4.3
1 .0004642141285359797... 1
7776
1
6
1
6 13
41
( )kk
1 .00047154865237655476... ( )
( )
( )7
127 7
128
1
2 1 70
kk
AS 23.2.20
1 .00049418860411946456... ( )11111
1
kk
1 .00056102267483432351...
1
3456
1
12
1
12 112
31
( )
( )kk
1 .000588927172046... r( )16 Berndt 8.14.1
5
82
1 2
4 2
2 2
16
2 2 2
2 2 2
2 2
16
2 2 2
2 2 2log
log( )log log
.00064675959761549737... 11
22
21
24
9
23
15
25
1
2
1 3
51
( ) ( ) ( ) ( )( )
( )
k
k
k
k
.00069563478192106151... ( )
( )
3
1728
1
12 31
kk
1 .00076240287712890293... ( )( )
12
21
2
1
kk
k
k k
.000807441557192450666...
022 1
)2cos(53
2
1
64
1dx
e
xxx
.00088110828277842903...
1
3456
11
12
1
12 12
31
( )
( )kk
.00091188196555451621... e7
1 .00097663219262887431... 15
1 k kk
1 .00099457512781808534... ( )10110
1
kk
.00099553002208977438... ( )10 11
kk
.000998407381073899767...
2210
1
k kk
=
2
1coth
2
1(cot21coth25
16
1
61
2 iii
.00101009962219514182... 13
124 6 4 1
1
110 4
2
( ) ( )kk kk k
.00102614809103260947...
2 25510
1
)(11)55(
k kk k
k
kkk
.001040161473295852296... 6
.0010598949016150109... 15
8 42 6 4 6 1
1
110 6
2
coth ( ) ( ) ( )k
k kk k
.00108225108225108225 =1
9241
362
7
kk
.00114484089113910728...
1
3456
5
6
1
12 22
31
( )
( )kk
1 .00115374373790910569... 1
124416
1
12
5
12
7
12
11
123 3 3 3 ( ) ( ) ( ) ( )
=( )
( )
( )
( )
1
6 5
1
6 1
1
4
1
41
k k
k k k
1 .001242978120195538...
1
1458
1
9
1
9 82
31
( )
( )kk
3 .00126030124568644264... 23
1)1(1
2 2
1
2
k
k
kk
k
.001264489267349618680...
1
02
44
1
)1(
7
22dx
x
xx
1 .00130144245443407196... 1
31457280
1
8
3
8
5
8
7
85 5 5 5 ( ) ( ) ( ) ( )
=( )
( )
( )
( )
1
4 3
1
4 1
1
6
1
61
k k
k k k
.001311794958417665... 197
27 64
3 3
32
1
2 8 30
( ) ( )
( )
k
k k
.00138916447355203054... e
e kk4
1
41
1
2
1
4 21
cos
( )!
3 .00139933814196238175... 2
1 .0013996605974732513... 3 3
62
3
4
3 1
312
log
log log( )( )r Bendt 8.14.1
.00142506011172369955... 1
102 k kk log
.00144498563598179575...
2
66
)2(
)6(
11
9601
k k
k
.00144687406599163371... ( )2
162
k
kk
1 .0014470766409421219... 6
61960
663 6
64
1
2 1
( )( )
( )kk
AS 23.2.20
=
2
6
)2(1
1
k k
k
.00146293937923861579...
1
8192
9
16
1
16 92
30
( )
( )kk
24 .00148639373646157098... ( ) ( )log4
4
21
2
k
kk
.00153432674083843739...
1
3456
3
4
1
12 32
31
( )
( )kk
.0015383430770987471... 930 5
1920 2 1
6
61
( )
( )
k
kk
.0015383956660985045... 4 3
256
1
4 8 30
( ) ( )
( )
k
k k
1 .001712244255991878...
1
1024
1
8
1
8 72
31
( )
( )kk
1 .00171408596632857117... 5 3
6
( )
1 .00181129167264869533... 1
1536
1
4
1
4 13
40
( )
( )
kk
=
1
04
3
1
log
6
1dx
x
x
.001815222323088761... 5
54
197
216
1
1 2 3
2
2 2 2 21
k k k kk ( ) ( ) ( ) LY 6.21
.001871372759366027379... 7
360
3
2
1
1
3
3 21
( )
k e kk
Ramanujan
=
12
3
1
)(
kke
k
.001871809161652456210... 1
12 21
22
1k e
k
ekk
kk
( )
.001872682449768546116... 1
121
12
1k e
k
ekk
kk
( )
.00187443047777494092...
1
20
12
)(
1
1
kk
kk e
k
e
.001877951661484336384... 7
02
6
8
)7(45
1
xe
dxx
.001912334879124743... 15 109
960
2
6
2
0
e
k
k
k
( )!
1 .001949804343... j9 J311 1 .00195748553... G9 J309
.001984126984126984126 = dxe
x
e
kx
kkk
k
0
2
5
1
51
1)1(
)1(
504
1
2 .001989185473323053259... 10/910/710/310/1
5)1(sin)1(sin)1(sin)1(sin
sinh
=
110
11
k k
1 .00200839282608221442... ( )9
.00201221758463940331... ( )9 11
119
2
kkk k
.00201403420628152655... 3
46
3
25
1
23
5
44
9
83
17
162
387
642 ( ) ( ) ( ) ( ) ( ) ( )
= H
kk
k
15
1 2( )
.00201605994409566126... ( )8 1 11
19
2
kk kk k
.00201681670359358... 3277
3375 32
1
2 7
3
30
( )
( )
k
k k
.00202379524407751486... ( )7 2 11
19 2
2
kk kk k
.002037712107418497211... )7()3()5()10(4
7)1,9( 2 MHS
.00203946556435117678... ( )6 3 11
19 3
2
kk kk k
.00205429646536063477... 7 3
4096
1
8192
1
2
1
16 82
30
( )
( )( )
kk
.002083
02
7
1480
1xe
dxx
.00210716107047916356... ( )5 4 11
19 4
2
kk kk k
.002124728805039749548...
244
1
4 111log
1)4(
sinh
4log
901
kk kkk
k
.0021368161356763007...
1
3456
2
3
1
12 42
31
( )
( )kk
.00213925209462515473... ii 2)24
1)5(
28
13
= ( )4 5 11
19 5
2
kk kk k
1 .0021511423251279551... 5
486
4
4
W J313
1 .002203585872759559997... 19
11520
2
96
2
48
3 2
8
5 3 2 4
log log ( ) log
=1
4 2 1
25
0k
k k
k
k( )
.00227204396400239669...
1
1458
8
9
1
9 12
31
( )
( )kk
.00228942996522361328...
3/13/1 )1(2))1(2326
1)6()3(2
i
=1
3 6 19 6
1 1k kk
k k
( ( ) )
.00234776738898358259... ( )
( )
3
512
1
8 31
kk
1 .00237931966451004015... log
logsin/2
4 2 0
4
G
xdx GR 4.224.2
=
log/ x
xdx
1 20
1 2
GR 2.241.6
1 .0024250560555062466... 2 2
5
5
4
3
2 5
1 5
210
log loglog ( )
r Berndt 8.14.1
.002450822732290039104...
dx
e
exx
x
2
4
5 14
3
.00245775047043566779... 171109
24 5
2
1
ek
kk
( )!
.002478752176666358... e6
2 .0025071947052832538... 6 2 8 2 62 2
22
0
log log( )
k H
kk
kk
.00254132611404785053...
02
4
5 14
)5(3xe
dxx
.002597505554170387853...
3
4
432
1 )2(
.00260416161616161616 =1
284
1
4 24
!
1 .00260426354837675067... 1
2
1
2
1
2
1
4 4
1
2
1
2
1
4 160
cos cosh cos( )!
e
e
k kk
.002608537189032796858... 1
8
5
3
1
4
1
4 7 12
log arctan
dx
x x
.002613604652091533982... dx
x72 1
.00262472616135652810...
276
63log2log
xx
dx
.00264711657512301799...
02
8
9 14
)9(315xe
dxx
.00264953284061841586...
5
6arccoth5210
5
2213arctan5210231log
5
2213arctan5252
20
1
=dx
x x7 22
1 .002651309852400201...
0
2/7
cossin
5
22dx
x
xxx
.00270640594149767080... 1
4
5
3
1
8 7 32
log
dx
x x
1 .00277832894710406108... cosh cos1 1
.0028437975415075410...
24724
1
3
7log2log
xx
dx
.00286715218149708931... 1
92 k kk log
.00286935966734132299...
1
3
1
)5(
)1(
4
)3(3
216000
195353
k
k
k
=
1
0
1
0
1
0
555
1dzdydx
xyz
zyx
5 .0029142333248880669... 2
10
k
k k !
.002928727553540092... ( ) ( )
( )
3
36 324
1
3
2 1
31
k
k
k
k
.00302826097746345169...
1
8192
7
16
1
16 72
30
( )
( )kk
.0031244926731183736...
1
3450
7
12
1
12 52
31
( )
( )kk
.00318531972162645303... 2
33
2 33 2
2ex
x x
sinsin
.00321603622589046372... loglog
23
2
9
24 7 52
dx
x x
.00326776364305338547... 5
.003286019155251709...
1
1458
7
9
1
9 22
31
( )
( )kk
.0033002236853241029... Re ( ) i
.0033104481564221302... 25645
7
16
1 4
.0033178346712119643...
0
3)62(
)1(
64
7
32
)3(3
k
k
k
=
1
0
1
0
1
0222
555
1dzdydx
zyx
zyx
24 .00333297476905227... e e e e
e e
k
ekk
( )
( ), ,
3 2
5
4
1
11 11 1
1
14 0
.0033697704469093... 15 5
16
31
32
1
3 50
( ) ( )
( )
k
k k
.00337787815787790335...
1
1024
7
8
1
8 12
31
( )
( )kk
.003472222222222222222 =1
288 1 2 3 4 51
k
k k k k kk ( )( )( )( )( )
.00364656750482608467... 1
27
3
36
1
3 3
1
31
( ) ( )
( )
k
K k
=
1
0
1
0
1
0333
555
1dzdydx
zyx
zyx
.00368755153479527...
3
2
36 3
91 3
216
1
432
1
6
( ) ( ) Berndt 7.12.3
= 1
6 5 31 ( )kk
.0037145966378051377... log( log( log(log ))) 2 1 .00372150430231012927... log( log(log ))2
.00374187319732128820... coth
12
12e
1 .00374187319732128820... coth
e e
e e
.00375000000000000000 = 3
800
1
2 3 4 51
( )( )( )( )k k k kk
1 .00375568639565501099... 19
2 2
1
4 3
1
4 1
5
12
1
51
1
5
( )
( )
( )
( )
k
k
k
k k J 326
=
15
2/
)12(
)1(
k
k
k Prud. 5.1.4.3
.0037878787878787878 =
1
91
)1(
)1(
264
1
kkk
k
e
k
1 .003795666917135695...
4 3 2 5
8 2 1 40
2arctan log dx
x
.00388173171751883... 32
256
1
4 2
3 1
31
( )
( )
k
k k
.0038845985984860509... 22
72
2
3
1
2 1
3
0
log( )
( )
k
kk
k
k
1 .0038848218538872141... arctan2
1 .003893431348... j8 J 311
1 .0039081319092642885... 14
1 k kk
1 .0039243189542013209... 656
6200145
8
8
G J 309
.00396825396825396825 =1
2525 1
252
6
( )kk
2 .00401700200153911987...
1
91
1
k k
= 9/43/19/29/1 )1()1()1()1(/1 9/89/73/29/5 )1()1()1()1(/1
.00402556575171262686... 7 4
8
7
8 4
18 4
2
( )
sinh
( )
k
k k k
=
powerintegertrivialnonaω
42
4 1
)1()()1(
k
k
k
k
.004031441804149936148...
02
2
3 18
1dx
e
exx
x
.00406079887448485317... 19450 8 1
88
( )
( )
.004061405366517830561...
primep k
kk
kp
1
8 )8(log)(
1 .0040773561979443394...
8
94508 ( )
1 .00407802749637265757...
8/18/78/38/14
)1(sinh)1(sin)1(sin)1(sin2
i
=
2
81k
k
.00409269829928628731... 15
16 8
2
8
2 2
2 2
cothsin sinh
cos cosh
= ( )8 11
118
2
kkk k
1 .00409337231139922783...
2
8
3
1
1
2cos2cosh
csch16
k k
.00410564209791538355... 10
.00410818476208031502... 17 1 18
2 1k kk
k k
( )
.00413957343311865398... 3
42
2 3
3
2
( ) tanh
= 16 2 18 2
2 1k kk
k k
( )
.004147210828822741353... 576768
35
512
319 42
=1
12 52 ( )kk
1 .00415141584497286361... dxx
x
1
0
22
1
arccos
2421616
.004166666666666666666 =
02
3
1)7(
240
1xe
dxx
.004170242045482641209... )6()3()5()4()7()2()9(4)1,8( MHS
.0042040099042187... 15 3 18 3
2 1k kk
k k
( )
.0043397425545712... 15
84
4 ( ) coth
= 14 4 18 4
2 14
1k kk
k
kk k k
( )( )
.00442764193431403234... ( )
( )( )
( )
2
2
7
1803 3 1
1
2
4 1 2
41
k
k
k
k
1 .00449344967863012707... dxx
x
kk
1
03
3
04
)3(
1
log
6
1
)13(
1
3
1
486
1
.00450339052463230128... ( )2
52
k
kk
.00452050160996510208... ( )2
152
k
kk
3 .00452229793774208627... 22cot4
22csc
22
2
= k kk
kk
k k
2 2 12
21
2
2 22
( )( )
1 .00452376279513961613... 31 5
325
1
2 1 50
( )( )
( )
kk
AS 23.2.20
.004548315464154810203... 3
256
1
9
1
2 1 2 1 2 3
2
2 2 21
( ) ( ) ( )k k kk
LY 6.21
.0045635776995554368... 12
3
64
1
2 3
3 1
31
G k
k
k
k
( )
( )
.00460551389327864275... log2661
960 7 62
dx
x x
.004641520347027523161...
3/13/1 )1(231)1(2316
1
3)2()5(2 ii
= 13 5 18 5
2 1k kk
k k
( )
.00475611798059498829...
1
8192
3
8
1
16 62
30
( )
( )kk
2 .0047586393290200803... Gx
xdx
log log( )2
2
1
1
2
20
1
GR 4.295.6
=
log sin
/ xdx
20
2
1 .004759800630060566114... log log ( )22
41 2 8 r
.0047648219275670510... 67
64 96 360
3
8
5
2
1
2
2 4
52
( ) ( )
( )k kk
1 .00483304056527195013...
1
3
1
3
13
)16(
)1(
)56(
)1(
216
7
k
kk
kk
.00486944347344743055... 7 3
1728
1
12 6 31
( )
( )
kk
.00491550994358136550... sin
2
2 4
2
22
1
kk
.004983200... mil/sec
.00506504565493641652...
1
1458
2
3
1
9 32
31
( )
( )kk
.00507713440452621921... 4
51
93 5
192 2 1
( )
( )
k
kk
2 .005110575642302907627... 4
3
10 3
81
=kk
k
F Fk
2
12 1 2 12
1
22 2
3
2
1
4
1
33 3
5
2
1
4
, , , , , ,
.00513064033265867701...
1
3
1
)4(
)1(
1728
1549
4
)3(3
k
k
k
=
1
0
1
0
1
0
444
1dzdydx
xyz
zyx
.005161189593421394431...
1 16
1)(
2coth
830
13
kk
k
.0051783527503297262... 7 3
128 512
1
1024
3
4
1
8 2
32
31
( )
( )( )
kk
.0053996374561862323... 15
42 4 6 2 6 1
1
( ) ( ) ( ) ( )kk
=1
8 62 k kk
.005491980326903542827...
dxe
xxx
022 1
2/cos
8
413
Prud. 2.5.38.10
.005555555555555555555 =1
180 2 3 4 51
k
k k k kk ( )( )( )( )
6 .005561414313810886999... 2
2 23
2
0log
x
dxx
.00567775514336992633... ( , )5 3
1 .00570213235367585932... 1
2
2
3
3
4
1
216 63 21
log log
k kk
Ramanujan] Berndt Ch. 2
.005787328838611000493... 1
81 k kk log
.005899759143515937... 45 7
8 6
( )
Berndt 9.27.12
1 .005912144457743732... dxx
x
kk
1
05
2
13
)2(
1
log
2
1
)45(
1
5
1
250
1
1 .0059138090647037913...
3
1 33
243
91 3
486
1
324
1
6
1
17496
1
6
( ) ( ) ( )
=( )
( )
2 1
6 1
2
40
k
kk
.005983183296406418... 3
3032
26
27
1
2 5
( )
( )
k
k k
.00603351696087563779...
2
7
log)7(
k k
k
.00605652195492690338...
1
4096
7
162
2
8 81
( ) log x
x xdx
.00609098301750747937...
1
3375
7
152
2
8 71
( ) log x
x xdx
.00613294338346731778...
( ) log( )3
196
1
2744
1
22
2
8 61
x
x xdx
1 .00615220919858899772...
2 )(
)1(
)1(
)(
k k
k
k
k
2 .00615265522741426944... k
kk2
1 1
.00618459927831671541...
1
2197
7
132
2
8 51
( ) log x
x xdx
.00624898534623674710...
1
1728
7
122
2
8 41
( ) log x
x x
.00625941279499622882...
1
42
2
16
5 2 2
5 2 2arctan log
( )
arctan arctanlog1 2
8
2
81 2 1 2
3
8
2
16 6 22
dx
x x
31 .0062766802998201758... 3 2
0 1
log( )
xx
x xdx GR 4.261.10
= x e
edx
x
x
2 2
1
/
GR 3.4.11.16
.00628052611482317663... 2 1 2
3148
2
8
3 3
32
1
2 2
log ( ) ( )
( )
k
k
k
k
.00629484324054357244... 3312
21log
3
5arctan
32
1
=1
160
5
61
11
6
1
64 12 1 62
Fdx
x, , ,
.00633038459106079398...
1
1331
7
112
2
8 31
( ) log x
x xdx
.00634973966291606023... log log21
531 6
2
dx
x x
.006400000000000000000 = 4
6251
41
3
1
( )kk
k
k
.00643499287419092255...
1
1000
7
102
2
8 21
( ) log x
x x
.006476876667430481... log arctan3
4
2
2
1
2 4 6 22
dx
x x
6 .006512796636760148...
0
3
4
2
0,3,1
)1(
14
kke
k
ee
eee
102 .00656159509381775425... 5
3
.006572038310503418...
1
729
7
92
2
8 11
( ) log x
x xdx
.0066004473706482057... ( ) ( )i i
.0066201284733196607... dxx
x
1
05)1(
log
6
1
4
2log
.00664014983663361581...
( )2 1 1
2 11
kk
k
.0067379469990854671... e5
.00675575631575580671...
1
512
7
8 12
2
81
( ) log x
xdx
.0067771148086586796... ( )( )
( )4 1
12
422
24
1
jk
f k
kji k
.00677885613384857755... 3
46 5
1
23 4 3 2 52 ( ) ( ) ( ) ( ) ( ) ( )
= H
kk
k
15
1 1( )
.0067875324087718381... log
arctan7
6
1
3
5
3
3
6
1
8 6 32
dx
x x
.00695061706903604783... 108
11
2
1arctan
8
2
1 .00698048496251558702...
962 2 8 2 12 3
1
4 2 1
22 3
40
log log ( )( )
kk k
k
k
.00700907815253407747... 2 3
343
2
81
( ) log
x
x xdx
.00702160271746929417... 7
48 180
3 3
4
23
8
1
1 2
2 4
41
( )
( )( )k k kk
.00709355636667009771... 16
3
128 1
3 2
2 31
Gx
xdx
log
( )
.00711283900899634188... 9
.0071205588285576784... 3
8
1 1
4
1
1
e k
k
k
( )
( )!
6 .00714826080960966608... ( ) ( )
1 2 11 4
1
k
k
k
1 .007160669571457955175... 1
230 kk
.0071609986849739239... 1
2454 3 2 452 ( )
.0071918833558263656... )logsin()logcos(2/2
iiiie i
.00737103069590539413...
1
216
7
62
8 21
( ) dx
x x
2 .00737103069590539413...
1
216
1
62
1
216
7
6 12 2
2
60
1
( ) ( ) log x dx
x
.007455925513314550565... sech2
1
( )kk
.00747701837520388... 17
360
3 3
4
1
2
4 1
41
( ) ( )
( )
k
k
k
k
.00748486855363624861... Gk
k
k
10016
11025
1
2 7
1
21
( )
( )
.007511723574771330898... 1
6
1
22
1
csch ( )k
k
[Ramanujan] Berndt Ch. 26
.0075757575757575757575 = 1
1329 ( )
.00763947766738817903... log3
2
13
24 6 42
dx
x x
1 .00776347048... j7 J 311
1 .0078160724631199185...
17)!(
1
k k
.0078161009852685688... 1716 15428
15
2
135
1
12
4 6
7 71
k kk ( )
.0078437109295513413... 119
72 6
1
1 2 3
2
2 21
k k k kk ( ) ( ) ( )
1 .00788821... G7 J 309
.0079130289886268556...
1
125
7
52
2
8 31
( ) log x
x xdx
.00798381145026862428... 3 5
44
4 4
( )( ) c Berndt 9.27.12
.008062883608299872296...
dx
e
exx
x
2
2
3 14
1
.00808459936222125346... 4
1 22
2
0
2
log logx e x dxx
.0081426980566204283...
1
8192
5
16
1
16 52
30
( )
( )kk
2 .00815605449274531515... 11 1 1 14
1 8 3 8 5 8 7 8
sin( ( ) sin( ( ) sin( ( ) sin( ( )/ / / /
= 11
81
kk
.00824553665010360245... 1
614496 1536
1
4
3
43 3 3
G ( ) ( )
.00824937559196606688... ( )( )
3 13 3
32
f k
kk
Titchmarsh 1.2.14
.0082776188463344292...
1
3456
5
12
1
12 52
30
( )
( )kk
.00828014416155568957... )7(
11
.008283832856133592535...
1
7)7(log
)(1
kprimep
kk
k
p
.00830855678667219692...
1
1
)!23(
)1(
32
3cos
3
2
3
1
2
1
k
k
k
e
e
1 .00833608922584898177... sin sinh sin
( )!
1 1
2
1
2 4
1
4
1
4 10
e
e kk
.00833884527896069154... 1
81 1
1
8 4 11
cos sin( )!
e
k
kk
1 .00834927738192282684... ( )7
=
0
624
)42)(32)(22)(12(4
)2(
381
64
381
)5(64
1143
)3(8
kk kkkk
k
=
0
236
)72)(62)(52)(32)(12(4
)2()1175793942604248(
7560 kk kkkkk
kkkk
Cvijović 1997
=
primep pppppp
ppp654321
321
1
1)4(
=
primep pppppp
ppppp654321
54321
1
1)6(
.0083799853056448421... ( )8 1 11
17 1
2
kk kk k
.00838389135409569500... )11(2
331)9()2(84)7()4(21)6()5(4)4,7( MHS
9 .00839798699900984124... 4
.00841127767185517548... ( ) log( )6 1 1 1
1
6
2
k
k
k
kk k
.00847392921877574231... 1
4
2
3
1
3
1 3
2
1 3
2
i i
= 16 1 17
2 1k kk
k k
( )
.0085182264132904860...
1
1458
5
9
1
9 42
31
( )
( )kk
.00860324727442514399... 15 2 17 2
2 1k kk
k k
( )
.008650529099561105501... )5()3(7560
)1,7(8
MHS
.00866072400601531257...
1
1024
5
8
1
8 32
31
( )
( )kk
.00877956993120154517...
1
64
7
4
1
64
3
4
2
272 2
2
8 41
( ) ( ) log x
x xdx
.00887608960241063354... 7
83
2
1
4
( ) ( )i i
= 13 4 17 4
2 1k kk
k k
( )
1 .00891931476945941517...
( )
( )
( )6
7 71
k
kk
Titchmarsh 1.2.13
5 .00898008076228346631... 1
20 ( )!kk
.00898329102112942789... iie 32/3
.00915872712911285823... 91 3
216 36 3
1
432
5
6
1
6 1
32
31
( )
( )( )
kk
.009197703611557574338... 3
8 4 3 3 3
4 1
91
cot coth
( )kk
k
.009358682075964994622...
02 )7)(4(!
2)1(
4
7
4
52
k
kk
kkke
.0094497563422752927... 3 3 23
1 12
21
2
31
( ) log ( )
( )
k
k
k
k
.00948402884097088413... 5
3
2
3 90
1
3
3 3
2
3 3
2
4
i i
= 13 4 17 4
2 1k kk
k k
( )
.00953282949724591758... 7
45 16
15
16
7 4
8
15
16
1
3
4
40
( ) ( )
( )
k
k k
2 .00956571464674566328...
1
04
21
2
1
log
4
3
4
1
64
16
x
dxxG
.00958490817198552203...
96 2 2
1
8 7 8 5 8 3 8 11
( )( )( )( )k k k kk
J243
.00960000000000000000 = 6
625
3
61
log x
xdx
.00961645522527675428... ( )
( )
3
125
1
5 31
kk
.009632112894949285675... 64
9
8
)3(
=
1
0
1
0
1
0222
555
1dzdydx
zyx
zyx
2 .00965179272267959834...
12 3
11sinh3csccosh3
k kh
2 .00966608113054390026... 7
108 1
3 2
60
log x
xdx
.009692044900729199735...
02
2
3 14
)3(xe
dxx
153 .00984239264072663137... 5
2
.00988445549319781198... 9 3
961
2 2
21
31
( )( )
( )
k
k
k
k
.0099992027408679778... 1
3072
1
4
3
4 643 3
3
( ) ( )
.010007793923494035492... 2
33
1
6
1
320
csch
( )k
k k
2 .0100283440867821521... i i i i i2
4 2 2 2 21 1 2 2 ( ) ( )
2
2 2coth
=H
kk
k2 1
21
.010097259642724159142...
1 64
1)2(21
16126
61
kk
k
.01026128066531735403... 3 3
2
1549
864
2
6 51
( ) log
x
x xdx
.01026598225468433519... 4
.01031008886672620565...
1
27
7
32
2
8 51
( ) log x
x xdx
.01036989801169337524... 69
16
3
8
3
2
1
1 2
2
2 31
( )
( ) ( )k k kk
1 .01037296826200719010... 7 3
16 64
1
128
1
4
32
( ) ( )
Berndt 7.12.3
=
1
1024
1
8
5
8
1
4 32 2
31
( ) ( )
( )kk
.01037718999605679651... ( )k
kLi
k kk k
1 1 16
26
2
.01041511660008006356... 11
2
1
2
1
4 4
1
1
cos cosh( )
( )!
k
kk k
GR 1.413.2
.01041666666666666666 = 1
96 1 2 3 4 5
2
1
k
k k k k kk ( )( )( )( )( )
1 .01044926723267323166... 41
41
1
16 2 21
sinh
kk
.010466588676332594722...
02
2
3 1
log)3(2)3()2log232(
8
1xe
dxxx
.01049303297362745807... 21log8
2
2
2log
56
1
8
7cot
16
=1
64 8
1
822 2k k
k
kk
k
( )
.01049435966734132299...
0
3)4(
)1(
4
)3(3
216
197
k
k
k
=
1
0
1
0
1
0
333
1dzdydx
xyz
zyx
1 .01050894057394275299... 1
24576
1
8
3
8
5
8
7
83 3 3 3 ( ) ( ) ( ) ( )
=( )
( )
( )
( )
1
4 3
1
4 1
1
4
1
41
k k
k k k
=
04
2/
)12(
)1(
k
k
k
.01065796470989861952... 2 5 2 33
24
5
43
9
82
81
16 ( ) ( ) ( ) ( ) ( ) ( )
= H
kk
k
14
1 2( )
.01077586137283670648... 13
8
2
4
3
12
3 1
3 1 8
3 1
3 1
log loglog
=1
12
1
12
1
12
1
121
1221 1k k
hgk
k
kk
k
( )( )
.01085551389327864275... log2131
192 6 52
dx
x x
.010864219555727274797...
1
2244
1)1(71csc6
12
1
k
k
kk
6 .01086775003589217196...
1
04
3
04
)3(
1
log
)1(
16
4
1
256
1
x
dxx
kk
.01097098867094099842... 60
49
3600
1
2 1 2 2 2 6 2 70
( )
( )( )( )( )
k
k k k k k
.01101534169703578827... 9
43 5
17 5
2
( ) ( )k kk
= 1
1 22 5 15
1 1k k kk
k k( ) ( )( )
.01105544825889466389... 11
1536
1
4
3
4
1
2 33 3
40
( ) ( ) ( )
( )
k
k k
1 .0110826597832979188... 1
2 1 255
11 ( )( )
k kkk k
k
1 .011119930921598195267... 1
22 2cosh cos
=
sin sin sin
( )sin
( )/ /1 11 2
2
1 2
21 4 3 4 i i
= 114 4
1
kk
7 .01114252399629706133... 3 12 2 9 31
12
130
log log( )( )k kk
.01121802464158449834... 1coth1cot24
1
= csc
( )cos cos sin sin
1
4 11 1 3 1 12
2 2
ee e
=1
14 41 kk
1 .01130477870987731055... ( )jkkj
3 115
.01134023029066286157... 1
36 6
1
8
23
0
( )k
kk
k k
k
1 .01140426470735171864... log log
( )3
2
2 2
36 r
1 .01143540792841495861... 32 2 8 2 4 3 2 4 51
2 6 51
log ( ) ( ) ( ) ( )
k kk
=( )k
kk
5
21
.01147971498443532465... 720 7 11
6
0
( )( )
x
e edxx x
726 .01147971498443532465... 720 7 17 ( ) ( )( )
1 .01151515992746256836...
012
10
012 1112
cos31323 x
dxx
x
dx
2 .01171825870300665613... 21
71
/k
k k
.0117335296527018915...
sinh2
1
2 4
2
21
k
kk
=
1
02)1(
)log2cos(dx
x
x GR 3.883.1
.01174227447013291929... 117
2 7k ks ds
k slog( )
.01175544134736910981... coth
2
12e
2 .01182428891548746447...
1
05
2
13
)2(
1
log
)45(
1
5
1
125
1
x
dxx
kk
.0118259564890464719... 2
114449
864 1 2 3 4
H
k k k k kk
k ( )( )( )( )
.01184674928339526517... 3 3
2 22 2 4
1
1 2
2 1
31
( )log
( )
( )( )
k
k
k
k k
.01196636659281283504... 3 2
6 4116
52
27
log x
x xdx
.0121188016321911204... 7
5760
5
192
93
256
1
1
4 2
2 42
( )
( )
k
k k
.01215800309158281960... e
k
k
k
2
0
7
32
2
5
( )!
.01227184630308512984...
256 162 20
dx
x( )
.01228702536616798183...
12244
11cos31sin2
6
1csc
k kk
.012307165328788035744...
1
0
1
0
1
0222
333
132
)3(3
8
1dzdydx
zyx
zyx
.01232267147533101011... 8 1 .01232891528356646996... 5 2 6 3 ( ) ( )
.01236617586807707787... 4 2
2 41
30
768
1
2
1
4 1
( )kk
J373
.01243125524701590587... 18 13
32
9 2
32
13
128
G
log
J385
.0124515659374656125... 80
3
8
45320 2 48 3 560
1
2 1
2 4
4 41
log ( )( )k kk
.01250606673761113456... 1 4 4 8 6 3 12 52
3
61
( ) ( ) ( ) ( )( )
k
kk
1 .01251960216027655609... 3 3
8 2
9 3
82
1
2 1 4 1 6 10
loglog
( )( )( )k k kk
.01256429785698989747... 160 44 215
24 80 2 20 3
1
2 13 41
( ) ( ) log ( )( )k kk
.01261355010396303533... 256
163
96
5
720
24
=1
12 42 ( )kk
.012625017203357165027... 2
)3(7
6
1
9
4 2 Borwein-Devlin, p. 56
.012665147955292221430...
022 18
1dx
e
exx
x
.012706630570239252660...
025
14
)5(155/1xe
dx
2 .01279324663285485357...
1
00 !!1515
5
1
k
k
kk
FII
1 .01284424247706555529... 3)2()2(
6
1
6
5
432
1
108
)3(91W
.01285216513179572508...
( )log
66
1
k
kk
.01286673993154335922... 1
1536
3
4
1
4 33
40
( )
( )
kk
.01288135922899929762... 31 3
108
1
1 2 3
2
21
k k k kk ( )( )( )
.0128978828974203426... 3 3
44 2
24
13
4
1
1 2
2 1
3 21
( )log
( )
( ) ( )
k
k k k k
6 .01290685395076773064... 5
81 1
4 3
60
log x dx
x
.01292329471166987383... 209
225
1
2 5
1
21
Gk
k
k
( )
( )
.012928306560301890832...
1
2
2
42
223sin3csch2
k kk
kk
1 .01295375333486096191...
21log
21log3log
2
1
2
3log2arccot
ii
ii
=
222
0
2)14(16
1
xx
dx
kkk
.01303955989106413812... 110
2
.01312244314988494075... 1
216
4
3
5
62 2
2
6 31
( ) ( ) log
x dx
x x
1 .01321183515... ( )( )
1 12
1
k
k
k
k
5 .01325654926200100483... 8
1 .01330230301225999528...
0 2
12sin
2
1
2
1sin
1cos45
6
kk
k
.01352550645521592538... 7 3
4
7054
3375
1
8
7
22
2
8 61
( ) log( )
x
x xdx
.01358311877719198759... 1
43 3 12 2 2
1
12
1 3
30
( ) log( )
( )
k
k
k
k
.01361111111111111111 =49
3600
1
2 1 2 2 2 6 2 70
( )( )( )( )k k k kk
.01375102324650707998...
2 4 2
4132
3
128
21
163 1
2 3
( )( )
k
kk
.01387346911505558557... ( ) ( )( )
2 6 465
8
1 4
1
2
42
k k
k kk
= ( ) ( )
1 2 13
1
k
k
k k
1 .01389384950459390246... cos sin( )!( )
1 11 1
4 2 10
e k kk
.01389695950004687007... 1
512
9
8
5
82 2
2
6 21
( ) ( ) log
x
x xdx
1 .01395913236076850429... iLi e Li e
k
ki i
k2 2 2 21
sin
.013983641297271329... e ie ie /2
.014063214331492929178...
2
4 2 1coth ( ) ( )
= ( ) ( )
1 2 4 111
16 4
2
k
k k
kk k
1 .01406980328946130324... dxxx
x
0
222
)3)(1(
log3log
12
3log
.01408660433390149553... 3 3
256
1
4 30
( ) ( )
( )
k
k k k
.01417763649649285569... 2 3
27
7
18 12 31
dx
x x( )
.0142857142857142857 =1
701
162
5
kk
2 .01432273354831573658...
1
2/)51(5
!5
1
k
k
k
Fe
e
5021 .01432933734541210554... 127
240 1
8 7
0
1
log x dx
x GR 4.266.1
3 .0143592174654142724...
1)51/(2
5 1
5
1
k kFe
e
.01442468283791513142... 3 3
250
2
61
( ) log
xdx
x x
1 .0145709024296292091... e k
k
( )
1
1
1
1 .01460720903672859326... e EikH
kkk
k
12
1
2
1
2
2
21
21
log
!
.01462276787138248... ( )2
142
k
kk
1 .01467803160419205455...
4
4196
7 4
84
1
2 1
( )( )
( )kk
AS 23.2.20, J342
.01476275322760796269... 27
28
8
)3(7
=
1
0
1
0
1
0222
444
1dzdydx
zyx
zyx
.01479302112711200034... 1
1728
11
12
5
12 12 2
2
61
( ) ( ) log
x dx
x
6 .01484567464494796828... 2 2
12
2
1
k
k
k
k
k
ke
k
( )
!/
1 .014877322758252568429...
1
12
332log
4
331
8
27log
4
1
kkkH
1 .01503923463935248777... e
erf e x x dxx4
12
0
( ) sinh cosh
1 .01508008428612328783... 2 3
44
Berndt 8.17.16
.01520739899850402833... 17
16 48 180
3
4
1
2
2 4
41
( )
( )k kk
.01521987685245300483... ( ) ( ) ( ) ( )( )
4 3 6 3 5 71
3
71
k
kk
.01523087098933542997... 2 2
40648
2 1
6 3
( )
( )
k
kk
.01529385672063108627... log ( )
( )( )
2
8 16
1
8
1
4 2 4 4
1
1
k
k k k
.01534419711274536931...
2
3 21
5
43
1
1 2
( )( ) ( )k k kk
.015439636746... j6 J311
.01555356082097039944...
( )36
3
1 3
6
3 3
2
1 3
6
3 3
2
i i i i
= ( ) ( )
1 3 3 111
16 3
2
k
k k
kk k
.015603887018732408909... 45 5
484 3 504 2 462
1
1
1
6 61
( )( ) log
( )
( )
k
k k k
2 .01564147795560999654... ( )8
1 .01564643870362333537... 1
61 ( !)kk
.01564678558976431415... 2 6 42 4 252 2 4621
16 61
( ) ( ) ( )( )
k kk
1 .01567586490084791444... 13
1 k kk
.0157870776290938623...
1
8192
1
4
1
16 42
30
( )
( )kk
1 .0158084056589008209... log
log log( )2
2
1
2
2 1
2
2 1
2
2
21
k
kk
.01584647647919286276... dx
x52 1
.01593429224574911842...
( )4 1
4 14 11
kk
k
1 .01594752966348281716... 104
98415
6
6
G J309
.01594968819427129848...
1
3456
1
3
243 32 3
124416
1
3456
5
32
32
( ) ( )
= 1
12 8
13 3
1728 2592 331
3
( )
( )
kk
.01600000000000000000 = 2
125
2
61
log x
xdx
1 .01609803521875068827... 1
41 k k
k
.01613463028439279292...
254
15log2log
xx
dx
.01614992628790568331... 92
32 3
1
1
2
2 32
( )( )
k
k kk
= ( ) ( )
1 3 11 2
1
k
k
k k
.0161516179237856869... 10163
6 4
2
1
ek
kk
( )!
.0162406578502357031... 31 5
8
7 3
32
2
48 2 1
2 4
41
( ) ( ) log
( )
H
kk
k
2 .01627911832363534251... 2
71
k
k k k!
1 .01635441905116282737... 1
51 k kk !
.0163897496007479593... 1 22 16
21
2 1
3
32
Gk k
k
k
log
( )
( )
.016406142123916351060... 2
6
2 2
2 2
1
2
sin sinh
= 11 4 2 16 2
2
1
1k kk
k
k
k
( ) ( )
.0164149791532220206... 6 2 3 4 5 6 ( ) ( ) ( ) ( ) ( )
= 1 1
16 17 6
26
1 1k k k kk
k k k
( )
( )
.01643437172288507812... 7 3
512
1
8 4 31
( )
( )
kk
.0166317317921115726... ( ) ( )
13
2
k
k
k
k
.01666666666666666666 =
02 160
14/1xe
dx
2 .016707017649831720444...
1
71k
k
))1(())1(())1(())1(())1(())1((/1 7/67/57/47/37/27/1
.016733900585645006255...
1 1!
)(
k k
k
.01684918394299926361... log
arctan7
6
1
2
3
6
3
3
5
3 5 22
dx
x x
.01686369315675441582... 11 5 1 16
2
1
1k kk
k
k
k
( ) ( )
.01687709709059856862... 6155 63
1080
4 3
5 41
log x dx
x x
.01691147786855766268... 2
2112
29
36
1
1 2 3
k k k kk ( )( ) ( )
1 .016942819805640384240...
1 2
)(
kk
knfac
.0170474077354195802... 1945 6 1
66
( )
( )
.017070086850636512954...
1
6)6(log
)(1
kprimep
kk
k
p
.01709452908541040576... 13 3
216 324 3
1
432
2
3
1
6 2
32
31
( )
( )( )
kk
.017100734033216426153...
6
16
1 3
2
1 3
2coth cot
i i
12
2 2 32
2 3i icot
= 1
11 6 16
2
1
1kk
k
k
k
( ) ( )
1 .01714084272965151097... 3 2 2
8
3
2 12 3
0
dx
ex /
1 .01729042976987861804... 6 22
4 221
2 32
log( )
k k
k
kk
k
1 .01734306198444913971...
6
9456 ( )
1 .01735897287127529182... 1
120 1
1
5
5
2
5
2
log
( ) !( )( )k
k km
k m
1 .01736033215192280239... 1
22 22 k kk
1 .01736613603473227207...
1
6
)6(12
)12(691
703
638512875
703
k
k
k
H
1 .01737041750471453179... sinhcosh cos
4
3 11
3 62
kk
.01740454950499025083...
2
)1(coth
2
)1(cot2
16
1coth
816
3 iii
= 18 2 16 2
2 1k kk
k k
( )
.01745240643728351282... sin1
.01745329251994329577...
180 , the number of radians in one degree
.01745506492821758577... tan1
.01746673680343934044... 17 1 16 1
2 1k kk
k k
( )
.01746739278296249906... ( )
log( )6 1
11
6
2
k
kk
k k
.01758720202617971085... ( ) ( )( )
3 3 433
16 3 41
k
kk
.01759302638532157621... 11
12 2 3
3
2
1
16 16
2 1
tanh ( )
kk
k k
.01759537266628388376... ( ) log( )5 1 1 1
1
5
2
k
k
k
kk k
1 .01762083982618704928...
26
622
12
3sech6
k k
k
.01765194199771948957... 2
21
8
144
1
12 3
G
kk ( )
.017716230658780156731...
4
401440
215
164
2 1
2 1log ( )
log( )
( )
k
kk
Prud. 5.5.1.5
.01772979515817195598... 1
6
11 2
18
3
4 1 2 31
log log
( )( )( )
dx
e e e ex x x x
1 .017739549085798905616... 912
5
12
1
72
1
9
10 2)1()1(
G
=1
144
1
12
5
12
7
12
11
121 1 1 1 ( ) ( ) ( ) ( )
=( )
( )
( )
( )
1
6 5
1
6 1
1
2
1
21
k k
k k k
.017796701498469380675... )2(62log)2(18)2(2log182log12
1 )3(232
=
12
31 log)1(
k
k
k
k
.01785302516320311736...
12
61)15(
1
kk
kkk
10 .01787492740990189897... sinh( )!
32
3
2 1
3 3 2 1
0
e e
k
k
k
AS 4.5.62
.01793192101883973082... ( )
( )arctan log
1
2 2 1 93
1
3
10
91
1
1
k
kk k k
2 .01820187236017492412...
1 2
22 112log22log26
72
1
k kB
1 .01829649667637074039... ( )5
.01831563888873418029... e4
.018355928317494465988...
16)1(
)1,6()4()3()5()2()7(3k
k
k
HMHS
.018399137729688594828...
2
)1)2()2(k
kk
.018399139592798207065...
2
)1)2()2(k
kk
.01840295688606415059... 7
82
4
14 2 16 2
2 1
( ) coth ( )k k
kk k
.01840776945462769476... 3
512 42 30
dx
x( )
.0185185185185185185185 =
1 )4)(3)(2)(1(54
1
k
k
kkkk
H
.01856087933022647259... 7 3
16 192 2 1
4
41
( )
( )
k
kk
1 .01868112699866705508...
2
1)()(k
kk
.01874823366450212957... 5
33
1
2
1
4 2 1 2 31
arctanhk
k k k( )( )
.01878213911186866071... ( )
( )
3
64
1
4 31
kk
.01884103622589046372... loglog
23
2
1
8 5 32
dx
x x
.01891895796178806334... log log2
3
3
4
1
30 4 3
1
36 622
k kk
=( )k
kk
1
62
1 .01898180765636222260...
4/14/14/14/1 22
1
22
1
4
1 iHH
iHH
= ( )4 1
2
1
215
1
k
k kkk k
.01923291045055350857... 2 3
125
2
61
( ) log
x
x xdx
1 .01925082490926758147...
( )
( )
( )5
6 61
k
kk
Titchmarsh 1.2.13
3 .01925198919165474986... e dx
e xx
2 2 4 1
20
( )
.019300234779277614553...
1
)1()1(
64
)12(
8
9
8
7
32
1
kk
kk
.01931537584252291596... 3
4
2
3
2
22 3
3
2
1
3
1
8
2 2
2loglog
log loglog
Li
= ( )
( )
1
2 4 22
k
kk k k
.019378701873... poweregertrivialnonak k
int
3
1
.01938167868472179014... 112
3 3
2
1
2
2 1
31
( ) ( )
( )
k
k
k
k
.01939671967440508500... G k
k
k
k2 16 128
1
2 1
3 1 2
31
( )
( )
.01942719099991587856... 2
5
7
8
1 2 1
2 42
( ) ( )!!
( )!
k
kk
k
k
.019534640602786820175...
1
12
)4(3)(2)(1()3(7210219
1728
1
k
kk
kkkkk
HH
.01956335398266840592... J3 1( )
.01959332075386166857... 14
96
2
32
1
2
1
2 12
1 3
21
log( )
( )Li
k
k
k
kk
.01963249194967275299... 4
33
1
61 3
3 3
21 3
3 3
2
( ) i
ii
i
= 13 3 1
1
13 16 3
2 13
2k kk
kk k k
( ) ( )
=
13
powerintegertrivialnonaω
3
)(
1
1
k k
k
306 .01968478528145326274... 5
.01972588517509853131... 1
9 108
1
3 3
2 1
21
( )
( )
k
k k
.01982323371113819152...
4
4190
17
164 3
1
2
( , )( )kk
.019860385419958982063... 3 2
4
1
2
1
64 43 21
log
k kk
[Ramanujan] Berndt Ch. 2
.0198626889385327809... 3
8
7
1920
4
5 31
dx
x x
.01990480570764553805... 3
64
3
32
3
41
31
2
1
log ( )k kk
k
k H
1 .0199234266990522067... log
log ( )5
2
1
5
1 5
25
r
1 .0199340668482264365...
2 2
2 22
2
16
5
8
1
12 2 2
k
kk k k
k k( )( ) ( )
.02000000000000000000 = 1
50
1 .02002080065254276947... 1
22 1 30 k
kk
k
( )
.02005429455890484031... 2
9
7 2
241
1
1 141
42
log
log( )
logx
dx
x
x
x
dx
x
.02034431798198963504... 10
9
1
3
2 1
2 1
1
2 31
2
( )!!
( )!!
k
k kk
J385
.02040816326530612245... 1
49
1 .02046858442680849893... 62 5
63 61
( ) ( )
a k
kk
Titchmarsh 1.2.13
.02047498888852122466... 4 5 4 4 3 12
2
51
( ) ( ) ( )( )
k
kk
.020491954149337372830...
( )
( )
( )
( ( ) )3
3
6
1 13
14
k
k nk n not squarefree
Berndt 6.30
1 .0206002693428741088... 1
9
8
9 190
csc
dx
x
.02074593652401438021... 7 3
8 322 2
1
64
1
4
32
2
6 21
( ) log( )
x
x xdx
2 .02074593652401438021... 7 3
8 32
1
64
1
4 1
32
2
40
1
( ) log( )
x
xdx
.020747041268399142...
1
2
4)!2(4
5
12 kk
k
k
B
e
e
1 .02078004443336310282... 13 3
27
2
81 3
1
54
1
3
1
3 2
32
31
( )
( )( )
kk
Berndt 7.12.3
=
1
03
2
1
log
2
1dx
x
x
33 .02078982774748560951... 416
3
3
2
2
0
( !)
( )!
k
k
k
k
.020833333333333333333 = 1
48
1 .020837513725616828...
2 3 11
21
1
2
1
2 5 31
( )k kk
= ( )2 3
21
kk
k
1 .02083953399112117969... 1
2
1
2
1
2
1
4 24 40
cos cosh( )!
k k
k
.020839548935264637084... 1
6 4 2 2 2
4 1
41
cot coth
( )kk
k
3 .02087086732388650526... ( )
!
k
kk
1
2 .02093595742098418518... 3
22
50125 5
1 25 1
csc cotlog
x
xdx
=
4 2 25 3 5
125 5 5
3
5 2
/
.02098871933468264598... 197
108
3 3
2
2
5 41
( ) log x dx
x x
1 .02100284076754382916... 1
8
1
23
1
2 2
2
20
1
, ,log
x
xdx
=1
2
1
2
1
23 3Li Li
.02103789210760432503...
9
1
18 6 3
9 3
6
1
18 6 3
9 3
6
i i i i
=1
3 1 131 ( )kk
.02108081020345922845... ( )k
kLi
k kk k
1 1 15
25
2
.021092796092796092796 =691
3276011 ( )
.02127659574468085106... 1
47
1 .021291803119572469996...
112
)1(
3
)1(
)13(
3
3
2
3
1
kk
k
kk
k
.02129496275413122889... ( )
logk
k k kkk k
1
4
1
41
1
42 1
.02134847029391195214... 7
.021491109059244877187...
1
2
2
32
223sin2csch
2
3
k kk
kk
.021566684583513603674...
0
2
2
4
1
4
sech
k k
k
1 .021610099269294925320...
0
4
433 arctan
122log
62log2)3(
4
3dx
x
x
1 .02165124753198136641...
01 )12(16
1
5
22
4
1arctanh4
3
5log2
kk k
Li
.0217391304347826087... 1
46
60 .02182669455857804485... 42 26 22
3
1
logk Hk
kk
1 .02210057524924977233...
k
k
kkk
2
)12(32
1
22
1arcsin22
0
.02219608194134170119... 7
30720
4 3
51
log x dx
x x
.02222222222222222222 =1
45
.02222900171596697005... 2
218
3
2
29
16
1
1 3
( )
( )( )k k kk
.02224289490044537014...
2
54
2
9
1
9 3 3
5 2
2
1
9 3 3
5 2
2
i i i i
1
18 6 3
5 2
2
1
18 6 3
5 2
21 1i i i i
( ) ( )
= 1
11 3 1
3 22 1k
k kk k
( ) ( )
.022436419216546791773...
12 )32)(22(4
)2(
2
)3(
12
1
kk kk
k
=
12
232
4
11log12
2
1arctanh48124
6
1
k kk
kkk
.02258872223978123767... 4 211
4
1
1 2 3
1
1
log( )
( )( )( )
k
k
k
k k k
.02271572685331510314... log log
( )
2
3
5
24 1 40
1
x
xdx
.02272727272727272727 =1
44
.0227426994406353720... 11
42 4
1 1
1 26 42
41
( ) ( )( ) ( )k k k k kk k
.02277922916008203587... 17
16
3
22 4 4
0
4
log sin tan/
x x dx
.022891267882240749138... )3(7560
37)5(
120
11
2
)7()9(
24
197)3( 64
23
16
3
)1(k
k
k
H Borwein-Devlin, p. 63
.02289908811502851203... 9
8 12
3 2
4
2
2
1
2
2 2
4 22
log log
( )kk k k
.02291843300216453709... 3
23
13
8
1
9
2 1 1
932 1
log( )
k k
k
kk
k
1 .0229247413409167683... 1
1542 kk
1 .0229259639348026338... 1
2 1 244
11 ( )( )
k kkk k
k
.02297330144608439494... 50 43
128
28 2
128
71
1536
2 1
2
1
2 61 3
2G k
k kk k
log ( )!!
( )!!,
1 .02307878236628305099...
64
2 2 22
1
2 1 22
2 21
sin sec(( ) )kk
.02309570896612103381... )(ofzeroa,1
2log12
log
2s
Edwards 3.8.4
1 .02310387968177884861... 31 5
8
7 3
32
7 3
4
2
48 48 42 2
2 4 4 ( ) ( ) ( ) loglog
=H
kk
k ( )2 1 41
1 .02313872642793929553...
12
2 12
1)2(
2
2log
1
log
kk k
k
k
k
=
2112
1arctanh
1
2
2log1)2(
4
2log
kkk kk
kH
.023148148148148148148 = 5
216
1
5
1 3
1
( )k
kk
k
1 .02316291876308017646...
122
2
)1112(
1
)112(
1
36
)32(
k kk
=log x dx
x120 1
.02317871150152727739... 233
1920
2
32
1
1643
coth
kk
.02325581395348837209... 1
43
1 .0233110122363703231... 3 2
3148
2
4
1
4 2 1
2
log
( )kk k
k
k
= arcsin logx x
xdx
0
1
.02334259570291055312... 3 2 3 3 2 3 2 31
2
0
( ) ( ) ( )logx x
e edx
x x
1 .02335074992098443049... 27
218 2
1
1 9
1
21
log( )
( / )
k
k k k
.02336435879556467065...
12 )189(2
1
6
2log
36
5
kk kk
=( )
( )( )( )( )
12 1 2 2 2 4 2 50
k
k k k k k K ex 107b
.02343750000000000000 = 3
128
1
3
3
1
3
51
( ) logk
kk
k x
xdx
.02346705930540378299... 2
2 4
2
21
sinh
k
kk
.02351246536656649216...
3
5
27
1
4
1
381
4
27
)3(26 )2(3
=
2
3173
2
3134
)3(6
1
381
8
3
)3(233
3 iLii
iLii
i
= dxxx
x
136
2log
.02353047298585523747... 2 328567
1200072
2
8 71
( )log( )
x
x xdx
.02375236632261848595... 2log)3(2124
2log
2
1
24
2log
903
4
224
Li
=
3
1
!
)3,(
k kk
kS
.0238095238095238095 =1
42 3 B
.02385592231300210713...
1
560
2
2
)5(
)1(
144
115
12 xx
dx
kk
k
1 .023884750384121991821...
2
2 2 211
1
21
1
2 1
cosh( )kk
1 .02394696928321706214... ( )jkkj
2 114
.02402666421487355693... 1
162 6k k
s dxk slog
( ( ) )
.02408451239117029266... ( ) ( )
( ) /
1 2 1
21
1
21 20 1
3k
k kk
kk
k HW Thm. 357
1 .02418158153589244247... 16 2 4 2 2 3 41
2 5 41
log ( ) ( ) ( )
k kk
=( )k
kk
4
21
2 .02423156789529607325... 21
61
/k
k k
1 .02434757508833937419... G k
kk8 64
1
256
1
4
1
6144
1
4
2 1
4 1
22 3
2
40
( ) ( ) ( )
( )
=G
8 64 128
7 3
32
1
6144
1
4
2 33
( ) ( )
.02439024390243902439 =1
41
.02439486612255724836... 1728
2035)3()5,3(
=
1
0
1
0
1
0
444
1dzdydx
xyz
zyx
3 .02440121749914375973... ( )
!
/k
k
e
kk
k
k
3
0
1
31
.02461433065757607149... 4 3 3
16
2
5 31
( ) log x dx
x x
1 .02464056431481555547... 2 1 272
64 2 12 2 22 32 1
24
23 2
1
( log ) log log ( )( )
H
k kk
k
.02475518879810900399... 61 6
72
1
1 2 3
2
21
k k k kk ( ) ( )( )
.02481113982432774736...
1 )3)(2)(1(2kk
k
kkkk
H
7 .02481473104072639316... 5
.0249410205141828796... log( )
5
6 6 62
k
kkk
.024991759249294471972...
2 )1(2
1)()1()1(3
2
log3log3
12
2log432
4 kk
k
kk
k
.02500000000000000000 = 1
40
.0252003973997703426... ( ) ( ) ( )2 3 43
2
15 4
2
k kk
= ( ) ( )
1 4 11
1
k
k
k
.02533029591058444286... 1
4 2
.02540005080010160 =1000
3937 = meters/inch
.02553208190869141210... ( )3 1
8
1
8 113
2
k
kkk k
2 .02560585276634659351... 4430
21871
21
8
( )k
kk
k
.025641025641025641 =39
1
1 .02571087174644665009... 23
7
3
2
1 32
1
1
arcsinh
( )k k
kk
k
k
1 .02582798387385004740...
122
22
)12(
1
2
1
2cot
242csc
8 k k
J840
=
112
)22(
kk
kk
1 .02583714140153201337... ( ) ( )
( )6 7 1
71
k
kk
HW Thm. 290
.02584363938422348153... dxxkk
k
k
6
526
1)(log
1
.0258653637084026479... 2 5 2 3 4 3 2 42 4
1
( ) ( ) ( ) ( ) ( ) ( )( )
H
kk
k
.02597027551574003216... 1
2 162 1
1
8 1 8 11
( )
( )( )k kk
GR 0.239.7
1 .02617215297703088887...
08 18
csc8224 x
dx
.026227956526019053046... 161
900 6 0
1
G
x xdxx arccot5 log
.026242459644248737219...
1 64
)2(4221
128 kk
kk
.0263157894736842105 =1
38
.02632650867185557837... 1
2
7
21
2
12
2
0
e
k
kk
k
k
( )( )!
241 .02644307825201289114... 1233
2
1
ek
k
k
k
!
.02648051389327864275...
245
14
1 )1()248(2
1
3
22log
xx
dx
xx
dx
kkk
.02649027604107518271... 3 1
3164 2
1
2 1
G k
k
k
k
( )
( )
.02654267736969571405... 3 3
4
7
8
1
3 30
( ) ( )
( )
k
k k
=
1
0
1
0
1
0
222
1dzdydx
xyz
zyx
1 .026697888169033169415...
1 2
)2(
2csc
2log
kk k
k
=
122
11log
k k
1 .02670520569941729272... 4 7 2 5 3 46
1
( ) ( ) ( ) ( ) ( )
H
kk
k
.02671848900111377452... ),(
.02681802747491541739...
3
2
6
7
216
1
18
)3(13
381
52 )2()2(
3
=
2
3132
2
314
)3(3
1
2
)3(
381
102 33
3 iLii
iLii
i
=
125
2log
xx
dxx
.026883740540405483253...
1
1
)12(62
)1(
2
3log
5
121
k k
k
kk
k
6 .02696069807178076242...
1
03
3
04
)3(
1
log
)13(
16
3
1
81
1
x
dxx
kk
.027027027027207207027 =1
37
.02707670528833012617... Gk
x dx
x x
k
k
8
9
1
2 5 20
6 41
( )
( )
log
.02715493933253428647... ( )
( )!cosh
k
k k kk k
1
21
1
2
1
2 2
.02737781481649722160... 2 4
4164 768
1
2 2 1
k k
kk
( )
( ) J349
.02737844362978441947... 3
2
5
324 2
0
4
sin tan/
x x dx
.02742569312329810612...
1 .0275046341122482764...
0
22 )!12(
)1(4
1
k
k
k
BeLi
.02753234031735682758... dxx
xx
0
2
2
cosh
log2log
2
2log3
.02755619219153047054...
( )
( )
log
( )
log5
5 1
1
55 51
p
p
k
kp prime k
=
16
)(
k k
k
3 .0275963897427775429... cosh( )!
e e
k
k
k2 20
GR 1.411.4
.02768166089965397924... 8
289 40
x x
edxx
sin
.02772025198050593623... ( )k
kLi
k kk k6
26
1
1 1
1 .02772258593685856788...
1
3
1
3
13
)14(
)1(
)34(
)1(
264
3
k
kk
kk
J326, J340
=
03
2/
)12(
)1(
k
k
k Prud. 5.1.4.3
.02777777777777777777 = 1
36
1
2 1 2 2 2 4 2 50
( )( )( )( )k k k kk
K ex. 107c
= k
k k k kk ( )( )( )( )
1 2 3 40
.027855841799040462140... ( )
( )
( ) ( )
( )
1
4 4 1 4 4 12 1 1
k
kk k kk
k
k
k
k k
.02788022955309069406... 115 5
16
1
2 50
( ) ( )
( )
k
k k
1 .02797279921331664752... 7 183
2
0
ek
k kk
!( )
2 .02797279921331664752... 7 172
4
0
ek
kk
( )!
19 .02797279921331664752... 7e
1 .0281424933935921577... 1)2(sinh
cscsinlog2
11
1
kk
ke
e k
.02817320866780299106... 3 3
128 1
2
51
2
40
( ) log
xdx
x x
x dx
e x
1 .02819907222448865115... 1
.02835017583077031521...
21log222log8
8
7cot
16
1
=
12
2 864
1
8
)(
kkk kk
k
.02839399521893587901...
145
21
2
cos
1cos
)1(cos1
51cos4
1cos21
kk
k GR 1.447.1
.02848223531423071362... 3
2
4 1
3
1
1
e
k
k
k
k
( )
( )!
.0285714285714285714 =1
35
.02857378050946295008...
( )log
55
1
k
kk
.02874500302121581899... 1
0
222
arctanlog16
)3(
2
log
27
2log
216108
19
54dxxxx
1 .02876865332981955131... ( )ck
kLi
kkcc
kc
1 12
122
21
.028775355933941373288...
1
31
2)!1(
)1(
4
132
kk
k
k
k
e
.02883143502699708281... 3 3
22
4 1
2 3
41
( )log
log
( )
x
xdx
18 .02883964878196093788... log log11
30
12
x
xdx
= 122
34 2
1
6
7
6
2
3
1
72
2
3
7
61 1 2 2
log ( ) ( ) ( ) ( )
.0290373315797836370... 1
1000
9
10
2
5 12 2
2
51
( ) ( ) log
x dx
x
.029076134702187599206...
023
14
)3(33/1xe
dx
.0290888208665721596...
108 92 20
dx
x( )
3 .02916182001228824928... sinh
( )
3
7 31
3
2 21
kk
.029280631484544157522...
2 )1(2
1)(2)1(6log632log19
12
1
kk kk
k
=
2 2
11log)12(
4
11
k kk
k
.02929178853562162658... 1
48
1
16
1 2
420
e k
k k
k
( )
( )!
.029292727281254639...
25 1
log
k k
k
9 .02932234585424892342... 4 1 12
2
k
k
k( ( ) )
.02941176470588235294... 1
34
.029524682084021688... c34 2 21
2412 2 8 3 3 2 6 4 ( ) ( ) ( ) ( )
Patterson ex. A4.2
.0295255064552159253... 7 3
4
56
272
1
2 1 32
( )
( )
kk
= log2
6 41
x
x xdx
= x x
xdx
4 2
20
1
1
log
GR 4.261.13
.02956409407168559657... 1
4
1
36
5
6
1
6
1
3 21 1
1
21
( ) ( ) ( )
( )
k
k k
= 1
36
5
6
4
31 1
6 31
( ) ( ) log
x dx
x x
1 .02958415460387793411...
0 2)12(22
1
8
1,
2
3,
2
3,1,1,
2
1
k k kk
kHypPFQ
.029648609903896948156...
12 )32)(12(4
)2(
8
)3(9
6
1
kk kk
k
1 .02967959373171800359...
1
04
12/3
1
log
4
3
4
1
16 x
dxx
.029700538379251529... 2
3
9
8
2 1
2 42
( )!!
( )!
k
k kk
.029749605548352127353...
2
2
4
)1()(
4
2log9
24
9
kk
kk
1 .0297616606527557773...
3 33
3 3 1330
log
x dx
e x GR 3.456.1
=
log x dx
x1 330
1
GR 4.244.2
.0299011002723396547...
2
214
39
16
1
1 2
k k kk ( )( )
J241, K ex. 111
.0299472777990186765... 3 2
2
2
2
5
4 2 1 2 3
2
1
log log
( )( )( )
H
k k kk
kk
.0300344524524566209... 2
27
3
21
2
1
log ( )
k kk
k
H k
2
.0300690429769907224... log ( )( )
6
5 51
52
kk
k
k
k
=
1 5
11log
5
1
k kk
1 .0300784692787049755...
1 )12(!!)!12(
!)!2(11
21,0,
2
3
k kkk
kerf
ee
2 .0300784692787049755...
0 )!12(
4!1
2 k
k
k
kerf
e
.03019091763558444752... ( )31
2
1
21 1 i i
= )2()2(2
1)3( ii
=
1
1
235
1)32()1(1
k
k
k
kkk
.03027956707060529314... 3 2
4 401024
x dx
e ex x
1 .030296955002304903887...
2/1
2/12
22 arcsin32log
3
2
93
16
x
dxxG
.03030303030303030303 = 1
33
1 .03034552421621083244... 6 3
7
( )
1 .03034552421621083244... e
e
.03044845705839327078...
( )( )
3
42
2 2 c Berndt 9.27
1 .03055941076... j5 J314
.03060200404263650785... 1
64
1
4
1
16
1
4
1
2 16
1 2
1
cos sin( )
( )!
k
kk
k
k
2 .03068839422549073862...
2 2
322
1 2 35 5 2
1
( )( )
k k
k kk
= ( ) ( )( ( ) )
1 1 12
2
k
k
k k
.030690333275592300473...
12
11
1
2
log ( )( )
kk
k
k
k
.03087121264167682182... dxee
xxx
022
34
18
3
240
.03095238095238095238 = 13
420
1
2 1 2 5 2 90
( )( )( )k k kk
.03105385374063061952... 132
1
2 1
3 1
31
( )
( )
k
k k
.03111190295912383842... 33
48 48
9 3
24
1
2
2
31
( ) ( )
( )
k
k k k
.031125140377033351778... 15 5
8
45 3
2140 2 126
1
1
1
5 51
( ) ( )log
( )
( )
k
k k k
.03117884541829663158... log
( )( )
( )( )
2
4
1
61
1
1 22
k
k k kk
5 .03122263757840562919... 2 2
2 1
1 22 1
1 1
k
k k
k
k k k
( )
( )!sinh
.03122842796811705531... 2 1
216
2 2 31
1 2
log( )
( )( )
k
k
k
k k
.03124726104771826202... 1
32 84
1
16
1
21
csch
( )k
k k
.03125002980232238769...
115 132
)5(
2
1
kk
k
kk
.03125655699572841608... 23
2 2 81
2
22
3
log Li
1 .03137872644997944832... 1
51 ( !)kk
.03138298351276753177... 12635
3 9
1
1
2 4
5 51
k kk ( )
1 .03141309987957317616... cosh( )!
1
4
1
2 160
k kk
.03145390200081834392... 1
642
5
82
9
8
1
64
5
8
9
81 1 , , ( ) ( )
= ( )
( )
1
4 1
1
21
k
k k
= log x dx
x x6 21
.031667884637975851999... dxx
xxci
1
2
sincos21sin1cos
.03202517057518619422... 12
2
1
11
12 2
2
logsinh
coth log
k kk
= ( ) ( )
11
2 11
k
k
k
kk
.032049101862383653901...
0
2327
3 8
116876141)(
32
k
k
kkkkr
)1(
1
2
1)(
kkkrwhere
Borwein-Devlin. p. 36
.032067910200377349495... dxxx
x
14
2
)1()1(
log
12
2log5
646
1
.032080636429384244827...
4
1
4
3)5(297664
4096
1 )3()3(2 G
= 4/
0
3 tanlog
dxxx
.032092269954397683... 527
7350
2 2
35
1
2 7
1
1
log ( )
( )( )
k
k k k k
.03215000000000000000 = 1
32
2
51
log x
xdx
.03224796396463925040... 123
2
29
6
1
2 2 6
1
0
log( )
( )
k
kk
k
k
.03225153443319948918... 1
3
.032258064516129 =1
31
.032309028991669881698... )6(48
203)3(3)1,2,3( 2 MHS
.03233839744888501383... dxxxe x
0
log2log222
.032355341393960463350... dxx
xx
0
8
2
1
)1(
8224
8
.032411999117656606947... 4/
0
22 coslog24)3(92log486144
1 dxxxG
.03241526628556134792... ( ) ( ) ( ) ( )5 3 3 2 4 5 2 15
=1
12 51 k kk ( )
.03259889972766034529... 5
2 4
1
1 2
2
2 21
k k kk ( ) ( )
1 .03263195574407147268... log
( )
2
4 2 2
2 1
8 2 102
G k
k kkk Berndt 9.6.13
1 .0326605627353725192...
15
155 )13(
1
)13(
11
243
)3(242
kk kkG
J309
1 .03268544015795488313... 1
31 k k
k
.03269207045110531343... 6
5
4
3
.03272492347489367957... 96 646
0
dx
x
.03283982870224045242... 19
72
2
3
1
2 4
1
1
log ( )
( )( )
kk
k
H
k k
.03289868133696452873... 2
61300
log x dx
x x
.03302166401177456807... J
k kk
k
k1
0
4
41
4
1
( )!( )!
.0331678473115512076... 365
48
1
4 1 3
1
0
log( )
( )( )
k
kk
k
k k
1 .03323891093441852714... 7 3
16 192 2 1
4
41
( )
( )
k
kk
2 .03326096482301094329... 2
61
k
k k k!
.033290128624160141531...
3
3
2 6
1
61 3
3 3
21 3
3 3
2
2
( ) ( )i
ii
i
= ( ) ( )
1 3 2 111
15 2
2
k
k k
kk k
.0333333 = 1
30 2 4 B B
1 .03348486773424213825... 1
41 k kk !
1 .03354255600999400585... 3
30
.03365098983432689468... 1
27
1
3
13 3
81
2 3
7291
3
( ) ( )
= 1
1626
1
3
1
3 3 11 2
31
( ) ( )
( )
k
kk
.033749911073187769232...
1
02
6
)1(
)1log(2log3
34
5dx
x
x
.03375804113767116028... 5 2 3 4 51
1 51
( ) ( ) ( ) ( )( )k kk
= 1
6 52 k kk
.03388563588508097179... ( ) ( )k k
kk
0
1 2 1
1 .033885641474380667... pq k
kk
( )
21
.03397268964697463845... 3
2 12
2
2
2
2
1
2
2 2
3
log logLi
.03398554396910802706... ( )
( )!sinh
2 1 1
2 1
1 1
1 2
k
k k kk k
.03399571980756843415... J Fk k
k
k4 0 1
0
21
245 1
1
4( ) , ,
( )
!( )!
LY 6.117
.03401821139589475518... 1
144
5
12
11
12 11 1
61
( ) ( ) log
xdx
x
.03407359027997265471...
4/
0
4
1
tansin)168(2
1
16
5
2
2log
dxxxkk
k
1 .03421011232779908843... 2
0 )1(16
12
4
1,2,2,1,1,
2
1
kk
kHypPFQ
kk
.03421279412205515593... 3 3
22 2 3
1
36 31
loglog
k kk
J376
1 .034274685075807177321... 1
2
11 1
12
1
ee
erf erfix
dx
sinh
1 .03433631351651708203... 4
2 34
2 2 11
3
4
2
20
1log( !)
( )!( )
k
k k
k
kk Berndt Ch. 9
1 .03437605526679648295... 7
6
7 170
csc
dx
x
.03448275862068965517... 1
29
.03454610783810898826... 1 1
3
1
3
1
1e k
k
k
( )
( )!
2 .034580859539757236081... 12 2 21
1 340
log( )( )
k kk
.034629994934526885352... 3
2 122 2
3 3
4
12
4 32
log
( ) ( )k
k k k
.03467106568688200291... 1
1024
7
8
3
8
1
4 12 2
1
31
( ) ( ) ( )
( )
k
k k
1 .03468944145535238591... sin
arctan
x
xdx
0
1
2 .03474083500942906359...
22
3coscosh)(sinh
= sinhcosh cos cosh cos
4
1 1 3 1 1 33
= 11
61
kk
Berndt 4.15.1
1 .03476368168463671401... 1 11 1
1
2 12
2
2
e
k k e
k
kk
k
kk
k
k
/
/ ( )!
( )
=
Re( ) 2 1
1
k
ikk
2 .03476368168463671401... 1 11
11
1
2 11 2
e
k k e
k
kk
k
kk
k
k
/
/ ( )( )!
( )
.03492413042327437913... Ai ( )2
1 .03495812334779877266...
1 1
)3()4(
044 )1(2
)2(
2)1(2
1
2
12
k kk
k
k
k
kk kk
kHH
kLi
.035034887349803758539...
1
2
1
4
1
2
1
2
1
2
1
2
i i i i
= ( ) ( )
1 4 1 111
15
2
k
k k
kk k
.035083836969886080624... 1
21 1
2
log( )
k
kkk
.0352082979998841309... ( ) ( )( ) ( )
( ) ( )( )
( )2 34
4
3 6
42 3
3
22 5
2
=H
k kk
k ( )
1 51
1 .0352761804100830494... 2 3 15
12 csc
AS 4.3.46, CGF D1
.03532486246595754316... 11
32
1
1
2
2 32
( )
( )
k
k k
.03536776513153229684... 1
9
.03539816339744830962...
3
4
1
2 2 1 2 2
1
1
( )
( )( )( )
k
k k k k J244, K ex. 109c
.03544092566737307688...
1
125
4
52 ( )
.03561265957073754087... ( )
( )
5 1
5
.035683076611937861335...
1
12
)12()1(
2
1
kk
k kii
.03571428571428571428 =1
28
.035755017483924257133... 1
55
1p
k
kk
p prime k
( )log ( )
.03577708763999663514... 2
25 5
1 2 1 22
21
3
21
( ) ( )!
( !)
( ) ( )!
( !)
k
k
k
k
k k
k
k k
k
1 .03586852496020436212...
1
4)14()4()14()24(k
kkkkk
.03596284319022298703... 1
11 5 15
2
1
1kk
k
k
k
( ) ( )
.03619475030276797061... 2 4 2
4132 384
7 3
16 2 1
( )
( )
k
kk
.03619529870877200907... 2 7 2
81
1
12 31
log
log( )x
dx
x
.03635239099919388379... 1
2
1
2
1
2 2 12 11
arctan( )
( )
k
kk k
.036382191549123483046... dxx
xxx
0
2/3)2(sin
cossin21log
2
1
2442
1
Prud. 2.5.29.27
.036441086350966020557... k
kk
k
k
k6
2
1
111 6 1 1
( ) ( )
.03646431135879092524... 1
5
6
5
1
5 611
log( )
H nk
kkn
.03648997397857652056... 2 2 23
2
1
2
log
.036557856669845028621...
2 2 6
11
2
1arctanh2
)12(2
1)(
k kk kk
kk
k
.03661654541953101666... log( )
log4
5 5 51
1
5
1
52 1
k
k k kkk k
1 .03662536367637941437... 1
36
1
6
1
6 5 11
21
60
1
( )
( )
log
k
x dx
xk
.03663127777746836059... 2 1 24
2 1
0e k
k k
k
( )
!
.036683562016736660896...
1
1
27
2
1)27()1(1 k
k
k
kk
k
.03676084783888546123... 3 1
2 31128
3
32
1
2
1
4 1
( )
( )
k
k k
.03681553890925538951...
032 )4(256
3
x
dx
.03686080059124710454...
12 )22)(12(4
)2(
4
)3(7
4
1
kk kk
k
.03690039313500120008...
2log2
25
1
1 .03692775514336992633...
primep pppp
ppp4321
321
1
1)4()5(
1
12
1 3
1
2 5
1 12)1(
2
52)1(
2k
jk
k
k
k
j
k
kk
k
kk
Borwein-Devlin, p. 42
primep pppp
ppp4321
321
1
1)4(
.03698525801019933286... 6
.037037037037037037037 = 1
27
.03705094729389941980...
1235
1)38(1
kk
kkk
.037087842300593465162...
113 127
)3(
3
1
kk
k
kk
.03717576492828151482...
1225
1)27(1
kk
kkk
.03717742536040556757... ( )
log( )6 1 1
11
6
2
k
kk k
k k
2 .03718327157626029784...
21
3
5
32
.03722251190098927492...
03
)2(
)316(
1
16
3
8192
1
k k
.037286142336031613937...
1
2
2
22
223sincsch
3
1
k kk
kk
.03729649324336985945...
12
2 )84(3
1
)1(2
)1(
8
7
2
3log
4
9
kk
kk
k
kkk
1 .03731472072754809588... 22
22
1tanh2csch2coth
ee
ee
=1
2
16
2 20
k
kk k
B( )!
AS 4.5.67
.03736229369893631474... 1
2
3
64
1
4 1
2
2 31
( )kk
J373, K ex. 109d
.03741043573731790237... 1 2 4 32 4
1
( ) ( )( )
k
kk
.03742122104674100704...
13
)2(3
)912(
1
4
1
3456
1
432
)3(7
1728 k k
.03742970205727879551...
2
33
2
33
6
1
4
1
3
ii
=
1215
1)16(1
kk
kkk
=
24
1
1 1
2
11)13()1(
2
1
kk
k
kkk
.03743018074474338727...
21
1!
1)5( 5
k
k
k
ek
k
.03743548339675301485... ( )
log( )5 1
11
5
2
k
kk
k k
.03744053288131686313... 3 2
4
19
20 8
1
4 4 11
1
42
1
2
log
( )( )
( )
k k
k
k
kk
k
.03756427822373732142... dxe
xdx
xx
xx
04
2
15
2
1
log
32
)3(
.03778748675487953855...
032
1 )3(21
16348 x
dx
k
kk
27 .03779975589821437814... 6
.03780666068310387820...
13
)2(
)69(
1
3
1
1458
1
k k
.037823888735468111... 16 29
3
1
4 20e k
k
kk
( )
( )!
.037837038489586332884... 3 3
42 2
9
4
15 3
2
( )log
( )
k
k k k
=
13
1
)2()1(
)1(
k
k
kkk
=
12 6932
1
kk kk
.037901879626703127055... 1
2
2 2
3
1
27 331
log ( )k
k k k [Ramanujan] Berndt Ch. 2
.0379539032344025358...
12
51)5(
1
1
kk
kk
.03797486594036412107... ( ) log( )4 1 1 1
1
4
2
k
k
k
kk k
.03803647743448792694...
1
2)12(
812cos
7
1
k k
.03804894384475990627...
13
)2(
)58(
1
8
3
1024
1
k k
.03810537240924724234...
0
sinlog4
2log
22
1dxxxex x
.038106810383263487993... G
k k
k
k8
11
144
1
2 1 2 52 20
( )
( ) ( ) Prud. 5.1.21.42
1 . 03810687225724331102...
1
5
)5(2)5()10(
2
1
k
k
k
H
1 .0381450173309993166...
55259
4
155259
4
1ii
2
55259
4
1
2
55259
4
1ii
= k
kk
5
52 1
.0381763098076249602... 80 4 48 2 6 31
2 12
3 31
log ( )( )k kk
.03835660243000683632...
12
12 )88(2
1
1632
1
8
2log1
kk
k kkkk
= dxx
x
1
02)22(
)1log( GR 4.201.14
1 .03837874511212489202...
0
1
1112
)1(
k
k
k
.0384615384615384615 =
0
5cos26
1dxex x
.0385018768656984607... cossin
cos sin cos22
22 1 1
=
1
21
2/32/1 )!2(
4)1()2()2(
k
kk
k
kJJ
31 .03852821473301966466... 3 3
.0386078324507664303...
022 )1)(1(2
1
2
1xex
dxx Andrews p. 87
= 1log4
log1
022 x
dx
x
x
GR 4.282.1
.038622955050564662843...
12
1)1()1(
)13(
)1(
399
2log
3
2
6
1
108
1
k
k
k
k
.03864407768699789287... dxxx
x
14
2
)1(
log
36
331
1 .03866317920497040846... dxxx 22/
0
23
sin488
28 .03871670431402642487... 7
8 162
21 3
2
64 20 8 1
2 2 1
2 4 3 2
42
log( )
( )
k k k
k kk
=k k
kk
3
2 2
( )
.03876335736944358027...
1
3
1
)2(
)1(2)3(3)2(2
k
k
k
k
=
122
21
02
22
)1(
log
)1(
log
xx
dxx
x
dxxx
.038768179602916798941... 3
)3(
.03879365883434284452... 1log16
112log2
28
12
1
022
x
dx
x GR 4.282.9
.03886699365871711031... 21
2
1
2
5
41
1
2 2 1
1
2 11
arctan log( )
( )
k
kk k k
.03895554778757944443... 7 3
216
1
6 3 21
( )
( )
kk
.038978174603174603 =3929
100800
1
5 81
k k kk ( )( )
.0390514150076132319... 3
11
!
1)23(
22
1
k
kk
ekk
k
.03906700723799508106... 1
83 4 2 2 ( ) ( )i i
= 5
8 2
1
42 2
15
2
( ) ( )i i
k kk
= ( )4 1 11
kk
.0391481803713593579... ( ) log( )3 2 1 1
1
3
22
k
k
k
kk k
.03915058967111741409... 116
21
163
2 3
2
31
( )( )
k
kk
.039155126213126751348... 1
32 48 3 216
1
3 1 3 5
2
2 20
( )
( ) ( )
k
k k k
.03929439142762194708... 25 5
225 2
11
4
1
4 1 2
1
1
loglog
( )
( )( )
k
kk k k k
=
1 )42(5
1
kk k
.03939972493191508632...
232
2
)1(
1
32
221
k k
.03945177035019706261...
1
0
32
12
12
log)1log()33(
)1(2
54
12dxxxx
kk
k
.039459771698800348287... 1)2(1
1
4
9log2
2
kk
k
k
=
2
222
2
)1log(2
1
2
k
kkk
k
1 .03952133779748813524... 2 1cos e ei i
3 .03963550927013314332... 30 5
22
( )
1 .03967414409134496043... 1cschcoth2
1 22
=
2222
1
1
1
212(2)1(
kk
k
kkkk
1 .03972077083991796413... 3 2
23
1
9 2 10
log
( )
arctanh1
3 kk k
= 1 21
64 43 21
k kk
[Ramanujan] Berndt Ch. 2
.03985132614857383264...
02
)2(
)35(
1
5
3
250
1
k k
.03986813369645287294...
1
12
1)3()1(4
13
3 k
k kk
=
1222 )2()1(
1
3
4
k kkk
2 .03986813369645287294... dxx
xx
1
0
22
1
log)1(
4
5
3
.04000000000000000000 = 1
25
.0400198661225572484...
145
2log
2
1
216
251)3()4,3( dx
xx
x
=
1
0
1
0
1
0
333
1dzdydx
xyz
zyx
.04024715402277316314... ( )2 1
21
kk
k
1 .04034765040881319401... )4(103
4 .04050965431073002425... 7
.040536897271519737829...
15
2
)1()1,5(
2
)3(
4
)6(3
k
k
k
HMHS
.040634251634609178526...
122 16
1)2(
116
1
830
13
kk
k
k
k
=
0 72
)1(
2
1
k
k
k
.04063425765901664222... sin cos( )
( )!
1
2
1
2
1
2
1
2 1 41
k
kk
k
k
1 .04077007007755756663... 23 1 5 1
161
2 1
4
1
sin cos( )
( )!
k
k
k
k
.04082699211444566311... ( )( )
3 12 2
22
f k
kk
Titchmarsh 1.2.14
= 1
322 ( )jkkk
.04084802658776967803... 13
2
3
4
2
3
1
6
1
6
1
621 2
log log ( ) ( )
k k
k
k
k
kk
.04101994916929233268... 1681
292
1
2
383
2
1
2 4 2
1
1
cos sin( )
( )! ( )
k
kk k k
1 .04108686858170726181... ii eLieLi 21
21
.04113780498294783562...
54 2 8140
dx
x
.04132352701659985... B
kk
kk
2
1 2 2( )!
.04142383216636282681...
1
1)1(12
tanh2
1
2
1
k
kk ee
e
J944
.04142682200263780962... 7 3
16 64
1
128
3
4
1
4 1
32
31
( )
( )( )
kk
1 .04145958644193544755...
3
1
3
12
2
3log
6 22
32
LiLi
.0414685293809035563... 11
3
2
3
3
2 3
1
3 1
1
1
e
e
k
k
k
cos( )
( )!
.041518439084820200289... 112 4 2 2
12
4 22
tanh coth
( )k
k k k
.04156008886672620565... 26 3
27
4
81 32
1
27
4
3
32
2
5 21
( ) log( )
x
x xdx
2 .04156008886672620565...
13
)2(3
)23(
12
3
1
27
1
381
4
27
)3(26
k k
= dxx
x
1
03
2
1
log
= x dx
e ex x
2
20
.04156927549039744949... 13
21
6 2 3
3 3
2
1
6 2 3
3 3
2
( )
i i i i
= 1
5 22 k kk
.04160256000655360000...
12
12 15
)2(
5
1
kk
k
kk
.04161706975489941139... 1
4 2
1
2
1
2
1
2
1
2
1
4
1
1
cosh sin cos sinh( )
( )!
k
k
k
k
.04164186716699298379... cos cosh( )
( )!
1
2
1
21
1
4
1
1
k
k k GR 1.413.2
2 .04166194600313495569... k
k Hkk !
1
.04166666666666666666 = 1
24
1
3 1 3 4 3 70
( )( )( )k k kk
C132
=
1
1
)1()1(
kkk
k
e
k Prud. 5.3.1.2
=
1
131
)1()1(
kkk
k
e
k
1 .04166666666666666666 = 25
24
1
221
k kk ( )
1 .04166942239863686003... 12
1 ( )!kk
1 .04169147034169174794... 1
21 1
1
2 4
1
4
1
40
cos coshcos
( )!
e
e kk
.04169422398598624210... ( )
! ( )!
k
k kkj
jk
2 22
1
.04171115653476388938... 1
42
2
16
5 2 2
5 2 2
3
8arctan log
log
( )
arctan arctan2 1
8
2
81 2 1 2 4 4
2
dx
x x
.04171627610448811878...
1 )!4(8
1sin
16
1
168
1sin1sinh
k k
k
e
e
1 .0418653550989098463... e
e kk3
2
3 3
3
2
1
3 10
cos( )!
J804
.04188573804681767961... 1
6
9
7 4 22
log
dx
x x
.04189489811963856197... ( )
log
15
2
k
k k k
721 .04189518147832777266... 3
4
6
.04201950582536896173...
1
22 2 1 2
1
2 1
log cos log sincosk
kk
GR 1.441.2
.0421357083215798130... 4 2 29
4
1
2 32
1
1
log log( )
( )( )
k
k
k
H
k k
1 .0421906109874947232...
0
2/12/1
4)!12(
1)(2
1
2
1sinh2
kkk
eee
e
= 11
4 2 21
kk
GR 1.431.2
.04221270263816624726...
1
2 4 2 2 2
4 1
24 4 41
cot coth
( )kk
k
=
24 12
1
k k
1 .04221270263816624726... ( )4
2
1
2 114
1
k
kkk k
=
444 2
coth2
cot242
1
1 .042226738809353184449...
1)1(
122
22
12
2
1
12
1
kkkk
kkk
[Ramanujan] Berndt IV, p. 404
2 .04227160072224093168... 16 21
1 4
1
1 42
1
2
1
21
( )
( / )
( )
( / )
k k
k k k
.0422873220524388879... B k
kk
k
k
2 2
0 !
.04238710124041160909... log
( )( )( )( )
2
4 24
1
4 3 4 1 4 1 4 31
k k k kk
J242
.04252787061758480159... 1 1
21
11
12
= 1
1
2 12 2
12 1
1k k
k
kk
k
( )
.04257821907383586345... 8
125
12
125
4
31
41
2
1
log ( )k kk
k
k H
.042704958078479727... 64
2 27 3
4
1
2 1
2
31
log
( )
( )k kk
.04271767710944303099... 1
2
1
2
1
2 2 1 41
cosh sinh( )!
k
k kk
.04282926766662436474... arctan log2
4 4
3
4 142
dx
x
.0428984325630382984...
2 2
31
102 2 3 4
( ) ( )
1 .042914821466744092886... 1
2
1
2 1 2csc
e dx
e
x
x Marsden Ex. 4.9
.04299728528618566720... 163
2
58
9
1
2 2
1 2
1
log( )
( )
k
kk
k
k
1 .04310526030709426603... dxx
x
1
06
3
1
)1(log
32
32log
2
32log53
.0431635415191573980... 1
6 12 3
3
4
1
3 2 3 1 3 3 11
log
( )( ) ( )k k k kk
GR 0.238.3, J252
5 .04316564336002865131... e
.0432018813143121202... ( )k
kLi
k kk k
1 1 14
24
2
.04321361686294489601... 1
2
2 1
2 1
4
4
Mel1 4.10.7
.043213918263772254977... )log2sin()log2cos()1(2 iiiie ii
.0432139182642977983... ( ) //
/ /( )2 1 1 4
2
1 4
11 4 3 42 1
1
2
e k
k
J114
.04321740560665400729... e k
k
2
1
.04324084828357017786... arctanh1 1
2 2
1
2 12 10e e kk
k
log tanh( )( )
1 .04331579784170500416...
0
3
)!13(
)!(
27
1,
3
4,
3
2,1,1,1
k k
kHypPFQ
1 .04338143704975653187...
3/43/43/1
2
311
2
31121
3
1 ii
=( )3 1
2
1
214
1
k
k kkk k
.04341079156485192712... 23
4
13
22
5 2
2 2 1 2 3
2
1
loglog
( ( )( )
kH
k k kk
kk
.043441426526436153273... 2
3
2
21
1
2 2 1 2 31
arcsinhk
k k k( )( )
.04344269231733307978... ( ) ( ) ( )
1 4 4 111
15
2
k
k k
k kk
k k
.0434782608695652173913 = 1
23
.04349162797695096933... ( )k
kLi
k kkk k
1
2
2 22
22
2
.043604011980378986376...
13 )12(3)12(27
1
3
2log
4
3log
k kk
[Ramanujan] Berndt Ch. 2
1 .043734674009950757246... arctan( )ee ex x 1
1
2 20
1
1 .04377882484348362176...
( )
( )
( )4
5 51
k
kk
Titchmarsh 1.2.13
.04382797038790149392... 3 2
4 8
1
12
1
4 4 1
1
42 2
log
( )
( )
k k
k
kk
k
1 .04386818141942574959... 16
15
1
4 2 1 4
2
0
arcsin( !)
( )!
k
k kk
.04390932788176551020... 23
6 2
8
33 2
0
4
sin tan/
x x dx
1 .04393676209874122833... 7 2161
1 6
1
1 63
1
3
1
31
( )
( / )
( )
( / )
k k
k k k
.04398180468826652940... ( )
( )! ( )
1
1 2
1
1
k
k k k Titchmarsh 14.32.1
1 .04405810996426632590... 1
22
1
120
csch( )k
k k
1 .04416123612010169174... ee
ke
k
k
!0
.04419417382415922028... 1
16 2
1
4
21 2
1
( )k
kk
k k
k
.044205282387762443861...
245
245
1)4()3()2(2log
3
14
xx
dx
kkk
.04427782599695232811... 2 2 2 32
34 2
2 1
16 2 11
log log( )
k
k k kkk
.04438324164397169544... 1
4 48
1
2 2
2 1
21
5 31
( )
( )
k
k k
dx
x x
9 .04441336393990804074... 22
1 1( ) ( )k kk
.044444444444444444444 = 2
45 1 41
arccosh x
x( )
.0445206260429479365... ( )
( )
3
27
1
3 31
kk
.0445243112740431161... 3
32
1
4 12 31
dx
x( )
.04456932603301417348... 9 29 3
2
5
4
1
4 1 21
loglog
( )( )
kk k k k
.04462871026284195115...
11
1 5
1
5
1
5
1
4
5log
5
1
kk
kk k
Li
.04482164619006956534... 3
8 2
1
4
1
4 2 1
21 2
1
arcsinh ( )
( )
k
kk
k
k
k
k
1 .04501440438616681983... ( )
logk
k k kkk k
1
2
11
1
21 1
.04507034144862798539... 13
.04510079681832739103... 144391
2
53
2 2 1 31
e
e k
k kk ( )!( )
1 .0451009147609597957...
0 )12(8
1
22
1arctanh22
kk k
1 .04516378011749278484... li( )2
2 .0451774444795624753... 8 27
2 22
1
log
Hkk
k
.04522874755778077232...
1
12 )12(2
1
2
3log32log1
kk kk
.04527529761904761904 =1217
26880
1
4 81
k k kk ( )( )
.04539547856776826518... ( ) ( )4 5
1 .04540769073144974624... 1
2 2
13 22
0
coth( )
/
kk
.04543145370663020628...
12 )126(2
1
)!2(
)!42(
12
5
3
2log2
kk
k kk
k
.045452882429679849... primep
pp 221log
.045454545454545454545 =1
22
.04547284088339866736... 1
7
.04549015212755020353... 7 4
8
3 3
4
1
1
1
41
( ) ( ) ( )
( )
k
k
k
k
.0455584583201640717... 17
83 2
1 31
log( )
dx
e ex x
.04569261296800628990... 2
21216
2
36
1
6
( )
( )kk
.0458716234174888797... 12log216
1
16
1
232
=log
log
x
x
dx
x 2 20
1
216 1 GR 4.282.10
.04596902017805633200...
9 18 3
3
6
3 3
18 3
7 3
6
3 3
18 3
7 3
6
log i i i i
=1
27 131 kk
.04599480480499238437... 10
3
3
2
47
36
1
3 1 31
log( )( )
kk k k k
.04603207980357005369... log ( )( )
5
4 41
42
kk
k
k
k Dingle 3.37
= ( )( )
log
14
1
41
1
42 1
kk
k k
k
k k k
.04607329257318598861... Ik kk
31
21
6
1
3( )
!( )!
1 .0460779964620999381... 1 22 2
1
2 4 21
( ) cot
k kk
=( )2 2
21
kk
k
.0461254914187515001... 8
2
2
1
4 2 4 1 41
log
( )( )k k kk
J251
= 1
4 1 1211 ( )k j
kj
J1120
1 .046250624110635744578...
112 1)2(
2
5tan
52
1
kk kF
.046413566780796640461... 5
6
4 2
3 6
3 3
4
5 3
4
14
2
logcot cot
( ) i i
k k
k
k
2 .04646050781571420028... 1
3 130
kk
. 04647598131121680679...
0
2
)32)(12(16
2
8
23
kk kk
k
k
kG
.04655015894144553735... 1
2
3
12
1
6 1 6 11
( )( )k kk
.04659629166272193867...
1
2
12 )!22(
)!()1(
4
1,
2
1,2,2
k
k
k
kF
2 .04662202447274064617...
2 122
22 2
0
2
log logsin
/
x dx GR 4.224.7
= log2
20
1
1
xdx
x GR 4.261.9
.04667385288586553755... 11
41
2
1 2
0e
k
k k
k
k
( )
!( )
.04672871759112934131...
1
21
1
183012
1
93
)1(
3
2log
18
5
kk
k
kkk
.046748650147269892437... csc ( ) csc ( )/ /1 11 4 3 4
.04682145464761832657... H
kk
k
2
51 1( )
.046842645670287489702...
22
)(
kkk
k
1 .046854285756527732494... 1
1
1
2 22k k k k kkm
km( ) log log
=
2
1)(k
kdxx
1 .04685900516324149721... ( )
( )! !
k
k k kI
kk k
2
1 2
2 12
.046943690640669461497...
21 2
1
2
1
2 1
3
20
4
arctansin
cos
/ x
xdx
.04704000268662240737... sin
/
3
3
0
3
8 .0471895621705018730... 5 5log
.04719755119659774615... 3
12 1
2 4 2 1
2 1
16 2 11 1
( )!!
( )!! ( ) ( )
k
k k
k
k kkk k
k
1 .04719755119659774615... 3
1
21
1
6 5
1
6 11
1
arccos ( )k
k k k J328
=
0 )14)(12)(1(
1
k kkk
= 2 1
16 2 10
k
k kkk
( )
= dx
x
x dx
x
x dx
x60
4
60
1 2
301 1 1
/
.04732516031123848864... 1
4 ( ) ( )i i
1 .04740958101077889502... 4
5193
30 4
31
( ) ( )a k
kk
Titchmarsh 1.2.13
.047430757278760508088...
1
2
36
)2(332
72
1
kk
kk
93648 .04747608302097371669... 10
5 .04749726737091112617... 2
.04761864123807164496...
2 4 4
413
7
909 3
1
( )log
( )
x
xdx
.047619047619047619 =1
21
.04794309684040571460... 5
43
1 1
3 25 32
31
( )( ) ( )k k k k kk k
.04821311371171887295... log ( ) log( )
2 2 11
11 2 1
2 22 2
k k
k
kk k
.04828679513998632735...
22
11 )1(2
1
)88(2
1
)44)(24(
1
8
1
4
2log
kk
kk
k kkkkk
.048298943674391117963...
2 )1(2
1)()1()1(33log2
2
2log19
42
3
kk
k
k
k
.04831135561607547882...
1 1
222
2264
)!22(
)!(
2
1
18 k k
k
kkk
k
k
.04832930767207351026...
1
5
1
21
21
2
1
4 32
i i
k kk
= ( )( )
12 1 1
41
1
kk
k
k
1 .0483657685257442981... 1
2 1 233
11 ( )( )
k kkk k
k
.04848228658358495813... 1
36
2
3
7
6
1
3 41 1
20
( ) ( ) ( )
( )
k
k k
=log x dx
x x5 21
.04852740547184035013... ( ) ( ) ( ) ( )( )
3 3 5 3 4 61
3
61
k
kk
.04855875496950706743... 24 2 3 41
1081
22
0
1loglog( ) log
x x xdx
.04871474579978266535... 3 1 1
161
2 11
4
1
cos sin( )
( )!
k
k
k
k
.04878973224511449673... 2 32035
864
2
6 51
( )log
x
x xdx
1 .04884368944513054097... ( ) ( ) tanh2 3
3
3
2
15 4 3
1
k k kk
222 .04891426023005682... 8
364 2
33 2
22
0
log ( )/x Li x dx
.04904787316741500727... ( ) ( )( )( )
2 2 3 41 2 3
1
k
k kk
= ( ) ( )
1 2 11
1
2k
k
k k
1 .04904787316741500727... ( ) ( )( )
2 2 3 39 11 4
1
2
2 32
k k
k kk
= ( ) ( )
1 11 2
3
k
k
k k
3 .04904787316741500727... ( ) ( )2 2 3 1 12
1
H
kk
k
4 .04904787316741500727... ( ) ( )( )
( )2 2 31
11 13
2
2
1
k
kk k
k k
=
1
1
)1(k
kk
kk
HH
.04908738521234051935... 64 4
2
4 20
x dx
x( )
5 .049208891519142449724... 49
16 2
4
0
e k
k kk
!
.0493061443340548457... log ( )3
2
1
2
2 1
3
1
27 32 11
31
k
k kkk k
= dx
x x4 22
.049428148719873179229...
41 1 1
1
11 4 1 4 3 4
42
( ) csc ( ) csc ( )( )/ / /
ik
k
k
2 .04945101468877368602... 1
87 6 3 3 27 3 12
1
32 2 1
log log ( )
=H
k kk
k ( / )
1 31
.049472209004795786899...
24
124
log
)!1(
1
log
1)4(1
k
n
nk k
k
nkk
k
.04952247152318661102... 263
315 4
1
2 9
1
1
( )k
k k
.04966396370971660391... sech 3
2
13 3
e e
.04971844555217912281... 9
128 2
1
4 2 1
24
1
( )
( )
k
kk
k
k
k
k
.04972865765782647927...
1
12
1
3
1
3
1)1(
34
)1(
6
7
2
3log3
kk
kk
Likk
.04974894456184419506...
1
02
4
)1(
)1log(2log2
42dx
x
x
.04976039488335674102... 2
51log3
61 2
2
=
1
21
)12()!12(
)!()1(
4
1,
2
5,
2
5,2,
2
3,1
18
1
k
k
kk
kHypPFQ
.04978706836786394298... 13e
.04979482660943125711... 1423 1
2 2
1 2
1
e
k
k
k
kk
( )
( )!
.04984487268884331340... 5 2
6
19
36
1
2 2 31
log
( )( )
kk k k k
.04987030359909703846...
52
51)(
log
1dxx
kkk
.04987831118433198057... 2 1
2112
2 4 21
1 2
log( )
( ) ( )
k
k
k
k k
.05000000000000000000 =1
201
92
4
kk
= k k
kk
( )
( )
6
3 21
J1061
1 .05007513580866397878... 2 31 2 1 4 1 2/ / / CFG F5
.05008570429831642856... ( )3
24
.05023803170814624818... 51
48 24
3
2
1
2
2
31
( )
( )k kk
.05024420891858541301... ( ) log
( ) ( )
( )2 2
3 3
4
1
1
1 2
31
k
k
k
k
2 .05030233543049796872... 12
log ( ) kk
.05037557702545246723...
22
22
2 log)1(
2
)2(2log)2(2log
12 k
k
k
k
.05040224019441599976...
6
1
21 3
1 3
41 3
1 3
4
i
ii
i
= 1
2 1 131 ( )kk
.0504298428984897677... 8 22 4
3 2 12
4
0
1
G x xdx
log log( ) log
.05051422578989857135... ( ) log
( )
3 1
4 1
2
5 31
2
2 20
x
x xdx
x dx
e ex x
.05066059182116888572... 1
2 2
1 .05069508821695116493...
12
)1()1()1(
)45(
1
5
3
10
1
100
1
5
1
25
1
k k
= 2
125
1
16
1
25
9
5
21
( )
.05072856997918023824... Ik kk
40
21
4( )
!( )!
LY 6.114
2 .05095654798327076040... 2
21
1
2 21
sinh
kk
1 .05108978836728755075... cosh( )!
/ /1
2
1
2
1 1
20
e e
k kk
AS 4.5.63
2 .05120180057137778842... 21
51
/k
k k
.051297627469135685503... G 162log4)3(21128
1 2
= 4/
0
coslog
dxxx
.05138888888888888888 = 37
7201
12
113
logx
dx
x
.05140418958900707614... 2
51192
log xdx
x x
.0514373600236584036... 2
3 2
16 16
1
82 2
0
2loglog
x e x dxx GR 4.355.1
.05145657961424741951... 244
3
9
4
1
3 1 2
1
1
log( )
( )( )
k
kk k k k
1 .05146222423826721205... 22
5 5
3
5 csc
.05153205015748987317... ( ) ( ) ( )( )
2 6 3 6 47
8
4 1
1
2
42
k k
kk
= ( ) ( )
1 1 11 3
1
k
k
k k Berndt 5.8.5
1 .0515363785357774114... H kk
k
( ) ( ( ) )3
1
1 1
.05160484660389605576...
1
)1()1(
25
)12(
5
6
5
4
20
1
kk
kk
.05165595124019414132... 504013700
17
1
1
e
k
k kk
k
( )!( )
2 .05165596774770009480... 12
cot222
csc4
22
=1
2 2
2 2
24 2 121 1k k
k k
kk
k
( )
1 .05179979026464499973... 7 3
83
1
2 1 230
2
1
( )( )
( )
k
H
kk
kk
k
=
dxxx2/
0
tanlog
.05181848773650995348... 178
225
1
5
2 1
2
1
2 51
2
( )!!
( )!!
k
k kk
J385
.05208333333333333333 =5
96
1
2 4
1
1
( )
( )( )
k
k k k k
4 .05224025292035358157...
3 2 2
032
1
x xdx
x
sin
cos
.05225229129512139667... log log ( )3
4
2
3 4 3
1
36 6 621 2
k k
k
kk
k
=dx
x x x x x x6 5 4 3 21
2 .05226141406559553132...
4
0
2
31
3
6
( )
( )
( )
k
kk
Titchmarsh 1.7.10
.05226342542957716435... ek
kk
cos sin(sin )sin
!3
1
32
.05230604277964325414...
1
1
)12()!12(
)1(
2
1sin)1(
k
k
kk
ksi
.05238520628289183021...
1
1
)12()!12(
!)1(
4
1,
2
5,
2
5,
2
3,1
18
1
k
k
kk
kHypPFQ
.05239569649125595197... 1
1
1 1
3 33 31 1e e k kk
k k
cosh( ) sinh( )
.05256980729002050897... 5 6 2
16 1 32
log
logx
x
dx
x
4 .0525776663698564658... k jk k k
k kk
jkkj
kk2 1 2 2 22
1 11
0
1( )
( )
.052631578947368421 =1
19
.05268025782891315061... 1
2
10
9
1
19
1
19 2 12 10
log( )
arctanhk
k k K148
.05274561989207116079... 8
1
6
2
4
1
4 2 4 31
log
( )( )k kk
.052878790640516702047...
14
22 )1(2
)2(2log216)3(35
4
1
kk k
kG
.05290718550287165801... 3
3
12
2
3
2 1
41
log log ( )kk
k
.05296102778655728550... 2 24
3
1
4
2 1 1
432 1
log( )
k k
k
kk
k
=
1 )124(2
1
kk k
.05296717050275408242... 17
720
1
2
4
40
( )
( )
k
k k
.05298055208215036543... 3 2
2 2132 1
Gx
xdx
log
( )
1 .053029287545514884562...
1
222
22
)78(
1
)18(
1
8csc
642216 k kk
.05305164769729844526... 1
6
6 .0530651531362397781... 2 3 22
3
2 2
2 1 31
arcsin( )!!
( )!!
k
k
k
kk
.0530853547393914281... 3 3
2
7
42
1
1 32
( ) ( )
( )
k
k k
=log log2
4 31
2 2
0
1
1
xdx
x x
x xdx
x
1 .053183503816656749...
3 1
1)1(j k
jk
1 .053252407623515317... 8 2 2 2 31
2
3
24 31 1
log ( ) ( )( )
k k
k
kk
k
.05330017034343276914...
4
37
12
31
4
37
12
31
3
2log
6
iiii
=1
2 1 131 ( )kk
2 .05339577633806633883... 2 2 22
2
2 11
log log log csc( )
k
kkk
=
2 11
2 21
logkk
.053500905006153397... 140 ( )! !k kk
.05352816631052393921... 1
100
2
5
9
10 11 1
51
( ) ( ) log
x
xdx
.05357600055545395029... ( )jkkj
5 111
.05358069953485240581... 1
31 ( )! !k kk
.05361522790552251769...
22
sin2224
csc128
2
.05362304129893233991... ( )k
kk
1
2
32
.05362963066204526106... k
kk
!!
( )!
40
2 .05363698714066007036... 1
11 Fkk
5 .0536379309867527384... dxxxx
2
1)()1(
1 .0536825183947192278...
011 )14(4!
1
4
1,
4
5,
4
1
4
1,
4
3
4
1
4
5)1(
kk kk
FEii
.05391334289872537678... 2
32
7 3
16
1
27 2 1
( )
( )
k
kk
3 .05395433027011974380 =43867
1436417 ( )
.0539932027501803781... log log
( )
3
8
1
12 2 31
x
xdx
1 .05409883108112328648... 3
41
3
2 31
logkHk
kk
1 .05418751850940907596...
2
2
11( )
( )k
k
k
.054275793650793650 =5471
100800
1
3 81
k k kk ( )( )
.054298855026038404294...
2
2
3
)1()(3log
1833 kk
kk
.054612871757973595805...
12
2
4)22(2
)2(
8
1
8
)3(7
4
log
kkkk
k
.054617753653219488... 1
11
11e e
dx
e ex x
log( )
6 .0546468656033535950... 33
2 21
12
7
6
2
31 1
log ( ) ( )
=
log log1
13
0
1
xxdx
1 .05486710061482057896... 2 2
111
2 1
2
( )!!
( )!!
k
k kk
J166
.05490692361330598804... log ( )2
2
7
24
1
2 8
1
1
k
k k
1 .05491125747421709121... ( ) ( )
( )5 6 1
61
k
kk
HW Thm. 290
1 .05501887719852386306... 51
5
1
5 1 52 20
2
1
Lik
Hk
k
k
kk
( )
( )
.0550980286498659683... 284
38
4 1 30
log( )( )
k
k kkk
.05515890003816289835... log
log
2
4
1
4 12 20
1
2
x
dx
x GR 4.282.7
.0551766951906664843... ( )
!( )!
k
k kk I
k kk k
1
1
21
1
221
2
1 .05522743812929880804... ( )
( )!/2
11 1
1
2 1
1
2k
kk e
k
k
k
.05524271728019902534... 5
64 2
1 2
4
3
21
( ) ( )!
( !)
k
kk
k k
k
.05535964546107901888... 2
31
6 3
48 2 2
( )
( )
k
kk
9 .0553851381374166266... 82
1 .05538953448087917898...
2
22
7
6
2
21 2 2 12
2
1
1
coth ( ) ( )
k
kk
k k
k
2 .05544517187371713576... 3
32 2 1
3 2
40
2
2 20
log logx
xdx
x
x xdx
=x
xxdx
2
40
1
21
1
log GR 4.61.7
.0554563517369943079... 211
4 2
1
4 1
21 2
1
( )
( )
k
kk
k
k
k
k
.05555555555555555555 =1
18
1 .05563493972360084358... 1
22 2 2 2 2 2cos cosh sin sinh
=4
4 2 10
k
k k k( )!( )
.05566823333919918611... 1
18
2
93
1
920
csch( )k
k k
.055785887828552438942...
1 1
)(
54
1
4
5log
4
1
n kk
knH
2 .05583811130112521475... Hkk
k 2 11
.0558656982586571151... 1
42k
k k( )
38 .05594559842663329504... 14e
1 .05607186782993928953...
0
3/13/1
9)!2(
1
2
1
3
1cosh
kkk
ee
2 .05616758356028304559... 5
24
1
1
2 2
0
log( )
( )
x
x xdx
.05633047284042820302... k
k kx dx
k
1
152 4
5
log( )
.05639880925494618472...
4
141
5
145log25log2log82log81 22
22 LiLi
=( )
( )
1
4 1
1
21
k
kk k
=
1
0 4
logdx
x
xx
.05650534508283039223...
e
x x
xdx4 2
01 4
tan
GR 3.749.1
1 .05661186908589277541...
2
1
2
3log
2
3
3
1
2
3log
2
1
2
3log
2
322
2 LiLi
=
1 3
)1(
kk
k
k
kH
2 .056720205991584933132... 12 2 12
22
2
22
0
1
i i i i
Li e x x dxi log ( ) log( sin )
73 .05681827550182792733... 3
4
4
.05683697542290869392...
222
22
)1(
1
4
3csch
4coth
4 k k
.05685281944005469058... 3
42
1
1 3
2 3
21 2
log( )
( )( )
( )!
( )!
k
k kk k
k
k
=1
2 2 1 2 21 k k kk ( )( )
J249
= 1)2(1
1
kk
k
k
.05696447062846142723... 38 1
2
1 2
1
e
k
k
k
k
( )
( )!
.05699273625446670926... 1
8 8 2 2
1
142
csch sech
( )k
k k
.05712283631136784893... 1
22 3
14 3
2
( ) ( )k kk
= ( ) ( ) ( , )
1 3 1 21
1 1
k
k k
k dz k
.05719697698347550546... 7 675
120
4 3
4 31
log x
x x
.05724784963607618844... G xdx
e ex x16 4 40
1 .05725087537572851457... 1
21 1
1
2 1 2 10
Ei Eik kk
( ) ( )( )!( )
GR 8.432.1
= SinhIntegralx
xdx
x
dx
x( )
sinhsinh1
1
0
1
1
5 .0572927945177198187... 7 3
2
2
31
( ) ( )
r n
nk
.0573369014109454687... ( )
log( )3
42 2
7
720 124
14 2
5 42
k
k k k
.05754027657865082526... dxx
x
e
1
4cos24
GR 3.749.2
.0576131687242798354... 14
2431
21
5
1
( )kk
k
k
1 .05764667095646375314... 1
2 33
33 3
31
Li e Li ek
i ik
k
/ / cos
.05770877464577086297... log log ( )4 2
0
12
4
1
1
x
xdx
17 .05777785336906047201...
0
2100 )!(
55;1;)52(
k
k
kFI
1 .0578799592559688775... 7 6
432
2
51
( ) ( )
H
kk
k
Berndt 9.9.5
.05796575782920622441... log ( )
( )
2
2 2
2 1
2 2 1
1
2
1
22 11 1
k
k k kkk k
arctanh
= ( ) ( )
1 11
1
k
k
k
=log log
( )
xdx
x1 21
GR 4.325.3
.05800856534682915479...
( )
( )( )k
k
k
kLi
k kk k k5
25
25
1
5 11 1 1
.05807820472492238243... 3
2
7
6
2 1
2 32
( )!!
( )!
k
k kk
.058086078279268574188... 11
21 2 2 1
1
21
1
2
arccoth i
ii
iarctan arctan
= sin
sin
/ 3
20
4
1
x
xdx
.05815226940437519841... 3 3
2 3
( )
Berndt Ch. 5
2 .0581944359406266129...
1
1
( )!k
kk
.058365351918698456359...
12
21
2 )1(2
)1(
3
2
2
3log4
2
12
kk
k
k
kLi
1 .05843511582378211213... 2 10
11
4
5
4
1
4
1
4 4 1F
kkk
, , ,( )
.05856382356485817586...
( )2 1
2 11
kk
k
.05861382625420994135...
e
x x
xdx4 2
01 4
cot
GR 3.749.2
4 .05871212641676821819... 4
24 , volume of unit 4-sphere
.0588235294117647 = dxe
xx
k
kk
00
)1(42 4cos2)1(
4
1arctansin
17
1
.05882694094725862925...
142
log)(
)4(
)4(
k k
kk
.05886052973059857964... 1
18
1
9
2
3
1
9 2
1
21
21 2 1
1
log arctan ( )k kk
k
kH
.05889151782819172727... 1
2
9
8
1
17
1
17 2 12 10
log( )
arctanhk
k k
1 .058953464852310349... H k
kk
k
sin
1
.05897703250591215633... log( )
log3
4 4 4
1
41
1
42 1
k
k k kkk k
3 .05898844426198225439... 3 5 46
3 6 21
2 1
1 1 122
12 3 4
( ) ( ) log
( )
H k
k k k kk
k
.05918955184357786985... 7
11520
1
2
4 1
41
4
( )
( )Re ( )
k
k kLi i
.05936574836539082147...
1 )32)(12)(12(
1
3
1
8 k kkk
J240, J627
=
12 31616
1
k kk
1 .05940740534357614454... ee
erfkk
22
1
2
1
2
!!
3 .05940740534357614454...
10 !
!!
!!
1
2
1
2 kk k
k
kerf
ee
4 .05940740534357614454... ee
erfk
kk
12
1
2 0
!!
.059568942236683030025...
1 12
11
1
2
( )( )k
kk
k
1 .059638450439128955916... 3
4
9 3
44 2
1
1 2 1 3 10
loglog
( )( )( )k k kk
.059705906160195358363... log
( ) ( )( )
log2
43
3
43
13
1
k
k kk
.0597222222222222222222 = 43
720
1
1 42
k k kk ( )( )
.05973244312050459885...
2
0
2
161 2 2 2( log log ) log
/
x xdx
.05977030443557374251... 10 27 4
88 2
3 3
2
1
1
1
2 41
( )
( )log
( ) ( )
( )
k
k k k
1 .05997431887... j4 J311
.06000000000000000000 = 50
3
4 .06015693855740995108... e erfk
k kk
12
2 11
( )!!
( )!! !
=e dx
x
x
10
1
5 .06015693855740995108...
1 )!2(
4!1,0,
2
1
211
k
k
k
keerfe
.06017283993600197841... 5
18
1
2 2
1
2
1
2 2 1
1 2
0
arctan( )
( )
k
kk
k
k
.06017573696145124717... 2123
35280
1
3 71
k k kk ( )( )
.06034631805191526430... 5 1 6 1
161
2
4
0
sin cos( )
( )!
k
k
k
k
.06041521303050999709... 2 62
33
3
21
2 1 22 1
1
log log ( )( )( )
k kk
k
H
k k
1 .06042024132807165438...
2 22 1
1
1
1log
1)(
c ncc
c k nnk
ck
.06053729682182486115... 112 9 24 2
1081
22 2
0
1
loglog( ) logx x xdx
.06055913414121058628... 3
512
.060574229486305732161... )3(2
315
)(
log)(4
1
k kk
kk
5 .06058989694951351915... k k
kk
0
1 2 1
( )
1 .06066017177982128660... 3
2 21
2 1
2 91
( )!!
( )!!
k
k kk
CFG B4, J166
= 11
2 50
( )k
k k
.060930216216328763956... 23
2
3
4
1
3 1 21
log( )( )
kk k k k
=
1
1
)12(2
)1(
kk
k
k
.06097350783144797233... ( )( )
arctan
12 1 1
2 1
1 11
1 2
k
k k
k
k k k
.061208719054813641942... sin( )
( )!2
1
2 11
1
4
1
2 2
k
kk k
.061385355812633918024... 5
27
4
81 3
4 2
81
1
3 1 3 42 20
log ( )
( ) ( )
k
k k k
.06147271494065079891... 2
3sech
31
3
2log4
=( )
14
2
k
k k k
.06154053831284500882... 1
3
9 6 2 24 2 40
36 24 2 1
2
20
4
sin
( sin )
/ x
xdx
.06158863958792450009... 1
4
304
105
1
2 90
( )k
k k
1 .06160803906011000253...
122
)1()1(
)12(2
11
2
11
16
2
k k
k
=
112
)12(
kk
kk
.06160974642842427642...
3
4 4 22 23 4
1 4 1 4 /
/ /csc csch
= ( )
1
242
k
k k
.06163735046020626998... 9 81
24
1
2
1
2 4 1
1
1
cos sin( )
( )! ( )
k
kk k k
.061728395061728395 =5
811
51
4
1
( )kk
k
k
.06173140432470719352... ( )
( !)[{},{ , , }, ]
11 1 1 1
40
k
k kHypPFQ
.06177135864462191091... 35 40 2 6 31
1
1
4 41
log ( )( )
( )
k
k k k
.06196773353931867016... 2
25
3
51
2
61
1
2
( )k
kk
k
k
k
1 .06202877524308307692... arccsch4
.06210770748126123903... 216 1
3 2
4 21
2 2
20
1
log logx
x xdx
x x
xdx
8 .0622577482985496524... 65
.062346338241142771091... 1
16 4 22 2
1
8
1
21
csch
( )k
k k
.06238347847451615361... ( ) ( )k kk
1 2 12
.062400643626808666895...
2
44
12
log)()4()8(210
9450
k k
kk
.062500000000000000000 = 1
16
1
11 2 1 15 2 2
2 1
k k
k kk k( )
( ) ( )
=
1
3
)1(kkke
k
=xdx
x( )2 30 2
= log 11
4 91
x
dx
x
= dxe
xx
02 1
2cos
Prud. 2.5.38.9
1 .06250000000013113727... 1
1 k kk
k
.06251525878906250000... 1
221
2k
k
.062515258789062500054... 1
241
k
k
1 .06269354038321393057...
11
222
22
2log
12 kkkH
.06269871149984998792... ( ) ( ) ( )611
90
28
384 14 3 4 5
4 2
=1
16 41 k kk ( )
1 .062714148932708557698... dxxx2
1
log)(
.062783030996353104594... 1
0
12 )1log( dxx
4 .0628229009806368496... 3
22
3 2
4
2 2
2 2 22 1
1
log logkH k
kk
.06290376269772511922... 5635
6 185 3 5
1
1
2 4
5 41
( ) ( )( )k kk
.06310625275909953582... log
log( ) ( )2 1
1
2
22
1 1
1
k
k
k
k
.0632539690102041...
1
1
)1(
log)1(
k
k
kk
k
.063277280080096903577... 32 4
19
72 0
1
G
x xdxx arccot3 log
.06332780438680511248... 4 2
4 4145
10
335
1
1
k kk ( )
1 .06337350032394316468... 8 41
28
1
2
1
2 4 10
sinh cosh( )! ( )k kk
k
.0634173769752620034...
4
34
01536
1
1536
1
2
1
4 2
( )
( )kk
.0634363430819095293... 11
22
3
1
2 3 41 0
ek
k k k k kk k( )! !( )( )( )
1 .06345983311722793077... 8
3 32 3
12
2 1 20
Gk
kk
k
log
( )
Berndt Ch. 9
.06348038092346547368... 1
2 31
k kk
( )
1 .06348337074132351926... Ik k
k0 2
0
1
2
1
16
( !)
= ek
kk
( )!
( !)
1
22
0
.06364760900080611621... arctan( )
( )
1
2
2
5
1
4 2 1
1
0
k
kk
k
k
.06366197723675813431... 1
5
.0636697649553711265... 1
4
4
4 141
4( )
log ( )
( )
logk
k
p
pk p prime
=
14
)(
k k
k
.063827327695777400598... 2 212
1
2
12
3 22
log( )
k
k k k
1 .06387242824454660016... 21
2
4
62
( )
( )
3 .06387540935871740999... ( )1 2
3
12 .06387540935871740999...
3
1)1(
1 .06388710376241701175... 12
1 k kk
.0639129291353270933... 4
3164
7 3
42
2 1
( )log
( )
H
kk
k
.06400183190906028773...
1
01
2
)()1(
kk
k k
1 .06404394196508864918... ( ) ( )
( )
3 5
42
.06407528461049067828... 5
.06413507387794809562... 9
.06415002990995841828... 1
9 3
1 2
2
3
21
( ) ( )!
( !)
k
kk
k k
k
.064205801387968452356... 5125
log28
125
2
5125
2arccsch28
125
2
=
1
3
2)1(
k
k
k
kk
.06422229148740887296... 205
34 2 3
1
1
2
3 41
( ) ( )( )k kk
.064274370716884653601... 1)1()2)(1(
)1(
2
13log22log7
1
21
kkk
k
k
k
.06438239351998176981... log ( )2
3
1
6
1
3 90
k
k k
= log 11
6 71
x
dx
x
.06446776880150635838... 1
486
2
3
1
3 23
40
( )
( )
kk
.064483605572602798069...
2 2
)()1(
kk
k k
.06451797439929041936... 1
1
1
11
2 12 2
12
1e
kkk
k
( )
( )
.064710682787920926699...
08
22
1
)1(1224
4dx
x
xx
.06471696327987555495... 4
2 2 1 32
0
1
eEi
x x
edxx ( )
log
1 .06473417104350337039... 1
64
1
8
3
8
5
8
7
81 1 1 1 ( ) ( ) ( ) ( )
=1
32
1
8
3
8 81 1
2
( ) ( )
=
02
2/
12
1
2
1
)12(
)1(
)14(
)1(
)34(
)1(
k
k
k
kk
kkk
.06487901723815465273...
1
02
3
)1(
)1log(
2
2log3
332
1dx
x
x
.06498017172668905180... ( ) ( )4 6
90 945
14 6 2
61
k
kk
.06505432440650839036... Ei
e
k
kk
k
( )( )
( )
( )!
1 11 1
1
11
.06505816136788066186... log
( )2
41
4k
kk
.06506062829357551254... 11
12
3
18
3
2
1
9 31
1
322 2
log( )
( )
k k
k
k
kk
k
.06514255850624644549... log
loglog ( )2
2 3 3
2
4
3
5
2
2
2
3
1
2
2
3
1
2
1
3
3
16
Li Li Li
=log ( )
( )
2
0
1 1
4
x
x xdx
.06519779945532069058... 52 1 2
2
2 21
k
k kk ( ) ( )
1 .06519779945532069058... 63
2
2 1
2
3 22
( )!!
( )!! ( )
k
k k kk
.06528371538988535272... G
k
x
xdx
k
k2 8
1
2 1 121
2 21
( )
( )
log
( )
.06529212022186135839... 1
2
1
21
212
1
log sec ( )sin ( / )
k
k
k
k J523
.06530659712633423604... 10
61
2 21e
k
k
k
kk
( )
( )!
.06537259256... ( ) log
1 2
2
k
k
k
k
.06544513657702433167...
1
12
5
)1(
2
2log
3
2log8
144
241
k
kk
k
H
.065487862384390902356...
02
)1()1(
)(
)1(
2
1
24
1
k
k
k
.06549676183045346904...
3/1
6/13/13/1
6/113132
2
33233
3
1233
32
1 iii
=
23
1
1
13
1
3
1)3()1(
kkk
k
k
k
.06557201756856993666... 819 3
2
81 3
23 3
4
32 1
log( ) ( )
= 1
3 13 21 k kk ( )
.06567965740448090483... 2 2
3
19
36
1
2 3
1
1
log ( )
( )( )
k
k k k k
.06573748689154031248... 7 3
128
( )
.065774170245478381098... log( )( )csc ( )
1 1 1 2 12 2
1
k
kkk
.06579736267392905746... 2
21150
2
25
1
5
( )
( )kk
.0658223444747044061...
1
2
)3)(2)(1(24
5
36 k
k
kkkk
H
.06598267284983230587... e
k
k
k
2
016
19
48
2
4
( )!
.06598587683871048216...
12 21216
1
2
1
84
2log
k kk
=
13456 xxxx
dx
.06598803584531253708... e e
.06604212644480172199... 31
31
2 1
2 3 2 11
arcsin( )!!
( )!! ( )
k
k kkk
.0660913464480888534... 2
2 2120 5 4 5 5
1
2
1
5 1csc cot
( )
kk
J840
.0661733074232325082... 2 3 4 4 2 101
12 41
( ) ( ) ( )( )
k kk
.06620909207331664583... csc( )cos ( ) sin
1
2 21 1 3 1
1
122 2
2 21
kk
=
12
2 1)2(
2
1cot
)1)(1(2
3
kk
k
.06633268955336798335... 61
256
3
4
1
43 3
3
4 21
( ) ( ) log x
x x
.06637623973474297119... log9
10
.06640625023283064365... 1
22 p
p prime
.06642150902189314371... 1
4
2
16 1
4 2
4 121 1
4 21
k
kk
kk
k
k k
( ) ( )
.06649658092772603273... 2
3
3
4
1 2 1
2 22
( ) ( )!!
( )!
k
kk
k
k
.06660601672682232541... 1
1 eek
k
.06662631248095397825... 1 22
2
1
1
2 2
20
1
loglog log ( )
( )
x
xdx
2 .066646848615237039218...
12
11
12
5116
1728
1 )2()2(3
.066666666666666666666 = 1
15
= k k
k kk
( )
( )( )
6
2 41
J1061
1 .066666666666666666666 = 16
153
1
2
1
4 3
21
1
202
2
,( )k
k kk
k
kk
.06676569631226131157... 1
2
8
7
1
15log arctanh
=1
15 2 12 10
kk k
( ) K148
.06678093906442190474... ( ) log3
18 1
2
41
2
30
x
x xdx
x dx
e x
1 .06683721243768533963... k k
kk
1
1 3
( )
1 .06687278808178024267... 1
21 k k
k
.06687615866565699453... 2 2 3 4 3 4 2 72 1
16 11
log log( )
k
k k kkk
3 .06696274638750199199... 2 2
3
2 2
34
1
23 3
2 3
3
log log( )
Li
=log
( )( )
2
120
1
1
x
x xdx
2 .0670851120199880117... 3
15
.067091633096215553429... csc cot
( )
( )2 2
2 2 21
1
4
1
4 1
2 1
k kk
k
1 .06711089410508528045... 12
ii eEieEii
=
1 !
sin,0,0
21
k
ii
kk
kee
i
.067519906888675983757...
4
51arctan2coth)51cot(
5
2arcarc
=
1
11
)12(4
)1(
kk
kkk
k
FF
12 .06753397645603743945... 48
2 2 113
20
1
2Gx
xdx
log log log
.067592592592592952592 =169
2700
1
3 61
k k kk ( )( )
.06759686701139501638... 1
2 10 5
1
25 1
2
2521 1
cot
( )
k
k
kk
k
.0676183320811419985... 90
13311
101
2
1
( )kk
k
k
10 .06766199577776584195... cosh ( )( )!
31
2
9
23 3
0
e ek
k
k
AS 4.5.63
.067678158936930728162...
)6432(log332log123log92,
6
5611
72
1 2 H
=
1 )16)(16(k
k
kk
H
.06770937508392288924... ( ) ( )
( )
1 2 1
16
16
16 1
1
12 2
1
k
kk k
k k k
k
1 .06771840194669357587...
( ) ( )( )
( )2
11 1
1
1
2
1 1
k
kk
k
k kk k
.06775373784985458697...
e
e
x
xdx
2 2
2
12 20
1
cos( log )
.06788026407514834638... 2 1 2
0e
t
tdt
cos
AS 4.3.146
.06797300991731304833...
1
62 1
1
1 22
( )
( )
( )( )
k
k kk
Adamchik-Srivastava 2.22
.06804138174397716939... 1
6 6
1
8
21 2
0
( )k
kk
k k
k
.06808536722795788835... 1
4
1
1
2
2 2 21
log
e
e
dx
e ex x
.06814718055994530941... log( )
( )( )2
5
8
1
1 3
1
14 3
2
k
k
k
H
k k
dx
x x
=dx
x x( )
1 31
1 .06821393681276108077... ek e
eke
e
1
0 !
2 .06822709436512760885... 122
1)(
kk
Hk
.06826001937964526234...
2
4 21 26 2
11 2 1
coth ( ) ( )k k
kk
k
k
.06830988618379067154... 4
4 12 2 20
1
dx
x x( log )( ) GR 4.282.3
.06834108969889116766... 7 3 2
72 41
2
log log x
xdx
.06837797619047619048... 919
13440
1
2 81
k k kk ( )( )
.06838993007841232002... ( )k
kk 22
1 .06843424428255624376... cosh( )!
/ /1
2
1
2
1 1
20e
e e
k e
e e
kk
1 .06849502503075047591...
2
31
2
31
9
)3(833
iLi
iLi
.06853308726322481314... log loglog( )3
22
1
2
1
4
1
52 20
1
Li Li
x
x
.0685333821022915429...
2 12
2
2 12
2231
32
3sin
k k kk
kk
k
2 .0687365065558226299... 2
2 1 1 1 1 1 1 2 2 2 2 2 2 25
1
k
k k kHypPFQ
!{ , , , , , },{ , , , , , },
.06891126589612537985...
1
4
log)4(
k k
k
1 .068959332115595113425...
05 15
4csc
555
2
5
2
x
dx
.0689612184801364854... 8
7291
1
3 1
1
3 1
4
4 41
( ) ( )k kk
1 .0690449676496975387... 22
71
2 1
2 81
( )!!
( )!!
k
k kk
1 .06918195449309724743... 3
29
.0691955458920286028... 2 8
2
4 0
log log sinx x
edxx
.069210168362720747484...
log( ) ( )
log( )
( )
2
83
7
83
2 1
2 1 30
k
kk
Prud. 5.5.1.5
.0693063727312726561... 12 2 3 14
542
0
1logarcsin log
x x x dx
.06934213137376400583... 1
512
7
8
3
8 12 2
2
41
( ) ( ) log
x
x
.06935686230685420356...
3/1
6/13/16/13/1
6/113232
2
33233
2
33233
32
1 ii
ii
=( )3 1
3
1
3 113
2
k
kkk k
.069397608859770623540... 1},2,2,2,2{},1,1,1,1{!
1
13
HypPFQkkk
.06942004590872447261...
32 2 2
2
2 30
x dx
x( )
.0694398829909612926...
3 9 36
184
3
1
3 1
21
2 21
log( )
( )
k kk
.06946694693495230181...
1 25
)2()525(5
2
1
550 kk
kk
.06955957164158097295... 4
)3(
3
2log4
4
1
2
1
4
12log3log2log
3
322
LiLi
= log2
1
2
1
x
xdx
.069628025046703042819... 1
0
23
arctanlog4
7
3282dxxxx
G
.06973319205204841124... 2 3
27
1
3 12 21
dx
x x( )
1 .06973319205204841124... 2 3
27
2
3 21
22 2
3
2
1
402 1
kk
k
Fk
, , ,
3 .06998012383946546544... 3
.07009307638669401196... 5
12
2
2
1
2 81
1
201
2
log ( )( )
( )k
k
kk
kk
k
=
1 122 )63(2
1
12208
1
k kk kkkk
= log( )
11
1 31
x
dx
x
.07015057996275895686... 7
9720
4 3
41
log x
x xdx
.07023557414777806495... ( )
( )!cosh
k
k k kk k2
1 1
21
2 1
12 .070346316389634502865... e
e xdxx
1
2 0
sin
.0704887501485211468...
23
1
11log
3
1
3
1)3(
2
3coshlog3log
3
1
kk kk
k
.070541352850566718528...
4/
0
2)3()3(2 tanlog4
1
4
1192
3072
1 dxxxG
1 .07061055633093042768... 41
4
1
4 13
42 20
2
1
Lik
Hk
k
k
kk
( )
( )
1 .07065009697711308205... 1e
e
e
e
.0706857962810410866... 4 2 3 41
5 42
( ) ( ) ( )k kk
= 1
1 41 k kk ( )
= 1
0
3log)1log(6
1dxxx
.070698031526694030437...
3
2
2
9
2
14)2log3(log33log
2
3log
2
91
4
322
2
LiLi
=
1
1
)2)(1(2
)1(
kk
kk
kkk
H
179 .0707322923047554885... 2 2
2 2 5
0
23 2
18 2 1
loglog log
( )( )
xdx
x x GR 4.264.3
.07073553026306459368... 2
9
.0707963267948966192...
3
2 21 1
33 3
1
ie e
k
ki i
k
log logsin
1 .0707963267948966192...
1
2 1
sin k
kk
GR 1.441.1, GR 1.448.1
= arctansin
cos
sin1
1 1
2
21
k
kk
2 .0707963267948966192...
1
2
22 20
k
k k
k
.07084242088030439606... 1
1000
4
5
3
102 2
2
4 11
( ) ( ) log
x
x x
.07090116891887671352... 3
8
3
4
2
3
1
2 122
log( )
( )
k
kk k
.07101537321990997509... arctan log log sin2
5
5
20
2
5
2
0
x x
edxx
.07103320989006310636... ( ) ( )
1 2 112
kk
k
F k
3 .0710678118654752440... 5 2 48
2 12
0
k k
kkk
7 .0710678118654752440... 50 5 2
.07130217810980315986... 120326 1
60
e k k
k
k
( )
!( )
.071308742298512508874... ( )
log log log ( ) log3
48 2 2 2
66 12
22
0
1
x xdx
1 .07133434403336294393... iI e I e
k
ki i
k22 20 0 2
1
sin
( !)
.07134363852556480380...
1
21
4)1(
841
60
kkkk F
k
.0713495408493620774... 16
1
8
1
4 3 4 1 4 11
( )( )( )k k kk
J239
= log
log
x
x
dx
x 2 20
1
24 1 GR 4.282.8
.0714285714285714825 = 1
14
.0717737886118051393... 3 2
3 30432
x
e edxx x
.071791629712009706036...
2
33
1
3
11
3
11
2
1
2
1
k kk
.07179676972449082589... 7 4 31
16 1
2
1
kk k
k
k( )
.07189823963991674726... 1
4096
3
8
7
8 13 3
3
41
( ) ( ) log
x
x
.07190275490382753558... 10 3
82 2
0
4
sin tan/
x xdx
.07192051811294523186...
1 4
1
4
1
3
4log
4
1
kkLi
.071995909163798022179...
24 1
log
k k
k
.072000000000000000000 =9
251
91
2
1
( )kk
k
k
.07202849079249295866... ( )kk
16
2
.07215195811269160758... 8
.07246703342411321824... 2
20
4 22
9
12
1
3
1
( )
( )
( )k
k
k
kk k k
= ( )cos
1 12
21
k
k
k
k
=log
( )
x
x xdx3
1 1
=x
e edxx x3 2
0
GR 3.411.11
1 .07246703342411321824... 2 2
0
13
12
1 1
( ) log( )x x
xdx
.072700105960963691568...
12
1 22 8189
1
19
1
9
1)2(
368
3
kk kk kkk
k
.07270478199838776757... 1 21
2
1
4 2 1
1
1
arctan( )
( )
k
kk k
.07277708725845669284... 1
1000
3
10
4
53 3
3
4 11
( ) ( ) log
x
x x
.072815845483677... lim loglog
n k
n
nk
k
1
22
1
Berndt 8.17.2
.07284924190995026164...
11
1 1
1
12 2
2( ) k kk
=1 1
21 1
2
( )kk
k
.072900620536574583049...
1
0
2 3
log
3
11 dx
x
xxLi
.07296266134931404078...
1
11
)14(4
)1(2
4
3
4
5
21
kk
k
kk
k
1 .07318200714936437505...
0
2
64
12
4
12
kkk
kK
.07320598580844476003...
iiii
2
1
2
1)1()1(
4
1
=cos2
10
x
edxx
.0732233047033631189...
02 1
cos
8
22dx
e
xx
Prud. 2.5.38.9
.07325515198082261749... 1
20736
1
4
3
43 3
3
4 21
( ) ( ) log
x
x x
.073262555554936721175...
1
1
4 !
4)1(4
k
kk
k
k
e
.07329070292368558994...
kF
kk
k
kkk
k
2
1,3,2,1
8
1
2
1)()1( 11
22
2
=
16
9log1
2
1
1 .07330357886990841153... 120 2 2670 2 8520 2 8000 26 7 8 9I I I I( ) ( ) ( ) ( ) 3025 2 511 2 38 2 210 11 12 13I I I I( ) ( ) ( ) ( )
=k
k kk
4
21 5( !) ( )
755 .07330694419801687855... 7
4
.07344910388202908912...
2
2cosh
.07345979246907078119...
1
22
22
)512(
1
)712(
1
31218 k kk
J346
1 .07351706431139209227...
1
2
)!2(
)2()!(
k k
kk
.07355072789142418039...
3 3
2
3
3
10
1
3 80
log ( )k
k k
.073553956728532...
12
1
)12(
)1(
k
kk
k
H
.07363107781851077903... 3
128
1
4 2 1 2 3 2 5
2
0
kk k k k
k
k( )( )( )
.07366791204642548599... ci( )
.07387736114946630558... 5
2
4 1
2 2
1
1
e k
k
kk
( )
( )!
.07388002973920462501... 35
48 2 CFG B5
.07394180231599241053... 8
3
11
3 24
0
4
sin tan/
x xdx
.07399980675447242740...
2
2
2
21
2
1sinh
k
kk
1 .07399980675447242740... 12
21 1
sinh
( ) ( )( ) ( )i i
.07400348967151741917... 3 62
33
2
3 3 1 22
1
log log( )( )
H
k kk
kk
.07402868138609661224... log log log
(log )( )2 3
2 3
2 3
2
2 2
32
1
2
1
2
5 3
8
Li Li
= log ( )
( )( )
2
0
1 1
1 2
x
x x x
.074074074074074074074 = 2
271
21
22 2
2
0
3
02 3
( ) ( ) ( ) ( )kk
k
kk
k
k kLi Li
= dxx
xdx
x
x
1
4
3
14
2 loglog
1 .0741508456720383647...
2
51
5
10
2( )
( )
( )
k
k k Titchmarsh 1.2.8
1 .07418410593764418873...
1
3
45/11
5/11log
5
1
5
5log2
kk
kk
k
FF
.07418593219600418181... ( )( )
11
1 13
23
2
k
k kkk Li
k k
2 .07422504479637891391...
155/45/35/25/1
11
)1()1()1()1(
1
k k
5 .074304749728368200293...
1 )(dx
x
x
.07438118377140325192...
11 )23)(13(53
1
4
5log
3
1
xx
dx
kkk
.07442417922462205220... 1
2
14
7
7
2
1
2 720
csch ( )k
k k
1 .07442638721608040188... 2log)2(4
)3(7
3
2log3
=
1
)3(
133 22
12
2
12
kk
k
kk
H
kLi
=log2
21
2
02 2 1
x
x xdx
x
edxx
1 .07446819773248816683...
e ek e k
k
2 22
0
11
1log
( )
.07454093588033922743...
1
22
1
1
)1(
2
1csch
2
1
k
k
k
.07473988678897301721...
1
31
1)1()1(2
)1(
2
)3(7
9
4log
27
184
kk
k
kk
k
1 .0747946000082483594... Li2
4
5
1 .07483307215669442120...
2
12
04
0
1
16 2
1
16
1
4
1
4 1 1
G
k
x dx
xk
( )
( )
log
3 .07484406769435131887...
1 2
)1(
327
162
k
k
k
kk
.074850088435720511891... ( ) ( ( ) )( )
1 4 11
11
14 4 2
2
k
k k
k kk k
.074859404114557591023...
2
33
2
33
3
1
6
1
3
2 ii
=
24
3/23/1 1)1()1(
3
1
6
1
3
2
k kk
=
2
15
12
k kk
= ( ) ( )
1 3 1 11
1
k
k
k
.074877594399054501993... 35 3
72
1
41
2
21
3
0
1
k kx x xdx
k ( )log( ) log
.07500000000000000000 = 3
40
1 .07504760349992023872...
2
12/1K
e
.075075075075075075075 =5
66 5 B
2 .07510976937815191446... 1
40 ( !!)kk
.07512855644747464284... ( ) log( ) log3
16
1 4
0
1
x x
xdx
1 .07521916938666853671...
1
52
2 )()5(k
kkd Titchmarsh 1.2.2
.07522047803571148309... 1
4
1
2
1
2
1
2
1
2 1 2
1 2
1
cos cos( )
( )!
k
kk
k
k
1 .07548231525329211758... 5 3 3 2 ( ) ( )
.0757451952714038687... ( ) ( )( )
( ) ( ) ( )( )
2 34
42 3 2 5
1 41
H
k kk
k
.07580375925340625411... 14 3
22
1
27 3
1 3
31
log ( )k
k
k
k k Berndt 2.5.4
=
1
2
32
9
2
32
1
k k kk
.07581633246407917795... 1
8
1
2
1 3
1e
k
k
k
kk
( )
!
.07591836734693877551... 93
1225
1
2 71
k k kk ( )( )
.0760156614280973649... 2031
2
1
4 21 4
1
1
ek k
k
kk
/ ( )
! ( )
.07606159707840983298... 190 4 1
44
( )
( )
.07615901382553683827... e
erfik
k
k k
k
1 11 4
2
1
1
( ) !
( )!
1 .07615901382553683827... e
erfi
kk
k
dx
e x
k k
kx1
1 42 1
1
1 0
1
( )
!
.07621074481849448468... ( )
log2 1 1
2 1
1
2
1
1
1
1 2
k
k
k
k kk k
= 12
2
1
2
log
arccoth kkk
=
0
sinlog2 dxxxex x
.07646332214806367238... 1
41
21
2
3
2
3
2
i i i i
2
3
)1(2log1
k
k
kk
.07648051389327864275... log ( )( )
237
607
1
70
bk
k
k
GR 8.375
=
12 30274
1
k kk
= dx
x x8 71
1 .07667404746858117413... 12
1
2
1coth
2 2
e
J983
= 1
1
1
21 2 1 12
1
1
1kk i
k
k
k
( ) ( ) Im ( )
= sin xdx
ex
10
= dxx
x
1
0
1
1
logsin
2 .07667404746858117413...
2
1
2
1
2 12coth
e
= 1
1
3
21 2 12
0
1
1kk i
k
k
k
( ) ( ) Im ( )
.07668525568456209094... 1 5
25 2/
.07671320486001367265...
12
12 )44(2
1
816
1
4
2log1
kk
k kkkk
=
2/
02
3
sin1
)log(sinsin
dxx
xx GR 4.386.2
.07681755829903978052... 56
7291
81
2
1
( )kk
k
k
.076923076923076923 =1
13
.07697262446079313824... G G 2
.076993139764246844943... pk
kk
p prime k
4
1
4
( )
log ( )
.077022725309366050752...
2coth53351cot5272
4
51arctan5719
)53(20
1arcarc
=
1
11
)12)(12(4
)1(
kk
kkk
kk
FF
.0770569031595942854... ( ) ( , )( )
( , , )39
83 3
1
31 3 0
30
kk
=1
2 1
2 2
0
1 x x
xdx
log
GR 4.261.12
=
1
0
1
0
1
0
222
1dzdydx
xyz
zyx
.07707533991362915215... 1
2
7
6
1
13
1
13 2 12 10
log( )
arctanhk
k k
.07710628438351061421... 2 2
40128
2 1
4 2
( )
( )
k
kk
.077200000000000000000 =139
1800
1
3 51
k k kk ( )( )
.07720043958926015531... dxx
x
14
3)3(
4
1
log
4
1
256
1
16
.07721566490153286061...
1
2
1
12 20
1
log ( )x
x
xdx GR 4.283.4
= 21 12 2
0
xdx
x e x( )( )
Andrews p. 87
.0772540493193693949... 1 10
1
( )x dx
.07730880240793844225... 4
12604 T( )
.077403064078615306111... 9
162
3
2 12 1 1
2
2
log( )
( )k
kk
k
Adamchik-Srivastava 2.24
.07743798717441510249... 11
2
1
4 2 1
1
1
erfk k
k
kk
( )
! ( )
1 .07746324137541851817... 1
31 k kk !!
.07752071017393104727... log( cos ) ( )cos
2 1 121
1
k
k
k
k GR 1.441.4
.07755858036092111976... 53
2log
54
1
5
21
104
5log
=
12
2 525
1
5
)(
kkk kk
k
5 .07770625192980658253... 9 1 2 /
.07773398731743422062...
1
1
4)12(
)1(2
4
5log
2
1arctan4
kk
k
kk
3 .07775629309491924963... 0
2
1
0
112 2
( ) ( )k mnk
kmn
mn
.07777777777777777777 = 90
7
.07784610394702787502...
cos( )
!e
e
k
k k
k
1 1
0
.07792440345582405854... 2 2 1 1 11 2 11
cos sin ( )( )( )!
k
k
k
k k
1 .07792813674185519486... 10514
4
p
p prime
Berndt 5.28
=
( )
( )
( )4
8 41
k
kk
Titchmarsh 1.2.7
= qq squarefree
4
.07795867267256012493... 8
92
2
3
1
2 1
1 2
0
log( )
( )
k
kk
k
k
.07796606266976290757... 50 43
128
1
7
1
2 7
2 1
21
2G
k
k
kk
( )!!
( )!! J385
2 .0780869212350275376... 1
log
1 .07818872957581848276... ( )
!/k
ke
kk
k
k
2
1
1
11
.07820534411412707043...
2
3
13 20e
x
xdx
cos
.0782055828604531093... 1
12 2 2
12 22 2
1
2 2 21
csch sech
sinh( )
( )
k
k k k
1 .07844201083203592720...
2
1
2
1
21
21
iiii
.07844984105855446265...
4 3
3
8
1
3 2 1
1
1
( )
( )
k
kk
k
k
.07847757966713683832...
sin sinh
cosh cos
2 2
2 2 2 21
= 1
11 4 14
2
1
1kk
k
k
k
( ) ( )
.0785670194003527336860 =17819
226800
1
1 101
k k kk ( )( )
.078679152573139970497... 2
32
26 1
2 1 1
21
log ( )( )
k
kk
=
kk
kkk
32
2
11 1
2log
.07882641859684234467... ci x xdx( ) log cos log log sin1 2 1 2 20
1
.07889835002124918101... 121
kk
k
1 .07891552135194265398...
1 2
1)12()1(6
22log
3
2
k k
k
.07896210586005002361...
12
22
)2(2
1272log244
12
1
kk k
.07907564394558249581... 3 3
4 12
1
1
2 1
31
( ) ( )
( )
k
k
k
k
.079109873067335629765...
primep p 41
1log)4(log HW Sec. 17.7
. 07912958763598119159...
8
3)1(
512
3
8
1)1(
512
3)1(1log
64
3
8cot
51232
3 )1(4/3)1(4/14/323
G
2
)3(3)1(
2
3
8
7)1(
512
3
8
5)1(
512
3 4/33
)1(4/3)1(4/1
Li
8/
02
3
sin
dxx
x
.07921357989350166466... log ( )( )
4
3 31
32
kk
k
k
k Dingle 3.37
=
1 3
13log
3
1
k k
k
k
.07921666666666666666 = 7
961
12
19
logx
dx
x
.079221397565207165999... 2
)3(3
24
)6(53)2,1,3(
2MHS
.07924038639496914327... 1
2
1
22
1
2
1
2 1 21
cos sin( )
( )!
k
kk
k
k
.079332033399540433998...
1
1)3( dxx
.07944154167983592825... 3 2 21
1 21
log( )
( )( )
k
k
k
k k
=1
16
2 1
1631 1k k
k
kk
k
( )
=
1
1
)12(22
)1(
k k
k
kkk
k
.07957747154594766788... dxe
xx
02 1
)2/cos(
4
1
.07970265229714766528...
1
125
3
52 ( )
.07998284307169517701... 32 3
3 3
2
1
3
4
3
1
3 11
21
log
( )( )
k kk
= log( ) log1 3
0
1
x xdx
.080000000000000000000 = 2
25 10 11
( )kk
k
.080039732245114496725... dxxx
x
145
2)2( log
)4(108
251)3(2
= x x
xdx
3
0
1
1
log
2 .08008382305190411453... 3 92 3 3/
1 .08012344973464337183... 7
61
2 1
2 71
( )!!
( )!!
k
k kk
.08017209138601023641... 4 3
411215
log x
x xdx
.0803395836283518706... log
( )( )
3
2 6 3
1
6
1
3 2 3 31
k kk
= ( )
( )
k
k k
dx
x x xkk k
1
3
1
3 3 12 26 5 4
1
.08035760321666974058...
2
12 log)1(
2log2logk
k
k
k
.080362945260742636423...
1
12
1)4()1(
2cos2cosh
4log
k
k
kk
=
24
11log
k k
.08037423860937130786... 11 5 1 14 1
2
1
1k kk
k
k
k
( ) ( )
.080388196756309001668... 1
3696 2 18 2 55 1
42 1
1
log log ( )
k k
k
H
k
.0805396774868055356... 1
1 113 2
( ) ( )k mnk
kmn
mn
1 .08056937041937356833... log
( )
3
2 6 1
k
k kk
1 .08060461173627943480... 2 11
2cos e ei i
.08065155633076705272... 1
4
1
2 1 222
1
( ) ( )!e
k B
kk
k
.080836672802165433362... 10
50 10 5 51 2
25
0
1
x
xx dxarctan( )
.081045586232111422... ( )
log
14
2
k
k k k
.08106146679532725822... 1 21
1
1
21
2
log ( ) k k
kk
K ex. 125
= log ( ) x dx1
2
GR 6.441
=
1 1
1
1
20
1
log logx x
dx
x GR 4.283.2
.081019302162163287639...
1 3
1
5
1
2
3log
5
1
kk k
122 .081167438133896765742... 8
63120 6 1
1
65
5
0
1
( )log( ) x
xdx
.08126850326921835705... 13
15 4
1
2 708 6
1
6
0
4
( )
tan/k
k k
dx
x xxdx
.081317489046618966975...
1
1)12( dxk
.08134370606024783679... 6
coth
2
143
2cot43
2cot
6
ii
i
= k
kk
k
k
k
2
62
1
111 6 2 1
( ) ( )
.08135442427619763926... 9 21
31
2
2 1 9 2 11
arcsin( )!!
( )!! ( )
k
k kkk
2 .08136898100560779787... 1
2 220log
xdx
x
.08150589160708211343... 1
22
3
4 1 21
loglog
( )( )
dx
e e ex x x
1 .08163898093058134474... 3
3 30
3 3
321
1
2 1
1
2 2
( )( )
( ) ( )k
k k k
.081812308687234198918...
0
7
3)cos(sin
192
5dx
x
xxx Prud. 2.5.29.24
.08183283282780650656...
1
1
27
3
1)37()1(1 k
k
k
kk
k
1 .08194757986905618266...
0 2
1 16)(!
)1(
2
1sin
4
kk
k
kk
.08197662162246572967... 153
26
3 1 21
log( )( )
k
k kkk
.081976706869326424385...
1 )!2(
12
1coth
2
1
)1(2
3
2
1
1
1
k
k
k
B
e
e
e
1 .081976706869326424385... 1
2
1
2 2
1 2 1 2
1 2 1 2coth/ /
/ /
e e
e e
= B
kk
k
2
0 2( )!
AS 4.5.67
.082031250000000000000 = 21
2561
71
2
1
( )kk
k
k
.08217557586763947333... 1
2 6 3
2
6
1
6 809 3
1
log ( )k
k k
dx
x x
.08220097694658271483... log
log loglog2 2
2 2
2
22 3
3
2
1
3
1
2
Li Li
.0822405264650125059...
2
21
2
22
2
1
)242(2
1
2
1
2
2log
12 kk
kk kkk
1 .08232323371113819152... )1,1,2()4(90
4
MHS
=
1
)3(
)1(k
k
kk
H
=
1 4 2
1
17
36
k
k
kk
Borwein-Devlin, p. 42
=
1
12
1 21 4
1
2
19
2
13
k
jkk j
k
kk
k
kk
1 .082392200292393968799... 4 2 2 11
4 60
( )k
k k
.0824593807002189801... 7
.08248290463863016366... 52
34
1
8
2 1 1
8 1
22
0 0
( ) ( )
( )
k
kk
k
kk
k k
k
k
k
k
k
.08257027796642312563...
1
16 8
2
8
2 2
2 2
cothsin sinh
cos cosh
= ( )8 4 11
14 4
2
kk kk k
.08264462809917355372... 10
121
1
10
1
1
( )k
kk
k
.082681391910818588188...
1
1
1)1()2)(1(
)1(log2log44
k
k
kkk
k
1 .08269110766346817971...
1 !!2
1
k
ke
k
k
H
k
H
.082710571850225464607... dxe
xxx
022 1
log)2(
612log
24
1
1 .082762193260924580122...
nprimepn
pn
e 12
1log
1lim
6
6 .082762530298219680... 37
.08282126964669066634... ( )7 3 11
14 3
2
kk kk k
.08282567572904228772... ( )
log( )6 2 1
11
2 6
2
k
kk k
k k
.08285364400527561924... 328
)3(7
4
3
64
1 3)2(
= )()(28
)3(733 iLiiLi
i
= dxx
xdx
x
xx
14
21
04
22
1
log
1
log
.082936658236923116009...
4/14/14/14/14/1 2
1sinh
2
1cos
2
1sin
2
1cosh
216
1
4/14/1 2
1sinh
2
1sin
28
1
=
1
21
)!4(
2)1(
k
kk
k
k
15 .08305708802913518542... 41
11
12 1
1
7 6
1
14
1e
k k
k
k
kk
k
k
k
( )( )!
( )( )
( )! Berndt 2.9.7
122 .08307674455303501887... log
( )( )( )
5
2
5
21
k
k km
k m
.0831028751794280558... 5
2
5
6
1
5 12
1
1
log( )
( )
k
kk
k
H
k
.08320000000000000000 = 276
31251
4
4
1
( )kk
k
k
.08326043967407472341...
2 3
3
2
18
2
32
1
2
15
363
log log( )
Li
= log ( )
( )
2
0
1 1
3
x
x xdx
.083331464003213147...
12
4)4(log
)()4(
)(
kk
kk
k
k
k
.083333333333333333333 = 1
126 2 1
1
14 2
2
( )kk kk k
=
02 1xe
dx
Prud. 2.5.38.9
2 .083333333333333333333 = 25
124 2 1 14
1
H kk
k
( )
1 .08334160052948288729... k
kk
!!
( )!21
.08334985974162102786... k
kk
3
51 3
257
322 9 3 27 4 27 5
( )( ) ( ) ( ) ( )
1 .08334986772601479943... k
kk
!
( )!21
.08335608393974190903... log loglog( )5
32
2
3
1
3
1
42 20
1
Li Li
x
x
.08337091074632371852... ( )4 1 1
21
22
4
k
kLi
kk k
2 .08338294068338349588... cos cosh( )!
1 1 21
40
kk
.08339856386374852153... 1 22
1
k
k
1 .08343255638471000236... 1
22 2
2
44 4
0
cos cosh( )!
k
k k
.08357849190965680633...
primepp
k k
k
14
1)(
24
1 .08368031294113097595...
242
11
4
2cos2cosh
k k
.08371538929982973031... 47 2
2
21 2
2
85
4 2 1 2 3
2 2
1
log log
( )( )( )
k H
k k kk
kk
57 .08391839763994994257... 21e 8103 .083927575384007709997... e9
1 .08395487733873059... H ( )/
33 2
.0840344058227809849... 11
2 1 21
Gk
k
k
( )
( )
= k k k
kkk k
( )
( )
2 1
16
16
16 112 2
1
= 1
8 21
3
41
kkk
k
, Adamchik (29)
= log x
x xdx4 2
1
.08403901165073792345...
1
21
)!2()1(
2
1sin2log2
k
kk
kk
B AS 4.3.71
=
1
cos2
k k
k
.084069508727655996461...
1 !2
1
2
1
2
1
3
2
kk
k
ek
k
k
Borwein-Devlin, p. 82
3 .084251375340424568386... 5
16 1 4
2 4
2 31
k
kk ( / )
.08427394416468169797... 8 32
1
4 3 4 2 4 1
2
21
( )( ) ( )k k kk
J271
.08434693861468627076... ( )kk
2
2
1
.08439484441592982708... ( )5 1 11
14 1
2
kk kk k
.084403571029688132... ( )
!
4 11
4
21
k
kek
kk
.084410950559573886889...
1
2
cos)1(1cos
2dx
x
xsi
.08444796251617794106... logsinh
( )log( )
4 4 11
1
4
2
k
kk
k k
= ii 2log2log2log
.08452940066786559145... ( )3 1 1 1 1
21
2 32
k
k kLi
kk k
.084756139143774040366... 11
2
1
2
1
2
12
1
cot( )
( )!
kk
k
B
k
.0848049724711137773... e i ei i 2 24/ log log
.0848054232899400527...
4 2
24
415 42
2
26
1
2
21
43 2
log
log( ) logLi
=log ( )
( )
3
0
1 1
1
x
x xdx
.085000000000000000000 = 200
17
.08502158796965488786... 21
2 4 1
2
20
4
arctansin
cos
/
x
xdx
4 .0851130076928927352... k
dkk
2
1 .08514266435747008433... dx
x( )1
2
.08515182106984328682... 178
225
1
6
2 1
2
1
2 61
2
( )!!
( )!
k
k kk
J385
.085409916156959454015... 1
0
222
log)1log(2
)3(72log4
212 dxxx
.08541600000000000000 = 41
480
1
2 61
k k kk ( )( )
.08542507002951019287...
5
241
1
1
2 2
30
log( )
( )
x
x xdx
1 .08542507002951019287...
5
24
1
1
2 2
20
log( )
( )
x
x xdx
1 .08544164127260700187... 21
2
1
2 1 20
sinh( )!
k kk
= 11
2 2 21
kk
1 .085495292598484041001...
122
)1()1(2
1
log
12
11
12
53816
144
1
xx
dxxG
.0855033697020790202...
1 )14(4
1
kkk k
.0855209595315045441...
1
2
1
)2(!
)1()1(
3
1
4
5
k
k
kkEi
20 .08553692318766774093... e3
2 .08556188259741465263...
1)(2
1
kk
1 .08558178301061517156... 2
2 2 12
1
k
k kk ( )
Berndt Sec. 4.6
=
1
2
2
)(
kk
ku
.085590472112095541833... log k
k kk4 2
2
.085628268201073866915...
1
2
)3)(2)(1(48
5
242
)3(
k
kk
kkkk
HH
.08564884826472105334... log ( )2
3 3 3
3
4
1
3 708 5
1
k
k k
dx
x x
.08569886194979188645... log2
3
1
6
1 3
12
3 3
4
5 3
4
i i i
1 3
12
3 3
4
5 3
4
i i i
=( )
1
132
k
k k
.08580466432218341253... ( )
( )
1
22
k
kk k
1 .08586087978647216963...
3 2
2
3
4log GR 8.366.5
.08597257226217569949... 1
2
1
1 11
1
sinh
cosh cos
cos
k
ekk
.08612854633416616715...
3 1
31360
1
( )
sinh
k
k k k
1 .08616126963048755696... ee k kk
12 2 1 2
1
2 1 10
cosh( )!( )
3 .08616126963048755696...
0
2
)!2(
12
2
1sinh41cosh2
1
k kee
.086213731989392372877... 1 1
8
1
2
1
22
12
1
sincsc
( )
( )!
kk
k
k B
k
.08621540796174563565... dxxx2/
0
2223
sinsinlog4
2log
4
2log
848
.086266738334054414697... sech
1 .086405770512168056865... 7 3
41 2 2 2
64 4 2 1
4 2
31
( )( log ) log
( )
H
kk
k
.086413725487291025098... log cos sin
cos sin
/2
4 2 0
4
G x x
x xxdx
= dxxx
x
4/
02
2
)sin(cos
= log cos/
x dx0
4
GR 4.224.5
.086419753086419753 = 7
81 9 11
( )kk
k
.086642070528904716459... dxx
x
2 2
1)(
.086643397569993163677...
1
21
3 1632
1
2
1
8
2log
kkk kkk
.086662976265709412933... 7
8 4
1
14 1
42 1
coth ( )k
kk k
.086668878030070078995... ( )
!
3 1 1 11
1 2
3k
k ke
k k
k
1 .086766477496790350386... 83
8 2 4 22 1
4 1
22
03
log log
( )
k
k kkk
.086805555555555555555 =25
288 1 2 3 4 5
3
1
k
k k k k kk ( )( )( )( )( )
.08683457121199725450... 1
1 2 30 k k k kk ( )!( )!( )!
.08685154725406980213... dxxx
x
13
2
)1()1(
log
2
2log
328
1
.086863488525199606126... ( ) log3 1 1 1
1
3
2
k
k
k
kk k
1 .08688690244223048609... log csc( )
1
2
3
2
3
2
3
2
2 1
1
k
k
k
k
.087011745431634613805...
642 2 1 2 2
0
2
( log ) log(sin ) sin cos/
x x xdx
2 .087065228634532959845... ek e
ek
kk
2
0
2/
!
.08713340291649775042... 2 1 2
4
1
4 2 1
21
1
arcsinh
( )
( )
k
kk
k
k
k
k
.08744053288131686313... 18
3 2
4
1
4
1
41
421 2
log( )
( )
k k
k
k
kk
k
=dx
x x x x5 4 3 21
.087463556851311953353... 30
3431
61
2
1
( )kk
k
k
1 .087473569826260952066... ( )311
96
6 8
6 85 4 31
k
k k kk
.08757509762437455880... 1
2
1
22
1
2 2 1 21
cosh sinh( )!
k
k kk
1 .087750469214439700994...
1
4
)4(8
)8(12
13
113400
13
k
k
k
H
.087764352700090443963...
212 4
)4(
)14(4
1
14
1
kk
k
kkk
kk
.08777477515499598892...
12 )32(4
)2(
2
)3(3
18
1
3
log
kk kk
k
.087776475955337266058... logG
.087811230536858587545... 2 2 3 21 23
1
( ) ( )( ) ( )
k
k kk
2 .087811230536858587545... 2 2 33 1
2 1
1
23 21
24
( ) ( )
( )
( ) ( )
k
k k
k k
kk
k
.087824916296374931495... 2 2 71
2 2
12
2 2 11
arcsin
( )k
kk kkk
.087836323856249096291... 2465 1
50
4
21
e k k
x
xdx
k
k
e( )
!( )
log
.087981169118280642883... 6
2
3
2
31
16 4
1
log
logx
dx
x
1 .088000000000000000000 = 136
125
2 2
0
x x
edxx
sin
.08800306461253316452... J
k kk
k
k1
0
2 3
31
3
1
( )!( )!
.088011138960...
14
)(
k k
kpf
.08802954295211401722...
1
6
7
21
7
2 1 21
cos( )
kk
.08806818962515232728... 4 3
4 2116
6
log x
x xdx
6 .08806818962515232728... 4 3
40
3
20
1
16 1 1
log logx dx
x
x dx
x
=
0
3
xx ee
dxx
1 .08811621992853265180... 4
12 2 2
4 14
42 1
sinh
exp( )
k
ki i
k
kk k
.08825696421567695798... Y0 1( )
.08839538425779600797...
1
04
2
)1(
1)1log(
3
2log
72
23dx
x
xx GR 4.291.23
.088424302496173583342...
( ) 1
S
where S is the set of all non-trivial integer powers
.0884395847555822353... ( ) ( )k kk
12
.08848338245436871429...
1
24
1)2,4(
jk jkMHS
.088486758123283148089...
1
21
1)1()2)(1(
2)1(
6
19log
2
6log3
k
kk
kkk
k
.08854928745850453352... 1
25 1 3 1
1
2 2 3
1
1
cos sin( )
( )!( )
k
k
k
k k
.08856473461835375506... log tan ( )( )
( )!2
1
21
2 1
21
2 12
1
kk
k
k
B
k k AS 4.3.73
.08876648328794339088... 1
2 241
22
0
1
x x xdxlog( ) log
= x x x dxlog log( )10
1
1 .08879304515180106525... log
( )
2
2
2 1
4 2 102
k
k kkk Berndt 9.16.3
= arcsin
logsin/x
xdx xdx
0
1
0
2
= x x
xdx
arcsin
1 20
1
=
log x
xdx
1 20
1
GR 4.241.7
= log1
1
2
20
1
x
x
dx
x GR 4.298.8
= log( tan )/
20
2
x dx
GR 4.227.3
= log(tan cot )/
x x dx0
4
GR 4.227.15
= log(coth )coth
xdx
x0
GR 4.375.2
.088888888888888888888 = 45
4
.08894755261372487720... ( )
( )
k
k k k kkj j
jk
1
1
12 22
= ( ) ( )
1 121
k
mk
mk m
.08904125208589587299... 2 3
27 1
2
41
2
31
( ) log
x
x xdx
x dx
e x
= dxx
xx
1
03
22
1
log
.08904862254808623221...
1 1
43 cossincossin
2
1
16
3
k k k
kk
k
kk
.0890738558907803451... 11 1
2
21e
erfe
xdx
x
( )
.089175637887570703178... ( )
( )
k
kk Li
kk k
1
1
112
22
2
.089222988242575037903...
4
143log2log32log23log32log92
3 22
2
Li
= dxxx )2/1log(2/1log1
0
1 .0892917406337479395...
1110 )!1(!)!2(
!)!12(2
4!2
1
2
1
2 kkk kk
k
k
k
k
kII
e
.08936964648404897444... ( )( )
12
1
2
1
222
21
kk
k k
k
kLi
k k
1 .08942941322482232241... dxxK
1
0
41 12
1
3
1
2
3
4
3
4
5
3
2
3
2
.08944271909999158786... 1
5 5
1
16
2
0
( )k
kk
k k
k
.08948841356947412986... dx
x k
dx
xe
k ( ) ( ) ( )0 0 0
1
1
.08953865022801158500... 2 2
7
319
2940
1
7
1
21
log ( )
k
k k k
.08958558613365280915...
2)21log(
4
2
3
1
=( )
1
4 708 4
1
k
k k
dx
x x
.089612468468018277509...
12
0 152
45
2
153
10
)2(2
)1(
3
2log8
3
10
k kkk
k
kkk
k
1 .08970831204984968123... 90
180
1
2 44 41
( )
( )f k
kk
Titchmarsh 1.2.15
.089738549735380789530...
2 log)2)(1(
1
k kkkk
1 .08979902432874075413... ( )k
kLi
k kk k
1 1 13
23
2
7 .0898154036220641092... 4
1 .08997420836724444733... erfik kk
k
1
2
1
4 2 10
! ( )
.0900000000000000000000 =9
1001
91
( )k
kk
k
2 .09002880877233363966... ( )
( )log
k
k k kkk k2 1
21
1
222 1
.090029402624249725308... 1
2
2
2
1
8
1
2
1
2
log i i i i
= sin
)
2
0 1
x
e edxx x
.09005466571394460992... ( ) log( )
2 2 21
2 2 12
3 31
Lik
kkk
24 .09013647349572026364... dxx
x
04
45
1
log
2512
57
.09016994374947424102... 2
11 5 5
15
3 .09016994374947424102...
k
k
kk
12
5
15
5cos10
5
2cos10
2
155
0
11 .09016994374947424102... 11 5 5
25
.09018615277338802392...
1
20 20184
1
6
)1(2log
60
47
kk
k
kkk
=
167 xx
dx
.090277777777777777777 =13
144
1
3 41
k k kk ( )( )
.090308644735282059644...
4
51arctan5551coth52522cot5110
5120
1harcarc
=
1 )12)(12(4kk
kk
kk
FF
1 .09033141072736823003... coth2
1 .090354888959124950676... 16 2 102 2
2
2
log( )
k
kkk
.09039575736110422609...
1
1
12 )13(4
)1(21,
3
4,
2
1,
3
11
kk
k
kk
kF
.090397308105830379520... 3 3
343 7
.090411664984819155127...
Li kk
kk
( , ) ( )log1 0
01
1
2
1
2
.090430601039857489999... 1
9
2
3
1
4
1
3 11
22
( )
( )
kk
= ( ) ( )k k
kk
1 1
32
.09045749511556154465... 2 41
2
4
3
1
4 2 11
arctanh log( )k
k k k
.09049312475579154852... csch( )kk
1
.09060992991398834735... 1
2 2
1
22
log
( )k
kkk
=
log 11
2
1
22 k kk
5 .090678729317165623...
10112
2
!
2
)!1(
14,2,
2
1F)2(
k
k
k k
c
k
k
kIe
.09079510421563956943... 11
111
ee
dx
e ex xlog( )( )
.090857747672948409442... e e
e
k
e
k
kk
( )
( )
( )
1
1
13
1 2
1
.0909090909090909090909 =1
11
1 .0909090909090909090909 =
1
3
611
12
kkkF
.09094568176679733473... 2
7
.09095037993576241381... 2
31
7 3
16 2 1
( )
( )
k
kk
.091160778396977313106... 1
2
6
5
1
11
1
11 2 12 10
log( )
arctanhk
k k K148
1 .091230258953122953094...
02 33
1
5
1
2
21tan
21 k kk
.09128249034461018451...
1
222 1)4(csc12coth12415192
1
k
kkh
=
2
243 )1(
1
k kk
.091385225936012579804...
1
41 20
2
2 log2
16
)33(
)1(
108 xx
x
kk
kk kk
k
.09140914229522617680... 527
2 3
2
1
ek
kk
( )!
1 .091537956897273237144...
05
5sin
5
1
2
55
8
1dx
x
x
.09158433725626578896...
2
2
4
)1)((
16
8
36
7
84
2log3
kk
kkG
.09159548440788735838...
1 4
cos
1cos817
11cos4
kk
k
.09160199373136531664... 1
242 kk
1 .09174406370390610145... 1
G
2 .091772325701657557483...
1
21
2
2
1
11
1
2
2
3sech
2
7cosh
2
1
kk kkkk
kk
.09178052520743498628...
0 12
2sin
2
1
4
cothxx ee
x
.09180726255210907564... 1
32 2
3 3
2
3 3
2
i i
= 13 1 14
2 1k kk
k k
( )
.09189265634688859583... 1
83 2
1
22
1
2 2 1 2
2
1
sinh cosh( )!
k
k kk
.09197896550006800954... 1
25
4
5
1
5 1 11
21
51
( )
( )
k
dx
xk
.09199966835037523246... 1
120
cos x
ex
.092000000000000000000 =250
23
7 .09215686274509803 =3617
510 8 B
.09216017129775954758... log
log2
3
5
361
12
17
x
dx
x
.0922121261498864638... 245
64
5 4
4 21
log x
x x
.0922509828375030002... 4
2 11 2 10
log ( )( )( )
k
k
k
k k
=
122
1
02
2
)1(
11log
)1(
)1log(
x
dx
xdx
x
x
.09246968585321224373... log
( )( )
3
4
3
36
1
3
1
3 1 3 31
k kk
.0925925925925925925 = 5
541
51
2
1
( )kk
k
k
.09259647551900505426... ( )2
21
k kk
k
1 .092598427320056703295... ( )3 11
kk
.09262900490322795243...
1
1
)2)(12(
)1(
12
172log84
k
k
kkk
.092641694853720059006...
1
1)12(212
15log122log2
k
kk
kGlaisher
.09266474976763326697... ( ) log( )2 2 1 1
1
2
22
k
k
k
kk k
1 .09267147640207146772... 8
7
1
2 2 2 1 2
2
0
arcsin( !)
( )!
k
k kk
1 .09268740912914940258... 1
732 kk
1 .09295817894065074460...
02
2
)1(16
12416
kk kk
k
.09296716692904050529... 61
32
33
2
3 21
Lix
x xdx
log
6 .093047859391469... k k
k
e k
k kk
k
k
2
2
1
21
1 1 ( )
!
( )/
.09309836571057463192... 1
2 10 20 5
10
10
cot
log
= 1
5
2
5 5
1
5
4
5
2
5cos log sin cos log sin
= ( )
1
5 708 3
1
k
k k
dx
x x
.0932390333047333804...
k
k
ke
I
k
k 12
!
)1()4,2,
2
3(F
)2(
0112
2
.09328182845904523536... ek k kk
21
8
1
2 40 !( )( )
.09331639977950052414... 18 13
32
1
5
2 1
2
1
2 51
G k
k kk
( )!!
( )! J385
.09334770577173107510... 22
8 12
13
1e
k
k kk
k
( )!( )
2 .09341863762913429294...
0
3/1
3
)(
2
3
kk
kpfe
.09345743518225868261... 5 6 2
9
1
2 1 2 31
log
( )( )k k kk
.093750000000000000000 =3
321
33
8 11
2
12
1
( ) ( )( )k
kk
kk
kLi
k
.093840726775329790918... 1
232 5 1
1
12
log ( )( )
( )
k
k
k
k k
2 .09392390426793432431...
1
121
1 )!12(
)2(2)1(
2sin
1
k
kk
k k
k
kk
.094033185418749513878... 6
5
2
4
6
5
4360
11 234
= ( )cos
1 14
41
k
k
k
k
.094129189912233448980... 9 3
4
2
2
1
6
2
4 2 321
( ) log ( )
( )
k
kkk
.09415865279831080583...
1
122
)2)(1()1()2log1(2log2log21
k
kk
kk
H
.09419755288906060813... 1
2
1
2
1
1
1
2 21
csc ( )k
k k
1 .09421980761323831942... 1 4
16
4 1
4 1 1
4 1 1
4 1
4
2
2
21
2
2
/ ( )
( )
( )
( )
k
k
k
kk
J1058
.09432397959183673469... 1479
15680
1
1 81
k k kk ( )( )
2 .09439510239319549231...
2
1arccos
3
2
.09453489189183561802... 1
2
2
3
1
31
logk
kkk
J145
=
12
1
1
2 )22(3
1
)22(2
)1(
2
)1(
kk
kk
k
kk
k
kkkk
2 .09455148154232659148... real root of Wallis's equation, x x3 2 5 0
.09457763372636695976... ( )k k
kk
2
3 11
.094650320622476977272... Re ( )( ) ( )
ii i
2
2 .09471254726110129425... arccsch1
4
.094715265430648914224... 1
2
4 2 1
2
4 1
2 1 2 221
4
( )!!
( )! ( )( )
k
k
k
k kk
J393
.094993498849088801...
2222
log1)2(
k k
k
Berndt 9(27.15)
.09525289856156675428... 22
24 2 2 2 2
2 1
2 2 2 11
log log
( )!!
( )!! ( )
k
k k kkk
.095413366053212024... 8 21
2
1037
105
1
2 2 71
arctanh
kk k( )
1 .09541938988358739798...
1
2
3
4,
1 .095445115010332226914... 6
5
1
24
21
2 1
2 60 1
kk
kk
k
k
k
k
( )!!
( )!!
1 .0955050568967450275... ( )
121
k
k k
k
.095542551491662252... e Ei eEie
x
e xdxx( ) ( )
( )
2 11
1
2
0
1
10 .095597125427094081792... ( )1 1
3
11 .095597125427094081792... ( )1 4
3
.09569550305604923590...
( )3
2 11
kk
k
.09574501814355474193... 1
833 kk
.09587033987177004744... 5
3 2 41
tanh log xdx
x
.095874695709279958758... 2
2 2 4 1 2
0
1
G x xdxlog log( ) log
4 .09587469570927995876... 22
2 11
20
1
Gx
xdx
log log log
.09589402415059364248... 1
3
4
3
1
3 411
log( )
H nk
kkn
.096000000000000000000 =12
1251
41
2
1
( )kk
k
k
.09601182917994165985... 5 5
54
4
41
( ) log
x
x xdx
.096023709153821012933...
4
51arctan5551coth52522cot535
5320
1harcarc
=
1
1
)12)(12(4kk
kk
kk
FF
.096188073044321714312... 1
64
3
8
7
8 11 1
41
( ) ( ) log
x
xdx
.096202610816922143434... 6
1 .096383556589387327993... ( )3
.09655115998944373447... 2 5 2 31 4
1
( ) ( ) ( )( )
H
kk
k
=
2
1
14
24
11)1,1,3()1,4(
k
k
jk jkjkMHSMHS
.096573590279972654709... log ( )
( )
2
2
1
4
1
4 1
1
1
k
k k k J368
= ( ) ( )
( )( )
1
2 6
1
2 2 2 40 0
k
k
k
kk k k
=
1
2 )1(2
1
kk k
= 1
2 1 21k
k k k k( )( )
GR 1.513.7
= dx
x xx x dx xdx7 5
1
2
0
4
5
0
4
sin tan tan/ /
= log
( )
log
( )
xdx
x
x xdx
x
1 131
30
1
= log 11
4 51
x
dx
x
= log
( log )( )
xdx
x x 2 2 20
1
1 GR 4.282.4
.0966149347325... ( ) log
1
11
k
k
k
k
1 .09662271123215095765... 2
0
2
09 2 1 2 1
2
2 2
k
k k
k
kkk k
!
( )( )!!
( !)
( )! J277
= W2 J313
= 1
6 5
1
6 11 k kk
J338
=log x dx
x60 1
2 .09703831811430797744... 2 3
3 3
2
3
2
32
2
32
1
2
log log
Li Li
=log
( )( )
2
12
320
1 x
x xdx
.097208874698216937808... e
e
k
ekk
2
121
( )
.097249127272896739059... ( )
( )
1
2 2 1
1
1
k
k kk
k
1 .09726402473266255681... e
k k k
k
k
k
k
2
1 1
3
4
2
2
2
5
( )! !( )
2 .09726402473266255681... e
k k
k
k
2
0
1
4
2
2
!( )
1 .097265969028225659521... kk
k
k
kk
k 4
1arcsin
)14(
144
22log3
2)1(
12
1
.0972727780048164... 1
342 kk
.0974619609922506557... 1
37268
7
8
3
8 14 4
4
41
( ) ( ) log
x
x
.0976835606442167302...
12
1
24
)1(
4
2csc25
k
k
kk
.097777777777777777777 =22
225
1
2 51
k k kk ( )( )
.097869789464740451576... 9731
30
1
1 61
ek kk
( )!( )
.09802620939130142116...
iLi e Li e
k
ki i
k2
32
32
32
1
( ) ( )sin
.09809142489605085351... 312
3 21
4 2
2
3 21
log
k
k kk
= ( )( ) ( )
11
22
kk
k
k k
.0981747704246810387... 32 4
3 3
12 2
0
sin cos
( )
k k
k
dx
xk
1 .09817883607834832206... csc(log )
2ii i
.09827183642181316146...
log
5
4
.09837542259139526500... 166
3192
3
4 4 31
log( )
k
kkk
.09845732263030428595...
23
)1()3(1
4
)3(31
k
k
k
=
1
0
1
0
1
0 1dzdydx
xyz
zyx
.0984910322058240152...
( )k
k kk4 2
2
1 .098595665055303... ( ) ( )2 2 1
2
2
1
k kk
k
1 .0986122886881096914... log( )
3 21
2
1
4 2 10
arctanhk
k k
= 11
27 331
k kk
[Ramanjuan] Berndt Ch. 2, 2.2
= Lik
k
kk
11
2
3
2
3
J 74
= sin
cos
x
xdx
20
1 .09863968993119046285... 31
3
1
3 12
32 20
2
1
Lik
Hk
k
k
kk
( )
( )
1 .098641964394156... Paris constant
.09864675579932628786... 1
2
2
42
0
1
si
x xdx( )
log sin
1 .098685805525187... Lengyel constant
.0987335167120566091... 2
2 2224
5
16
1
1
( )
( )
k
k k J401
.098765432098765432 = 8
811
81
4
41
( )logk
kk
k x
xdx
.098814506322626381787... 8 4
1
2
1
16 121
coth
kk
1 .09888976333384898885... ( ) ( )k kk
3 12
5 .09901951359278483003... 26
.09902102579427790135...
12
1 1428
1
2
1
7
1
7
2log
kkk kkk
.099151048958717950329...
2
1)(11
1
kke
k
e
eH
e
.099151855335609679902... log
( )( )
( )
2
4
1
3 31
1
22
k
k kk
2 .09921712010141008052...
0 1
21
4
k k
ke
.09926678198501025408... ( )( ( ) )
11
2
2
2
kk
k
k
7 .0992851788909071140... k kk
kk
k
2
1
2
12 1 2
( )
29809 .09933344621166650940... 9
.0993486251243290835... 7715
17282 5 3
5 31
( ) ( )( )
k
kk
.099445877615933685923... primep pp 1
122
2 .099501138291619401819... log ( ) log3 3
21
1
3
3 1
30
k
kk
k
k
k Prud. 5.5.1.17
.099501341658675... ( )( )
( )!
12 1 1
21
1
k
k
k
k
1 .0995841579311920324... 16
9
2
34 2
14
341
log
( )!
( )!
k
k kk
.099586088986234725... 24
51
4 12
1
log ( )( )
k kk
k
H
k
.0996136275967890683...
4
2
8
1
2 23
0
4
log
tan/Gx xdx GR 3.839.2
.0996203229953586595... 32
1112
e Eik kk
( )( )!
.09963420383459667011...
0
2
2
44
14
2sinhhcsc
2
1
k k
k
1 .09975017029461646676... csc2
.0997670771886199093... 23
11
1 1
1 2
0
eEi
k
k k
k
k
( )( )
( )( )!
6 .0999422348144516324... e Ik
k
k
k
0
12
20
12
( )( )!
( !)
.09995405375460116541...
2
)(1log
kkk
k
.100000000000000000000 = 1
10
=
1 )3)(2(
)5(
k kk
kk J1061
8 .100000067076033613307... 0111213141234567891.0/1 , inverse of Champernowne’s number
.10000978265649614187... Likk
k10 10
1
1
10
1
1010 0
1
10
, ,
.10015243644321545885... | ( )|( ) k k
kk
1
91
.10017140859663285712... ( )3
12
.10031730167435392116... 3
2 12
1 1
2
11
12
32
2
( )
logk
k k k kk k
.100391216616618625526...
14
45
1
log
128
5
8
)5(93dx
x
x
.1004753... powerinteger
trivialnona2
1
, Goldbach’s Theorem
1 .10055082520425474103...
1
0
1
0
1
0222
2222
14
16
3dzdydx
zyx
zyx
3 .10062766802998201755... 3
10
.100637753119871153799... Likk
k4 4
1
1
10
1
104 0
1
10
, ,
.10068187878733366234... 8 2
3
4 0
1
G
x x x dxarctan log
.10085660243000683632... 3 2 2
16 1 30
1
log log
( )
x
xdx
.10096171...
22 1
)(
k k
k
.10098700926218576184... 2 2
3
13
36
1
1 3
1
1
log ( )
( )( )
k
k k k k
.1010205144336438036... 5 2 62 1
2 3 11
( )!!
( )! ( )
k
k kkk
1 .1010205144336438036... 6 2 61
12 1
2
0
kk k
k
k( )
.1010284515797971427... ( )3 1
2
.10109738718799412444... 12 2 2 2 6 2 6 13 2 3
0
1
log log log log ( ) x dx
.1011181472985272809... 2cschcoth23coth64
= k kk k
kk k
2
1
4 4
4 32
4 11
1 ( )
( )
( )
7 .10126282473794450598... K3 1( )
.10128868447922298962... Likk
k3 3
1
1
10
1
10
1
103 0
, ,
.101316578163504501886... 2
)2(
2
log)1( 2
22
k
k
k
k
.10132118364233777144... 2
.10148143668599877803... 23
4log
3
3
3)12(
)1(
1
1
k
k
k
kk
.101560303935709566... 1
11 11
3
1131
1
1k
k
k
kk
k
( )( )
.10159124502452357539... 1
10
2 5
55
1
520
csch ( )k
k k
.10169508169903635176... log ( ) 22
kk
2 .10175554773379178032... k
kk !
1
.10177395490857989056... G
k kk9
1
12 9
1
12 32 21
( ) ( )
=
0 12433
log
xx
dxx
ee
dxxxx
.10187654235021826540... 1
2
1
3
1
2 2 1
1
0
arctan( )
( )
1
2
k
kk
k
k
1 .10198672033465253884... 8
9
20
27
4
3 4
2
1
logk Hk
kk
.10201133481781036474... ( )k
kk 21
.102040816326530612245... =5
49 7 11
( )kk
k
.102129131964484887998... e e
e
e
k k
e
e
k k
k
1 1
220
( )
!( )
.1021331870465286303... 75161
3 2 3
3 3
0
e k
k kk ( )!
.10218644100153632824... 28
2
14
5
84
1
2 1 2 80
log ( )
( )( )
k
k k k
.10228427314668489686...
12
12
0 )(32
1
612
1
63
)1(
3
2log1
kk
kk
k
kkkkk
=
147 xx
dx
1 .10228427314668489686... 4 2
33
0
2
logsin( ) log tan
/
x x dx
.10246777930717411874... 1
2
2 1 2
21
2
0
1
cos log sinlog sin
e
x x
xdx
x x
edx
e
x
.10253725064595696207...
2
1)(2
1log2log42
2
1
kk
kk
k
.10253743088359253275... 1
101 ( )k kk
.10261779109939113111... Likk
k2 2
1
1
10
1
10
1
102 0
, ,
.10277777777777777777 =37
360
1
6
1
21
( )k
k k k
= x x dxx
dx
x5
0
1
171 1
1log( ) log
.10280837917801415228... 2
2196
2
16
1
4
( )
( )kk
.102953351968223325475...
4
1
4
3
256
)3(32log4248
6144
1 )3()3(2 G
= 4/
0
2 sinlog
dxxx
.103171586152707595114...
1
0
2 2
log
2
11 dx
x
xxLi
1 .10317951782010706548...
4/14/14/14/1 2
1
22
1
224
1H
iHH
iH
= ( )4 1
2 2 114
1
k k
kkk k
1 .10322736392868822057...
1 3
2
22
)3(352log216
1
k
k
kk
kG
2 .10329797038480241946... 5
55 3 13
2 1kk
k
k
k
( )
.10330865572984108688... 106
158 2
1
2
1
2 2 70
arctan( )
( )
k
kk k
1 .10358810559827853091... 1 21
log ( ) kk
.10359958052928999945...
( )
( )log3
42 2 3 2
2
4 21
2
31
x
x xdx
x dx
e ex x
= x x
xdx
2 2
20
1
1
log
GR 4.262.13
2 .10359958052928999945... 7 3
42 3 2
1
2 1 30
( )( )
( )
kk
= log2
20
1
1
x
xdx
GR 4.261.13
= x dx
e ex x
2
1
.10363832351432696479... 3
1 11
11
12
1
5
0e
k
k
k
kk
k
k
k
( )( )!
( )( )!
1 .10363832351432696479... 3
12 1
1
2
1e
k
kk
k
( )( )
( )!
.10365376431822690417...
2 1 20e
ex
xdxe
cos
AS 4.3.146
.10367765131532296838... 10
9
1
4
2 1
2
1
2 4
2
1
( )!!
( )!!
k
k kk
.10370336342207529206...
2
2
22
2
2 8 4
22
2loglog
log
= x e x dxx2 2
0
2
log
.10387370626474633199... 11
24
2 2 2
16
1
442
cot coth
kk
=
1
1 1)4(4k
k k
.1039321458392100935... 5 8 2 2 2
41
3
21
1
log log( )k k
k
H
k
.10399139854430475376... G k
k
k
k4
1
8
1
4 12 21
( )
( )
.10403647111890759328... dx
x x6 31 1
1 .1040997522396467546...
1 )1(
)4(
k kk
k
.10413006886308670891... 3
64 2 22 30
dx
x( )
1 .10421860013157296289... 1
3 3 10k
k k( )
.10438069727200191854... 2
21
1
3242 1
1
4 2 11
2
log( )
( )( )
k k
k
k
kk
k
=
1 )2)(1(2kk
k
kkk
H
15 .10441257307551529523... log !10
.10452866617695729446... 2
2 2
2
21sinh
k
kk
.10459370928947691247... 2 8
2
4
3
4
2
0
log log sinx x x
edxx
.10459978807807261686...
11 )24(
21
)!22(
)!1(!
2
1
33 kk kk
kk
kk
=
1345 xxx
dx
.1046120855592865083...
1
1
)12()!12(
)1(21sin)1(
k
k
kk
ksi
= dxxxx1
0
sinlog
.10464479648887394501...
18
1
8
1
21
4
1,2,
2
1
2 108/1
11 IIeF
=
1
21
21
)!1()!12(
)!()!(
k kk
kk
.10469603169456307070... 8
9
9
81
81
1
log ( )
k
k
kk
H
1 .1047379393598043171... 4
1 11
2 21
erf erfix
dx
x
cosh
= 3
14
1cosh2
x
dx
x
.104742177484448455438...
16
13
16
5214204
256
1 )1()1(2
=
144
log
xx
dxx
3814279 .1047602205922092196... eee
1 .104780232756763784223...
22 logk
k
kk
H
.10492617049176940466... ( ) ( ) )( )
( )
1 3 14 1
11
1
33 6 3
3 42
k
k k
k kk k k
k
.10493197278911564626... 617
5880
1
1 71
k k kk ( )( )
.10498535989448968551... 71 431 2 3
2
1
ek
k kkk ( )! ( )
.105005729382340648409... ( )
( )
k
kk Li
kk k
1
113
23
1
.10500911500948221002... log( )
log
2 2
2
161
1
1612
1
k
k kkk k
=
1
1 10
1 2
2
x
x
x
x
dx
xlog GR 4.267.5
1 .10503487957029846576... 11
121
k kk ( )
.10506593315177356353... 7
42
1 1
14 22
22
( )( )k k k kk k
= ( ) ( ) ( )2 1 1 2 12
1
1
k k kk
k
k
=
22
poweintegertrivialnonaω
2
)(
1
1
kr
k
k
5 .1050880620834140125... 13
8
.10513093186547389584...
( )
!
k
kk
1
2
21
.10513961458004101794... 5
8
3 2
4
1
2 122
log
( )kk k
.10516633568168574612... 10
1 .10517091807564762481... e1 10/
.10518387310098706333... sin ( )
( )!
1
8
1
2 1
1 3
1
k
k
k
k
.1052150134619642098...
ci x xdx
2 0
2
sin log/
.105263157894736842 =2
19
.10536051565782630123... log log , ,10 91
10
1
101 0
1
kk k
= 21
19
1
19 2 12 10
arctanh
kk k( )
1 .1053671693677152639... 1
2 1 222
11 ( )( )
k kkk k
k
1 .10550574317015055093... ( ) ( ) ( ) ( )( )
( )3 2 3 2 5
2
1 41
k k H
kk
k
.1055728090000841214... 12
5
1 2 1
2 4
1
1
( ) ( )!!
( )!
k
kk
k
k
.1056316660907490692... ( )3 1
2 2 113
2
k k
kkk k
.10569608377461678485...
0
3 log4
1 2
dxxex x
.105762192887320219473... ii i i
k i k ik61 1
1 13 34 4
1
( ) ( )( ) ( )( ) ( )
.10590811817894415833... 2 3
32 2 3 2
2 2
332 2log
log log loglog
log
= ( )
( )
1
2 1
1
1
kk
kk
kH
k
.10592205557311457100... 4 28
3
1
2 2 61
log( )
kk k
1 .10602621952710291551... 4
41
1
16 21
sinh
kk
.10610329539459689051... 1
3
.10612203480430171906... 10
2
5
26
751
12
16
log
logx
dx
x
.10618300668850053432...
22
1 1)1()1(
k
k
k
k
.10620077643952524149... 114
2 4k ks ds
k log( )
.10629208289690908211...
4 42 20e
x
xdx
cos
= cos( tan ) sin/
2 2
0
2
x xdx
GR 3.716.4
2 .10632856926531259962... k k
k
2 11
2
( )
.10634708332087194237... log log2
2
2
2
2
=( )
1
11
kk
k
kH
k
.10640018741773440887... loglog( )2
2
20
1
212
1
2
1
3
Li
x
x
.10642015279284592346... 1
82 2 2 21 1 2 2i i i i i i ( ) ( ) ( ) ( )( ) ( ) ( ) ( )
= ( ) ( )( )
( )
1 2 1 11
11 2
1
2
2 32
k
k k
k kk k
k
.10649228722599689650... 24 5 14 32
52 1
1 11 11
1
4 2 3
52
( ) ( ) ( )( )
k k k
kk
= ( ) ( )
1 1 14
1
k
k
k k
.10666666666666666666 =8
754
0
1
x x dxarccos
.10670678914431444686... 1
6
3
12
3
2
1
4 3
1
21
csch
( )k
k k
.106843054835371727673... 32log31526log
6
1
3
32log
=
1
1
)12)(1(2
2)1(
k
kk
kkk
kk
.106855714568623758486... ( )( )
arctan
11
2 1
11
1
32 1
k
k k
k
kk
k k
.10691675656662222620... 1
10
10
20
5
2
1
2 520
csch( )k
k k
1 .10714871779409050301... 12
2)1(
5
2arcsin2arctan
12
0
k
k
k
k
1 .1071585896687866549... )1)(2
xH x
.10730091830127584519... 1
2 8
1
4 1
1
2 21
( )
( )
k
k k GR 0.237.2
= 1
16 1
2
1621 1k
k
kk
k
( ) GR 0.232.1
= ( )
1
4 60
k
k k
= 1
7 31 x x
1 .10736702429642214912... 3
28
.10742579474316097693...
1
1
1 )1(4
)1(
)1(5
1
5
4log41
kk
k
kk kkk
.1076539192264845... one-ninth constant
.1076723757992311189... 10 62 21
ek
k kk ( )!
3 .1077987001308638468... e Ik
kk
0
12
20
1
2
( )!
( !)
.10801497980775036754...
4 2
1
21 1
21
2coth
i i
= k
k kk k
k
k
kk
1
4
1
42 2 13
1
1
1
( )( ) ( )
.10808308959542341351...
0
2
)!3(
2)1(
4
1sinh
8
1
k
kk
k
ee
.10819766216224657297... 3
2
3
2
1
2
1
2 22
log( )
( )
k
kk k k
.1082531754730548308... 3
16
1
12
2
0
( )k
kk
k k
k
.10830682115332050466... 432
9
1
3 2
1
21
Gk
k
k
( )
( / )
1 .10835994182903472933... 1
4 1 11k
k k ( )
1 .10854136010382960099... Hkk
k 2 11
.10874938913908651895...
161
14
2 2
0
2
ee x xdxx sin cos
.10900953585926706785... ( ( ) ) k
kk
1 2
22
.10905015894144553735... 27 4 3
481
13
15
logx
dx
x
.109096051791331928583... 71 6
108
1
31
2
21
2
0
1
k kx x x dx
k ( )log( ) log
.10922574351594719161...
12 )22(4
)2(
4
)3(7
4
1
2
log
kk kk
k
.10928875430471840710... x dx
x
6
120
1
1
.109375000000000000000 =7
641
71
1
( )kk
k
k
1 .10938951007484345298... 1
2
1
21
1 2
k
ke
k !sin sin sincos( / )
.10946549987757265695... 1
12
2
42
1
2 3
1
21
csch( )k
k k k
.10947570824873003253... 4 2 5
6
1
4 2
21
1
( )
( )
k
kk k
k
k
.10955986393152561224... ( ( ) ) k
kk
1
2
2
2
.109575787818588751201...
2
1 1
432 12 3
1
144
1
6
2
3
( ) ( )
= ( )
( ) ( )
1
3 1 3 32 20
k
k k k
.10960862136441250153...
123
2
3 2
log
2
142 dx
xx
xLi
.10965469252512710826... 1
212
0
12
Ei e Ei e dxe x
( ) ( )
.10967200...
22 1
)(
k k
k
2 .1097428012368919745... 1
22 k kk log
.109756097560975609756 =9
82
1
2 91 2 2 1 3
0
cosh log( ) ( ) logk k
k
e J943
.10981384722661197608...
1
561
20 12144
1
5
)1(
12
72log
xx
dx
kkk kk
k
.1098873045414083466...
4
2
2 1
2
20
4
arctan sin
sin
/ x
xdx
.109944999939506899848...
3
4log53511log5272511log11511512log5
515
1
=
1
11
)1(4
)1(
kk
kkk
kk
FF
.11005944310440489081... 1
132 k
k
kk
log
.11008536730149237917... 64
223
.11030407191369955112... 3 3
4 122 2 3
12
4 32
( )log
( )
k
k k k
.11037723272581821425... 1
256
1
8
3
8
5
8
7
8
1 2
4 21 1 1 1 ( ) ( ) ( ) ( ) log( )
=( )
( ) ( )
1
4 1 4 32 20
k
k k k
9 .1104335791442988819... 83
1 .11062653532614811718...
( )
( )
( )3
4 41
k
kk
Titchmarsh 1.2.13
1 .11072073453959156175...
2 21
2 1
2 2 2 1
2 1
8 2 11 1
( )!!
( )!! ( ) ( )
k
k k
k
k kkk
kk
=( ) ( )
1
4 3
1
4 1
1 1
1
k k
k k k J76, K ex. 108c
=
0
2/
12
)1(
k
k
k Prud. 5.1.4.3
= 11
2 30
( )k
k k
= dx
x40 1
Seaborn p. 136
=dx
x
dx
x
x dx
x20
20
2
402 2 1 1
= dx
x x( )1 12 20
1
= d
1 20
2
sin
/
.11074505351367928181... log( )
log2
3 3 31
1
3
1
12
k
k k kkkk
.11083778744223266... ( )
1
430
k
k k
.11094333469148877202... 1
10 11
3
1031
1
1k
k
k
kk
k
( )( )
.111001751290904447718... 9 4 6 3 265
16 3
2
41
( ) ( ) ( )( )
k
kk
.11100752891282199021... 4
.11100821732964315991... log3 2
3
.11100942712945975694... ( )
!
k
k kk3
2
.11105930793590622477... 2
3
1 .11110935160523173201... e Lie e kk
k2 2
0
1 1
1
( )
.11111111111111111111 =1
9 41 0
22
log
log(sin ) sin cos/x
xx x x dx
= ( ) ( ) ( )k k k
kk
kk
kk3 1 6 1 9 11 1 1
.11118875955726352102... 4/
0
22 tan)3(212log21664
1
dxxxG
.1112477908543131171...
12 )2)(1(
1)2()3(1252log4
8
1
k kkk
k
.111387978350712564526... 1)1()1(2
)1(
9
44
4
9log
4 1
212
kk
k
kk
k
.1114472084426055567...
1
21
1
21
1
21
=1
21
2 21
1
2
cot
=
3
1
2
1
2
1HH
=1
2
2 1 1
232 1k k
k
kk
k
( )
1 .1114472084426055567...
2
11
2
11
2
1
=1
2
2 1
231 1k k
k
kk
k
( )
.11145269365447544048...
2 3 2
36
2
3
2
32
1
2
3
23
log log( )Li
=log
( )( )
2
21 2 1
x
x x
.11157177565710487788... 1
2
5
4
1
9 2 12 10
log( )
arctanh1
9 kk k
= Im logii
12
.11157214582108777725...
1
1)1()1( )12(
9
)1(
31
31
12 kk
k
kkiii
.11163962923836248965... 1
541 kk
1 .1116439382240662949...
2
3
2
1
2
3
2
13333
333
1 iiii
i
= 3/23/13/13/2 )1()1(1)1()1(113
1
= 1 1
21 3 1 12 1
1
1
1k kk
k
k
k
( ) ( ) )
= 1 1 3 3 11
1
( ) ( ) ( ))k
k
k k
.11168642734821136196... ( )
1
5 607 2
1
k
k k
dx
x x
.1119002751525975723... Li k kk
41
1
9
1
9 4
.11191813909204296853... ( )kk
15
2
.11196542976354232023... 2
1
2 1 1
23 2 22 1k k
k
kk k( )
( )
.11207559668468037943... ( ) ( )
( )
2
32
2
16
16
16 11
2
2 21
k k k
kkk k
.11210169134154575458...
1
21 1
1
1 2 21
i i
k kk ( )
.11211354880511828217... ( )kk
k 2 111
1 .11225990481734629821... 1)1(1
)2(
kHk
k
6 .1123246470082041... e
F
k
kk
1
1
/
4 .11233516712056609118...
1
2
232 )(
)4(
)2(
12
5
k k
kd
Titchmarsh 1.2.9
=
p p
p22
2
)1(
1
.1123768504562466687... 8
2 2
1 1
2
12 9
15
1e
e
x
dx
x x
dx
x
sinh sinh
.11239173638994824403... 1
2421 3 2 22 csc
= 1
48 2 22 3 2 21 22csc sec sin sin
=( )
1
24 22
k
k k k
2 .11259134098889446139... 21
41
/k
k k
.11269283467121196426... 3 3
32
1
2 2 30
( ) ( )
( )
k
k k
=
1
2
1
23 31
Li i Li ik
k
k
( ) ( ) cos
=
1
0
1
0
1
02221
dzdydxzyx
xyz
.11270765259874453157... Likk
k3 3
1
1
9
1
9
.112741986560723906051... 2
2016 8
2
2 2
1
2 1 4 1 4 3
log
( ( )( )
G
k k kk
.11281706418194086693...
( )
( )
k
kk
2 1
2
.11281950399381730995... 1
22 1 1
1 1
2
2
0
1
log( )ee
e
edx
x
x
. 11292216109598113029...
1
12
)3)(2)(1(724
1
k
kk
kkkk
HH
4 .11292518301340488139...
1
1,,11
kk
k
.11304621735534278232... 3
43
4
3
1
3 1
1
1
log( )
( )
k
kk
k
k
1 .1131069910407629915...
2/3
1)( dxx
1 .11318390185276958041... 1
18 7 6 5 4 3 20 k k k k k k k kk
.11319164174034262221...
log
4
3
.11324478557577296923... ( )jkkj
4 111
4 .11325037878292751717... e 2
1 .113289355783182401858...
4 )4,(1
)1(
k
k
kS
.113513747770728380036...
1 6
1)1(
5
1
2
3
2
3log32log2
kk
k
.11356645044528407969... 5
361
5
61
51
1
log ( )k kk
k
H k
.11357028943968722004... ( )
11
2
k
kk k
.11360253356370630128... log
!
k
k kk 22
.11370563888010938117... 3
22 2
1
4 12 21
log( )k kk
J104, K ex. 110e
= 1
2 1 2 2
1
2 1 21 1k k k k kkk
k( )( ) ( )( )
= k
k k
k k
kk
k
1
4
2 2 1
431 1
( ) ( )
= ( )
12
2
k
k k k
= dx
x x
dx
e ex x( ) ( )2 21
21 1
.1138888888888986028...
| ( )|( ) k k
kk
1
9 11
.11392894125692285447... 616 1
40
3
21
e k k
x
xdx
k
k
e( )
!( )
log
.1139565939558891...
2 log)12(
)1(
k
k
kkk
.11422322878784232903...
9
112
7 2
2242 2
3 43 4 3 4
// /csc
csch
= ( )
1
842
k
k k
.11427266790258497564... 4 8 2 8 2 3 2 3 3 3 12log log log log log
= H
kk
kk 4 21 ( )
.1143602069785100306... Likk
k2 2
1
1
9
1
9
= 3log33
16 2
2
2
Li
2 .11450175075145702914... Ei Eik kk
( ) ( )( )!( )
1 1 21
2 1 2 10
=
4
1,
2
3,
2
3,
2
12)(2)1(2 HypPFQisiialSinhIntegr
.114583333333333333333 =11
96
1
2 41
k k kk ( )( )
.11466451051968119045... 4 2 4 22
32
2 22
2
21
log log( )
k
kkk
.11467552038971033281...
1
2
1
16
)1(
6csch
622
1
k
k
k
.11468222811783944732... 2 2 5 4 21 2 1
2 4
1
1
log log( ) ( )!!
( )!
k
kk
k
k k
.11477710244367366979... 1
16
1
2
1
8
1
2
1
2 4
1 2
1
cos sin( )
( )!
k
kk
k
k
3 .11481544930980446101... 2 1tan
3 .11489208086538667920... 1
2
11
1
Ei e Eie
k
k kk
( )sinh
!
.11490348493190048047... J2 1( )
.11508905442240150010...
23 1
)1(
k
k
k
.11512212943741039107... 1
10
3
5
4
54
6
5 5
cot cot cos logsin
1
104
8
5 54
12
5
2
5cos logsin cos logsin
1
104
16
5
2
5cos logsin
=
5
4
10
3
10
9
5
2
=1
5 1 5 21 ( )( )k kk
.11524253454462680448... dxex
x
e
e
1
02)(
log1)1log(
.11544313298030657212... 5
.1154773270741587992... log
log log (log )( )3
22 3
2
3
1
22 3 2
1
2
1
2
7 3
8
Li Li
=log ( )
( )
log2
0
1 2
21
21
2 1
x
x x
x
xdx
.11552453009332421824...
112 26
1
1224
1
6
2log
kk
k kkk
=
0 )4)(2)(1( xxx
dx
.11565942657526592131... 3
256
2
.11577782305257759047...
log log log2 63
2
2 3
2 31
16 2
1 x
dx
x
=
126
1
0
6 11log)1log(
x
dx
xdxx
.1157824102885971962... 2arccsch2
1
2
51log
2
1 22
=
1 2
1
022 2
4
)1(
22
1
!)!12(
!)!2(
2
)1(
k
k
kk
k
kk
kkk
k J143
.11591190225020152905... 1
4
1
2
1
2 2 12 11
arctan( )
( )
k
kk k
.11593151565841244881...
11 1
1)12(
2
)(2log
kkk k
kk
=
1 )12(4
)12(
kk k
k
=
kk kk
log 11 1
22
= cosxx
dx
x
1
1 20
1 .1159761639... j3 J311
.11621872996764452288... 2
312
9
4
27
12
3
sin k
kk
= iLi e Li ei i
2 33
33( ) ( ) GR 1.443.5
1 .1165422623587819774... 1
3
1
4 8
4
40
cossin x
xdx
.116565609128780240753... 7 3
4
2
2
1
4
2
4 2 221
( ) log ( )
( )
k
kkk
.1165733601435639903... arcsinh
2 2
2 1
2 31
2
2 2 11
1
( )( )
kk
k
k
k
k
k
.11683996854645729525... 7
8
8
7
1
7
1
1
log( )
k
kk
k
H
.11685027506808491368... 2
21
2 2116
1
2
1
2 1 2 1
1
4 1
(( )( )) ( )k k kk k
J247, J373
.117128467834480403863... )3()3()3(6)3()3()3(6)4(
1 )3(23
3
=
23
3log)(
k k
kk
.11718390123864249466... ( ) log
( )3 24 2
5
332
1
2 1
2
3 21
k kk
.117187500000000000000 =15
128
1
3
4
1
( )k
kk
k
.11735494335470499209...
12
13
90
1
2 1 2 70
( )
( )( )
k
k k k
.11736052233261182097... 2
3
1
2
1
4 2 10
arctanhk
k k( )
.11738089122341846026...
1
2
1
14
)1(3csc38
6
1
k
k
kk
.11738251378938078257... 2 1
2
1
3
2 1
2
1
2 3
2
1
G k
k kk
( )!!
( )!! J385
8 .11742425283353643637...
4
2112
4 2 2
( ) ( )( )
Lr k
kk
2 .11744314664488689721... 2 24 1
1
4( ) /
k
k
.117571778435605270... ( ) ( )
( )( )
3 244
32 2
1
H
kk
k
.1176470588235294 =2
17
.11778303565638345454... log( )
, ,9
8
1
9
1
8
1
91 0
1
1
1
kk
k
kkk k
= 21
172
1
17 2 12 10
arctanh
kk k( )
K148
1 .11779030404890075888...
1
1)2()2)(1(
2
1
k
kk
kk
.11785113019775792073... dxx
xdxxx
4/
0
34/
0
2
tan
sincossin
26
1
.117950148843131726911...
224
1
2
1222
iLi
iLiLi
1 .1180339887498948482... 5
2
1
2
2 1
200
k
k kk
=
1 121
1
k kF
.1182267093239637759... 3
21 1
1
2 2 21
cos sin( )
( )!( )
k
k k k
.11827223029970254620... 11
21 1 1 1 2 2 1 ( ) sin (cos ) log( cos )
=cos
( )
k
k kk
11
.11827410889083245278... G k
k
k
kk
log ( ) ( )
( )
2
4
4 1 2
16 2 11
= 21
22
1
41
kk kk
arctanh arctanh
.118333333333333333333 = 71
600
1
1 61
k k kk ( )( )
2 .11836707979750799405... 1 2
1 k
k
kk
log
.118379103687155739697... 7
5760
4
4 4
Li i Li i( ) ( )
.118438778425057529626...
3
1
6
5
432
1
36
)3(13
3162
5 )2()2(3
=
03)23(
)1(
k
k
k
.118598363315566157860...
53627
64log
8
511log511
)51(
527log5414
535
12
=
1
1
)1(4
)1(
kk
kkk
kk
FF
.11862116356640599538... 1 11
e x e dxe x,
.11862641298045697477... 1 1 2 1 11 2 1
2 2 1
1
1
log( ) ( )!!
( )! ( )arcsinh
k
k
k
k k J389
= ( )
( )
1
4 2 1
21
1
k
kk k
k
k
.11865060567398120259...
H kk
kkk k
1
2
2
22
2 11
1( ( ) )
log
1 .11866150546368657586... 2 3 11
3
30
e Eik
k kk
( )!( )
4 .11871837492687201437...
1
02
2
)1(
log
4
1
8
2dx
xx
x GR 4.241.11
.11873149673078164295...
4
2
3
1
2 506 4
1
4
0
4
( )
tan/k
k k
dx
x xxdx
.11877185156792292102... 8
1
22
1
2 1
2
0
1
logarctanx x
xdx
5 .11881481068515228908... k
kk
2
0 1( )!!
7 .11881481068515228908... ( )!!
!
k
kk
1
0
.1189394022668291491...
4
33
4 31
3
2015
51
83
1
( ) log x
x xdx
x
e edxx x
.11899805009931065100...
1
1)12( dxx
.11910825533882119347...
5
36 144
2
18
2
0
1loglogx arccot2 x xdx
.11911975130622439689... 2 1 1
2
2
21
sin log coslog
e
x x
xdx
e
.11920292202211755594... 1
1
12
1
21 0e e
ex
edx
k
kk
x
( ) cos( )
.11925098885450811785... ( ) ( )4 5 2
.119366207318921501827... 3
8
.1193689967602449311... ( )( )
12 1
1 1
1
kk
k
k
9 .11940551591183183145... 4 4
.1194258969062993801... 2
21
8
144
1
12 9
G
kk ( )
.11956681346419146407... cossin ( )
( )!1
1
2
1
2 2
1
1
k
k
k
k
.11973366944845609388... ( ) ( )( )
3 41 4
1
k
kk
.11985638465105075007... 1
4
1
2
1
2 4
1
1
sin( )
( )!
k
kk
k
k
.11995921833353320816... Ei
e
k
k
k
k
( ) ( ) ( )
!
1 1 11
1
.12000000000000000000 =3
25 5 11
( )kk
k
1 .12000000000000000000 =
1
2
425
28
kkkk FF
.12025202884329818022... 1
2
1
2
1
2
4
5
1
2 2 12 11
arctan log( )
( )
k
kk k k
.12025650515289120796... 7 3 3
15
2 1
2 31
1
k
k kk
k
k
( )
( )
.12041013026451893284... ( )
( ) ( )3
12
1
2 1 2
2
21
H
k k kk
k
.12044213230101764656... ( )
( !){},{ , ,},
11 1 1
30
k
k kHypPFQ
.1204749741031959545... e e ee
e
k
e k
k
kk
log( )( )
( )
11
1
10
1 .12049744137026227297...
2)1(coth
2122
1 i
k
H
k
k
= i
i i i i4
1 1 1 12 2 1 1 ( ) ( ) ( ) ( )
.1205008204107738885... ( log )
arcsin log1 2
8 0
1 x x x dx
.1205922055573114571... 2 2
5
47
300
1
1 5
1
50
1
21
log
( )( )
( )
k k k kk
k
k
= log log( )11
11
64
0
1
x
dx
xx x dx
1 .12065218367427254118... 3
21
230
log( )k
k k
1 .1207109299781110955...
112 8
)2()14(
188
1
22tan
24 kk
k
k
k
kk
=1
2 1
1
8 1
6
2 1 8 12 21
2
2 21k k
k
k kk k
( )( )
.12078223763524522235...
log log log
2
1
2
4 3
2
=
1
2
1
1 20 e
dx
e xx x GR 3.427.3
=0
2
2
sinh
cosh
x dx
xe xx GR 3.553.2
.1208515214587351411... 3 3
212 2 10
1
1
1
3 31
( )log
( )
( )
k
k k k
.12092539655805535367... 1
642 kk
.12111826828242117256... 3
30256
1
4 2
( )
( )
k
k k
.12112420800258050246...
( )
1
2
1 2
1 22
1
k
kk
k
kk
.12113906570177660197... 1
6
1
2 2
14
2
2 2 2 21
coth
( )k kk
.12114363133110502303... log5
6
.1213203435596425732... 3
22
1
4 1
21
0
( )
( )
k
kk
k
k
k
k
= sin tan/
x xdx2
0
4
.121606360073457727098... ( )2 1
4 11
kk
k
1 .12173301393634378687...
3
1
9
1
29
)2(4 )1(2 g
=log logx
xdx
x x
xdx3
0
1
311 1
.12179382823357308312... ( )3
2
.12186043243265752791...
0 )62(2
)1(3,1,
2
1
2
1
2
3
2
3log4
kk
k
k
=
123
1
1
2
3
33
1
)2(2
)1(
k kkk
k
kkkk
1 .12191977628228799971... 14
153
41
( )( )
a k
kk
Titchmarsh 1.2.13
.12200602389514317644...
332
1
31
31
2
1 iH
iH
ii
=
1
1
9
)12()1(
kk
k k
.12202746647028717052... 3
66 2
9 3
21
1
2 30
loglog
log( )x
dx
.12206661758682452713... 1
8 42
1
4
1
21
csch( )k
k k
=sin cosx x
edxx
10
55 .12212239940160755246...
03
31
)2(3
)(
12
3
1
9
34)3(26
k k
.122239373141852827246... ( ) ( )
1 3
9
1
9 1
1
13
1
k
kk k
k
k
1 .12229100107160344571... ( ) ( )
( )4 5 1
51
k
kk
HW Thm. 290
=
14
1
( )k
kk
.12235085376504869019... 12
21
2 22
log
( )k
k k k
2 .12240935830266022225... ( )k
k kk
22
.12241743810962728388... 21
4
1
2 42
1
1
sin( )
( )!
k
kk k
.12244897959183673469... 6
49
1
6
1
1
( )k
kk
k
1 .122462048309372981433... 2 11
6 11 6
1
1
/ ( )
k
k k
.122479299181446442339...
1 2
122
42
)1(
4
171log2
4
1arcsinh2
k k
k
kk
k
.12248933156343207709... 1
2
1
4
1
8
1
16 2 103 1
2
arctan( )
( )
k
kk k
dx
x x
.12263282904358114150... 2
2
0
18 2 11
361
1
loglog logx
xxdx
=log( ) logx x
xdx
14
1
.1228573089942169826... 7
21
1 2
1 2
0e
k
k
k
kk
( )
( )!
.12287969597930143319... loglog log ( ) log ( )
( )2 1
2
2
2
3
3
4
1
1
2 3 2
20
1
x
x xdx
.1229028434255558374... sin ( )
!( )
2 2
2
1 21
20
k k
k k k
.1229495015331819518... 325
48
15 1
12
51
( )
( )
k
k k
.1230769230769230769 =8
65
1
2 81
1
3 2 1 2
cosh log
( ) ( ) logk
k
ke J943
4 .12310562561766054982... 17
1 .12316448278630278663... 3
2
1
3
1
3 2 10
erfi
k kkk
! ( )
2 .12321607812022000506...
1
221 )12(
2
2
)12(
2
1
2
1
24
1
kkk k
kkk
.123456789101112131415... Champernowne number, integers written in sequence
.123456790123456790 = 10
81 10
1
21
4
1
k kk
k
k
kk
( )
.12346308879239152315... 37
226803
62 ( ) 18 MI 4, p. 15
= 2
36
1
32 4
1
32 2
13 2
2
41
( ) ( ) ( ) ( ) ( )( )
H
kk
k
.12360922914430632778... 1
3
2
9 121 1
k
kk
k
k
( )
.12370512629640090505... 21
22
1
2 4
1
1
cik k
k
kk
log( )
( )! ( )
.12374349124688633191... 355
1082 4 3
4 31
( ) ( )( )
k
kk
1 .123912033326422... dxx
1 )(
11
8 .1240384046359603605... 66
.12414146221522759619... 3
2
4
3
1
3 12
1
1
log( )
( )
k
kk
k
H
k
.12416465383603675513...
224
1
21
21
4
1
2
iH
iH
ii
=
12 )14(2
1
k kk
=( ) ( )
1 2 1
2
1
2 11
k
kk
k
529 .124242424242424242424 = 174611
330 20 B
.12427001649629550601...
2 41
1
4 coth ( ) ( )i i
= 311114
1 iHiiHi
=1
1
1 1
13 22
3 12
42k k k k k kk k k
.12429703178972643908... 4
251
4
5
1
4
1
1
log( )k
kk
k
kH
.12448261123279340605... 5
2
5
4 5 12
1
log( )
H
kk
kk
.124511599540703343791... ( )( )
18 1
1
1
kk
k
k
.12451690929714139346... 1
9 32
1
6 4 6 11
log( )( )k kk
J263
1 .124558536969386481587... dxx
x
1
1
33 arcsin)3(
8
92log
4
.1246189861593984036...
0 )3)(1(22
152log11
kk kk
k
1 .124913711872468221...
122
2
)12(
112
3
22log8
k kk
.12493815628119643158... dxx
x
1
04
2
16
log
4
1,3,
16
1
512
1
.12497309494932548316... ( )
1
2 1
1
1
k
kk
k
.12497829888022485654... sinh sin cos( )
cosh( )/ /2
4
2
4 2
1
2
1
2
1 4 1 4
i
=sin /
( )!
k
k kk
2
2 41
.12500000000000000000 = 1
8 4 12 21
k
kk ( ) J364
=
1 )3)(2)(1(k
k
kkk
H
=
1
5
)1(kkke
k
= ( )k
kk 4 11
= x x xdx3
0
1
arcsin arccos
= log 11
21
5
x
dx
x
1 .125000000000000000000 = 9
83
2 H ( )
4 .125000000000000000000 = 33
8 3
3
1
kk
k
.1250494112773596116... 1 2 12 4 6 3 8 52
3
51
( ) ( ) ( ) ( )( )
k
kk
.125116312213159864814...
2
2 21
26
152
2
4
1 1
( )( )
( )
( ( ) )k
k nk n not squarefree
.12519689192003630759... 1
6
4
9 3 2
1
2
2
20
2
( sin )
( sin )
/ x
xdx
.12521403053199898469...
1
8192
1
8
1
16 22
30
( )
( )kk
.12521985470651613593... ( )
( )! ( )
1
1 2 2
1
1
k
kk k k
Titchmarsh 14.32.1
1 .12522166627417648991... dxx
x
0
4
442
sinh)4()2(2
453
2 .125353851428293758868...
16
1
288
5
22680
97
4
)3(9)3(2)3(2417
962
)5(9 462
2
=
14
2
k
kk
k
HH
1 .1253860830832697192... 12 1
10
1
eEi Ei
e dx
x
x
( ) ( )( )
2 .12538708076642786114...
1
22
2
22log
6 kkkH
.12546068941849408713...
1
3456
1
6
1
12 102
31
( )
( )kk
1 .12547234373397543081... csc ( )3
6
1
18
1
9
1
2 21
k
k k
.12549268916822826564... 3
6
5
3
1
12
7
3
3
18 3 32
arctan log
dx
x x
.12552511502545031168... ( )k
kLi
k kk k4
24
1
1 1
.1255356315950551333...
2 22
)(11)()(1
k nk
k n
kkk
=
S )1(
1, where S is the set of all non-trivial integer powers
=
poweregernontriviala
int
11
3 .125552388163504646336...
1
2
)2(2
1
122
)3(3
k
kk
kk
HH
.12556728472287967689... 1
22 2
7
2
1
4
1
33 3
5
2
1
42 1 2 1F F, , , , , ,
=
2
51log
5
11
25
42arccsch
125
54
25
4
=
1
21
2)1(
k
k
k
kk
.12560247419757510714... 7
2 15 9459 3 3 5 28
1
1
2 4 6
6 31
( ) ( )( )k kk
.12573215444196347523... ( )k
kLi
k kkk k2
1
2
1
222
21
2 .12594525836597494819... 2
92 1
2
2 21
sinh( )
kk
.12596596230599036339...
1
1458
2
9
1
9 72
31
( )
( )kk
.12600168081459039851... Likk
k4 4
1
1
8
1
8
.12613687638920210969...
5252156
12log531log
20
1
5
2213arctan5210
5
2213arctan552
10
1
=dx
x x3 22
.1262315670845302317...
135
42
)1(
1)5()3(5
302
521
k kk
.12629662103275089876...
3
23
1512
7 3
128
1
1024
1
4
1
8 6
( )
( )( )
kk
.126321481706209036365...
3
2
2
31
3
1
1
13
20
1
log loglog
x x x
dx GR 4.325.5
.1264492721085758196... 1
2 3 3
2
3
1
3 506 3
1
log ( )k
k k
dx
x xdx
.12657458790630216782... 1 3 2 4 3 1 12
2
( ) ( ) ( ) ( ) ( )k
k
k k k
=4 2 1
1
3 2
2 31
k k k
k kk
( )
11 .12666675920208825192... 11 3 1
6 2 1 12 110
dx
x /
1 .12673386731705664643... 7
2
.12687268616381905856... 11 4 e
2 .12692801104297249644... log 1 2 e
.1269531324505805969... 1
2
3
8 131 1
k
kk
k
k
( )
1 .12700904300677231423...
4
34
3 3
4
3 3
11
3 1
2
21
i i kk ( )
.12702954097934859904... Likk
k3 3
1
1
8
1
8
.1271613037212348273... 3 2
4 8
1
16 4 421 2
log ( )
k k
k
kk
k
= dx
x x x x4 3 21
.12732395447351626862... 2
5
3 .12733564569914135311... 2 22 1
1
2( ) /
k
k
.127494719314267653131...
( )2 12
1
k
kkk
.12755009587421398544... ( ) ( ) ( )( )
4 10 2 2 3 151
14 31
k kk
.12759750555417038785... 13 3
216
4
1296 3
1
432
1
3
1
6 4
32
31
( )
( )( )
kk
1 .12762596520638078523... cosh( )!
/ /1
2 2
1
2 4
1 2 1 2
0
e e
k kk
AS 4.5.63
= 11
2 12 20
( )kk
J1079
.1276340777968094125...
1
1
11
1
122
)1(
12
)1(
kkk
k
kk
k
.12764596870103231042...
( )
( )
4 1
2 1
1 .1276546553915548012... 4 26
1
2
1
2
2
3 21 1
log( )
k k
k
kk
k
=2
1
0
22 )(
x
dxxLi
.12769462267733193161... 21
4
1 1
2 42
1
arcsin( )!( )!
( )!
k k
k kk
K ex. 123
.12770640594149767080... 1
4
5
3 142
log
xdx
x
.12807218723612184102... 120
1
( )kk
k
.12812726991641121051... 1
4
2
4
3
12 1 2 32 2 20
dx
x x x( )( )( )
28 .12820766517479018967... k kk
kk
k
3
1
3
12 1 2
( )
1 .1283791670955125739...
1 2
11
2
k k
k
.12850549763598564086... sin ( ) cos ( ) ( ) sin1 1 1 1 1 1ci si
= x x x dxlog sin( )10
1
1 .12852792472431008541... 4/34/14/1 )1(cot)1(cot4
)1( i
= k
kk
k
k
k
2
41
1
11
1
21 4 2 1
( ) ( )
.12876478703996353961...
112 )33(2
1
396
1
3
1
3
2log2
kk
k kkk
=4
13
11log
x
dx
x
1 .12878702990812596126... 5
6
.12893682721187715867... 1
2 21
k kk
( )
.1289432494744020511... Jk k
Fk
k3
00 12
1
34 1( )
( )
!( )!; ;
LY 6.117
.12895651449431342780... 1
250
2
5
1
5 22
30
( )
( )
kk
.12897955148524914959... 1
81 ( )k kk
.12904071920074026864... 888
5 51
2 1
161
1
( )kk
k
k
k
k
.129111538257100692534...
123
2
3 4
log)4(
8
1
2
1dx
xx
xLi
.12913986010995340567...
122 8
1
8
1
kk k
Li
1 .12917677626041007740... 12 1
1 k kk
.12920899385988654843... 27
23 1 3
1
9 5 31
log ( )
k kk
1 .12928728700635855478... 1
17 6 5 4 3 20 k k k k k k kk
.12931033647404047771... 243log331728log3124314
1
322
333)32log()32(64log31
4
3
=
1
1
12
)1()1(
kk
k k
.1293198528641679088... 2 1
2112
21
1
log( )
( )
k
k
k
k J347
= k
k k
k k
kk
k
1
4 2
1
23 21 2
( ) ( )
=log
( )
log
( )
log
( )
2
31
2
30
1
20
1
1 1 1
x
xdx
x x
xdx
x x
xdx
.12950217839233689628...
ee
k
k
k
1
41
/
.12958986973548106716... ( )3
12
1
4
2 12
6 5 4 32
k
k k k kk
= k k kk
( ) ( )2 2 12
2 .12970254898330641813... 1
1 1 13
0 ( !){},{ , },
kHypPFQ
k
3 .12988103563175856528...
iii
2
1
2
1tanh
.12988213255715796275... 13 3 13 2
2 1k k kk k
k k
( ) ( )
.12994946687227935132... 1
6 CFG A28
1 .13004260498008966987...
Gx
x xdx
2 8
7 3
42
1 1
2 2
20
1( )log
arccos
( )( )
.13010096960687982790... 18
7 3
4
2 1
2 3
2
30
( )
( )
k
kk
1 .13020115950688002... Hk
kk
3
1 3
.13027382637343684041... 1
4
1
2 2 40
sinh( )!
k
k kk
.130287258904574144837...
2
5553
2
552
51
1 )1()1(
=
1
1)1()1(k
kk kkL
.13030594417711116272... dxx
xx
081
)1(122
24
.13033070075390631148... 1
22 1
12
2
log( )
k
k kk
= 11
21
1
2
k
kk
kk
( ) log
.13039559891064138117... 101
12
3 31
k kk ( )
K ex. 127
.13039644878526240217... ( )
( ) ( )( )
k
k k k kmk m
kj j
jk mk
1
1
11
2 22 21
1 .13039644878526240217... 02 12
1 1 1( ) ( ) ( )k k jkk kj
=1
1
1
1 122 2 2k
k
k k
k
kjjk k k
( )
( )
( )
=
1 2
)(
m kmk
k
2 .13039644878526240217... 02
1( ) ( )k kk
.13039943558343319807...
4 32
2
2
22
0
4
log
tan/
x xdx GR 3.839.1
1 .13046409806996953523... log
( )
( ( )) logk
k k
k k
kk k2 2 2
22
21
1
=j k
k jkj
log2
21
2 .13068123163714450692... 3
4
1
4
1
20
e
e
k
kk
( )!
.13078948972335204627... 1)1(122log32log36
1
=
2
2
2
11log
2
1
3
1
2
1)()1(
kk
k
kkk
kk
k
.13086510517299719078... 8
49 281 2
4 2
7
2
71 2
( )
loglog( )
= 1
8 71 k kk ( )
Prud. 5.1.7.24
.13086676482635094575... 1
8 22
1
420
csch( )k
k k
1 .131074170771424458336... 2
3cosh
2
323cos
2
323cos
13
ii
=
1
33 )1(
11
k kk
1 .13114454049046657278... cos cosh
( )!
/ / 1 4 1 4
02 4
k
k k
.131263476981215714927... 2
2
0
1
44 6 G x xdxarcsin log
1 .131279000668548355421... i i( ) cos ( ) ( ) cos ( )/ / / /1 1 1 13 4 1 4 1 4 3 4
.13129209787766019855...
8
728log
7
8log
7
8log
2
1
8
7
8
1)3( 233 LiLiLi
=H
kk
kk 8 2
1
1 .13133829660062637151... 1
1 k kkk !
.131447068412064828263... )2,3()2,3()2()2(2
1 )2()2( iiii
=x x
e edxx x
2
0 1
cos
( )
.131474973783080624966... 7 3
64
1
4 3 31
( )
( )
kk
.13150733113734147474... ( )( )
11
2
3k
k kk
4 .1315925305995249344... 1
22
( )kk
.13178753240877183809... log
arctan7
6
3
6
3
3
5
3 132
dx
x
.13179532375909606051... k
k k k k k ks ds
k k
1 1 1
142
3 42 3
4
log log log( ) )
.131928586401657639217...
2
3 31 2
414
41
384
3
4
1
4
1
1 2
Gk
k
k
k
( ) ( ) ( )
( / )
.13197175367742096432... 4
2
21
1
2 2 2 30
log ( )
( )( )
k
k k k
=
12 168
1
k kk
= 12
( )kk
=
134
2/1
0
2 11log)41log(
x
dx
xdxx
1 .13197175367742096432... 4
2
2
1
2 2 1
1
1
log ( )
( )
k
k k k J154
=
12 3108
1
k kk
=
1
2/
1
)1(
k
k
k Prud. 5.1.2.5
= xx
dxlog 11
40
1
= log( )
11
2 10
x xdx
=
12
arctandx
x
x
.13197325257243247...
ieLi e e Li e
k
k
ii i i
k2 132
3 31
( ) ( )sin
( )
3 .13203378002080632299...
3 3
2 2 3
1
3
13
13
log ( )
( )
GR 8.336.3
.1321188648648523596...
1
1
2)1(
3
2log1
9
2
kk
kk kH
.1321205588285576784... 1
2
1 1
2
1
1
e k
k
k
( )
( )!
.13212915413764997511... 6
7
7
6
1
6
1
1
log( )
k
kk
k
H
.13223624503284413822... 1
742 kk
.13225425564190408112...
( )5
21
kk
k
.1323184679680565842...
1
3
31
33
32 cos
)1()(3)(4
1)(
8
5
k
kii
k
keLieLii
11 .13235425497340275579... 5815
729 33
6
0
ek
k kk
!
.13242435197214638289... log( ) 2
.13250071473061817597... 3
21
2 2
7
1
14
1
3 3 20
log ( )
( )( )
k
k k k
25 .13274122871834590770... 8
1 .132843018043786287416... ( )( )
12 1
1
1
kk
k
k
.13290111441703984606... 1 1
2 21
2 22 2 1
0e eek k
k
cosh( ) ( ) J943
1 .133003096319346347478...
4
9
.13301594051492813654... 2 24
3 2
23
1
2 1
2 2
2 21
loglog
( )
k
k k k
.13302701266008896243... 116 8
5
8
2
4 8
3
8
cot cot logsin logsin
= ( )
1
4 506 2
1
k
k k
dx
x x
1 .13309003545679845241...
4 2 2 20log
dx
x x
.13313701469403142513... ( )1 8
1 .13314845306682631683... ek kk
kk
k
8
0 0
1
8
1
2
! !!
.13314972493191508633... 1
0
2 arctanlog1216
1dxxhxx
.13316890350812216272... 1
2 2 5 5
1
5 1
1
21
csch
( )k
k k
.1332526249619079565... 2 2 2 22 1 2
2
1
log log( )( )
H
k kk
kk
1 .1334789151328136608... 3 5 2 3 41
( ) ( ) ( )
H
kk
k
.13348855233523269955... 5
12
2
32 1 2
2 1
4 2
1
1
log log( )
( )
k
k k k
k
kk
1 .13350690717842253363... ( ) ( )2 2 1 11
k kk
.13353139262452262315... log log8 71
8
1
811
Likk
k
= 21
152
1
15 2 12 11
arctanh
kk k( )
K148
.13355961141529107611... 3
161
3
4
1
3
1
1
log( )k
kk
k
kH
.13356187812884380949... ( ) log( ) log3
9
1 3
0
1
x x
xdx
6 .13357217028888628968... 2 1
1
k
k k kk
!
.13360524590502163581...
1
21
1 1
21
1 2 12
1
( )kk
k
=1
12 21 k kk ( )
.133790899699404867961... 1)1()2)(1(
)1(
9
4log
3
8
1
1
kkk
k
k
k
1 .13379961485058323867... 7
2
2
7
2 1
7 2 10
arcsin( )
k
k kkk
.1338487269990037604...
1
0
2sin
10
52sin22cos
5
2xe
dxx
e
.1339745962155613532... 13
2
1 2 1
2 3
1
1
( ) ( )!!
( )!
k
kk
k
k
.13402867288711745264... 1
1 122 ( )
logk
k
kk
.13429612527404824389... 1
162 3 3 83 2 2 2 coth cothcsch csch
=1
12 31 ( )kk
.13432868018188702397... coth
( )
2
4
1
8
3
8 1
1
4
4
4 2 21
e
e kk J951
.134355508461793914859...
03 27327
2
x
dx
.13451799537444075968... arctan12
2e
dx
e ex x
.13458167809871830285... ( )
1
3 2
1
1
k
kk
1 .13459265710651098406... 1
11
12
1arcsinh
arcsinh2
2
kk
1 .13476822975405664508... 1
2 11
2
0
1
kx dxk k
k
x
/ Prud. 2.3.18.1
2 .13493355566839176637... e e
xdx
x4 1 4
0
.13496703342411321824... 2
2 2112
11
16
1
1
( )kk
J400
= 1
2
1
12 21
4 2 22k k k kk k( ) ( )
= ( ) ( )k kk
1 2 12
1 .13500636856055755855... H
k
k
k
k
2
1 2
.13507557646703492935... cos ( )
( )!
1
4
1
2 1
2
1
k
k
k
k
.13515503603605479399...
11
1
3
1
3
1
2
)1(
2
3log
3
1
kk
kk
kEk
LiH
1 .1352918229484613014... dxx
xx
1
0
2
1
log
9
49
3
2
.13529241631288141552... Ai ( )1
20 .13532399155553168391... 2 3 3 3cosh e e
.1353352832366126919... 1 1 2 1 2 1 1 2 3
20
1
00e k
k
k
k
k
k k
k
k k k k
kk
( )
!
( ) ( )
!
( ) ( )
!
=( )
( )!
1 2
2
1
1
k k
k
k
k
=
03
3
)(
logdx
ex
x
1 .1353359783187...
( )
( )
k
kk4
2
.13550747569970917394... )12(2)1(cosh2
12
1
1
k
eek
k
k
=cos
cosh
2
0
x
xdx
GR 2.981.3
.135511632809070287634... 1
41
1
4
1
2
1
4 1 2
0
e dx
e
x
x
1 .13563527673789986838... 9 .13574766976703828118... I2 1( )
.13577015870381094462... 5
14
2 1sinh
2
1tanh1sinh
4
1
2
1sinh
2
1
4
11cosh
x
dx
x
.13579007871686364792... k
kk
!
!
10
.13583333333333333333 =163
1200
1
1 51
k k kk ( )( )
5 .13583989702753762283... k kk
3 2
1
1 ( )
.13587558784724491212...
1
1
1)1()2)(1(
2)1(
6
1log
2
5
k
kk
kkk
k
.13593397770140961534... 11 182 2
2
1
ek
k kk
( )!
.13601452749106658148... ii 11sinh2
=
12
1
22 22
)1(
1
)1(
k
k
k
k
kkk
=cos(log )
( )
x
xdx
1 20
1
GR 3.883.1
=cos(log )
( )
x
xdx
1 21
= dxee
xxx
0 )1
sin
.13607105874307714553... 16 23
4
1
41e k k
k
k
( )
!( )
1 .13607213441296927732... ( )k
kLi
kk k2
22
2
1
3 .13607554308348927218... 2 4 21
2 2 14
3
0
1
( )log
( )( )
Lix
x xdx
.13608276348795433879... 1
3 6
1
8
2
0
( )k
kk
k k
k
.13610110214420788621... 1
8 11
3
831
1
1k
k
k
k
kk
( )( )
=
4
33
)33(6
33
4
33
)33(6
32
3
2log
3
1
6
i
i
ii
i
1 .13610166675096593629... 4 11
1 41 4
1
ek k
k
/
( )!
1 .136257878761295049973...
0 0 24
1
4)!2(
!!
4
1arcsin
1515
16
15
16
k k kk
k
kk
kk
.13629436111989061883... 2 25
4
1 1
1 2 332 0
log( ) ( )
( )( )( )
k
k
k
kk k k k k GR 1.513.7
=
2
1
2
1)1(
)1(2
1
1
1
2k
k
k
kk
Lik
.13642582141524984691...
2122
)()1(111
1
kk
k
k
k
kk
2 .13649393614881149501... 11
1 1k
k k
( ) ( )
.13661977236758134308... 2 1
2
2 1
2
1
2 2
2
1
( )!!
( )!!
k
k kk
J385
= ( )( )!!
( )!! ( )( )
12 1
2
4 1
2 1 2 21
1
3
k
k
k
k
k
k k J391
.13675623268328324608... 13 3
27
2
81 3
1
54
2
3
1
3 2
32
30
( )
( )( )
kk
.13678737055089295255... 5
4 2 6
5
2
1
4
2 2
4 21
coth
k
k kk
= ( )( ) ( )
1
42 2 2
1
k
kk
k k
1 .13682347004754040317... k
kk
k
k
k
3
51
1
11
1
21 5 3 1
( ) ( )
.13685910370427359482... ( )
( )3
26
1
1
2
2 31
k kk
.13695378264465721768... 1 33
4
1
4 11
log( )k
k k k J149
.13706676420458308572...
22 1
21
3
3sin
k k
.13707783890401886971... 2
2172
1
6 3
( )kk
=1
12 3
1
12 92 21 ( ) ( )k kk
=
1 224
1
k kk
k
= log 11
61
x
dx
x
.13711720445599746947...
1
32
log)(
)3(
)3(
k k
kk
14 .13716694115406957308... 9
2
1 .137185713031541409889... 2 2 2
212
2
2
2
28 2
7 3
2 2 1
log loglog
( )
( )
H
k kk
k
.13737215566266794947...
( ) ( )
( )
3 5
3
.13744801974139900405... 2
9 12
2
2
1
8 61
log
( )k kk
.13756632099228274... ( )
1
232
k
k k
.13786028238589160388... 1
2 22 2 1
0sinh( )
e k
k
AS 4.5.62, J942
2 .13791866423119022685... 4
5
2
5 5
2
5
4
5 1 5 20
csc /
xdx
x
.13793103448275862069...
1
21
4
)1(
29
4
kk
kk F
1 .13793789723437361776... e
e xdxx1 4
02
2/
cosh
12 .13818191912955082028... dxx
0
121log312
.13822531568397836679... log log 11
111
1
ee ek ekk
= ( )k
e kkk
1
1 .13830174381437745969... I e
eF e
e
k k
k
k
2
0 10
2 1
23
2
; ,!( )!
1 .13838999497166186097... 1
11 k k
k
1 .13842023421981077330... 5
81 3
3
361
1
3 1
1
3 2
1
3 3
3
3 3 30
( )( )
( ) ( ) ( )k
k k k k
.1384389944122896311... 2 212
422
0
1
log ( ) log
K k
k
dk
k GR 8.145
1 .13847187367241658231... 9 3 9 3
4
13
231
log ( )!
( )!
k
k kk
.13862943611198906188... log 2
5
.13867292839014782042... 4 2 2 2 31
16
2
1
log log( )
kk k
k
k
=( )!!
( )!
2 1
2 41
k
k kkk
1 .13878400779449272069... 2 21 12 1
4 1
2
2 21
Gk
k k
k
k
log( ) ( )
( )
.1388400918174489452...
16 2 1640
dx
x
.138888888888888888888 = 5
36
1
2 3
1
51
1
1
k k k
k
k
k
kk( )( )
( )
= k
k k k kk
2
1 1 2 3 4( )( )( )( )
2 .13912111551783417702...
0 2
)(22
kk
kpfeee
.13918211807199192222...
1
1
1
2
)!12(
!!)1(
5
520
2
1arcsinh
5
41
k
k
kk k
kkk
.1391999869582814821... 2 1 2 1 2 2 2 2arcsinh log log
=( ) ( )!!
( )! ( )
1 2 1
2 2 1
1
1
k
k
k
k k k
.13920767329150225896... 1
31
2
4 2 16
2 2 2
20
4
log tan
cos
/
Gx x
xdx GR 3.839.4
.13929766616936680005... 1
6 2 6
3
2
1
2 3
1
21
csch( )k
k k
.13940279264033098825... 2
1 12
1
erf e dxx( )
.1394197585790642108... e
e x
dx
x x
dx
x4
5
41
1 1
2
12
17
14
sinh sinh
.13943035409132289... )3(2
5
2 .13946638410409682249...
4
1,
2
3,2,2
3
4
39
412 F
=
111
2
22
12)!12(
)!(
k
k
kk
k
k
k
kk
k
k
1 .13949392732454912231... sec ( )( )!
1
21
2 40
2
k
k
kk
E
k
.13949803613097262886... log log2
3
3
12 42 2
2
H kk
k
3 .13972346501305816133... log e e
.1398233212546408378... 1
5
1 2 1
5 2 20
1
cos sin sin
e
xdx
e x
.14000000000000000000 =7
50
=1
2 71
0
2 1 7
cosh log( ) ( ) log
k
k
ke J943
.14002410170685231710... 2/
0
324 sinlog)5(93)3(182log2128
1
dxxx
.14002478837788934398... 1
4
.14004960899154477194.. 1
121 e k
k
Berndt 6.14
.14018615277338802392...
01 4
)1(
)12(2
12log
6
5
k
k
k kkk
=
12 6104
1
k kk
= ( )( )
11
22
kk
k
k
=dx
x x5 41
1 .1405189944514195213...
0 )12(3
1)32log(
2
3
3
1arctanh3
kk k
.14062500000000000000 =9
64 91
kk
k
2 .14064147795560999654... ( )9 23 .14069263277926900573... e i i 2 Shown transcendental by Gelfond in 1934
18 .140717361492126458627...
2 2
2
1
sinh( )!
k
k
k
k
.14095448355614340534...
4
1
4
3)3(92log2248
2048
1 )3()3(2 G
= 4/
0
3
tan
dxx
x
2 .14106616355751237475... e
.14111423493159236387... 1
2 221
kk
k
.14114512672306050551...
1
33333 cos
1111log8
1
k
iiii
k
keeee
1 .14135809459005578983...
Lik k
k
kk
k
k2
22
2
2 2 2 1
( )
7 .1414284285428499980... 51
10 .141434622207162551... 2
32 2 4 3 4
1
2
22
2
2
12
30
1
log loglog
( )Li
x
xdx
. 14147106052612918735...
0
2
)2)(1(16
2
9
4
kk kk
k
k
k
2 .14148606390327757201... 3 2 32 3 1
1
1log log ( ) /
k k
k
.14155012612792688996... 323
2
77
6
1
2 50
log( )
( )
k
kk k
.1415926535798932384...
3 16
1
( )sink
k
k
k
1 .1415926535798932384...
2 14
1
( )sink
k
k
k
= arccos2
0
1
xdx
= x xdx2
0
2
sin/
3 .1415926535798932384...
= 1
2
1
2,
=
110443
1arctan48
239
1arctan20
57
1arctan128
49
1arctan48
Borwein-Devlin p. 74
=
12943
1arctan96
682
1arctan48
239
1arctan28
57
1arctan176
Borwein-Devlin p. 74
=
0 32
650
k k
k
k
k Borwein-Devlin p. 94
=3 1
41
8
4 1 4 31 1
k
kk k
kk k
( )( )( )
Vardi, p. 158
=dx
x
e dx
e
x
x1 12
2
/
=x
xdx
10 sin
=dx
x2 20 cos
= log log1 13
12
02
0
x dxx
dx
= log( )1 22
0
xdx
x GR 4.295.3
=
02
22
1
)1(logdx
x
ex
= log22
0
1
211
x
xx dx GR 4.298.20
=
02
11loglog dx
xx
=
02
22
1
))1(log(dx
x
ex
= 1
0
2logarccsc dxxx
6 .1415926535798932384...
3 2 23
2
1
2
222 1
1
Fkk
k
k
k
, , ,
7 .1415926535798932384...
1
1
22
4k
k
k
k
12 .14169600000000000000 = 122214
15625 6
6
1
kk
k
.141749006226296033507... log ( ) log2 2
0
1
1 x x dx
.14179882570451706824... log log2
2 32 4
2
8
2
=
40
4
x x xdxtan tan
/
GR 3.797.1
.14189224816475113638...
e
B
kk
k
k
1 0 ! J152
.14189705460416392281... arctan( )
( )
1
7
1
7 2 12 10
k
kk k
=
1
21
)2(
2arctan)1(
k
k
k
[Ramanujan] Berndt Ch. 2, Eq. 7.5
= arctan( )
1
2 3 21 kk
[Ramanujan] Berndt Ch. 2, Eq. 7.6
1 .142001361603259308316... 7 3
2 28
1 1
2
2
12
( ) ( )( ( ) )
k k kk
k
.142023809668304780503... ( )( )
17 1
1
1
kk
k
k
.1421792525356509979...
1
1
)14()!2(
!)!12(2log2
4
3
4
14
k kkk
k
1 .14239732857810663217... 4
11
.142514197935710915709... 6 42
4
2
46
1
2
21 3 2
4
2 3 4
4
( )log log ( ) log
Li
= log ( ) log3
0
1 3
1
21
1
x
xdx
x
xdx
.14260686462899218614...
1 2
)()(
kk
kk
1 .14267405371701917447... 2
22 2
125 5
1
5 1
1
5 4csc
( ) ( )
k kk
.14269908169872415481... 8
1
4 12 21
dx
x( )
.1428050537203828853...
1
3
2)1(
3
2log
27
2
81
14
kk
kk kH
.14280817593690705451... ( ) ( )
1 1 1 12
22
2
k
k k
k
k kLi
k
.142857142857142857 =1
7
1 .14327171004415288088...
3G
3 .1434583548180913141... log e e
.14354757722361027859... 3
216
.14359649906328252459... ( )
logk
k k k kk k
1
1
1 11
1
32
2
3 .14376428513040493119.... ( )k
Fkk
2
1 .143835643791640325907... e e ek
ke e
k
i icos cos(sin )cos
!1
0
11
2
GR 1.449.1
= )1sinh(cos)1cosh(cos)1cos(sin
.14384103622589046372... 1
2
4
3
1
7
1
7 2 12 10
log( )
arctanhk
k k K146
= dx
x x
dx
x x x32 0 1 2 3
( )( )( )
.14385069144662706314... 5
2
4
14 2 2
e
x
xdx
cos
( )
.143954174750746726243...
8
55log51125
5
22log55
4
5log5310
535
1
=
1 )1(4kk
kk
kk
FF
1 .14407812827660760234... Li2
5
6
1 .14411512680284093362...
1
2
2
)14(
18
4222log3
k kk
kG
= ( ) ( )k k
kk
1 1
41
.14417037552999332259... Likk
k4 4
1
1
7
1
7
.14426354954966209729... 23
2
2
3
1
2 1
1
1
log( )
( )
k
kk
k
k
.14430391622538321515... 4
4 2
0
e edx
xx x GR 3.469.3
=
e x xdxx 2
0
log
3 .14439095907643773669... 2 2
2
1( )kk
.14448481337205342354... e e ek
k ee e
k
kk
1 1
1
1
11
1/ / ( )
( )!
.14449574963534399955... H
kk
kk
( )
( )
3
1 3 2 1
2 .14457270072683366581... 2
41
k
k k k!
.144713210092427431599...
123
2
3 3
log)3(3
9
2
xx
dxxLi
.144729885849400174143...
12 )2(4
)2(loglog1
kk kk
k
e
1 .144729885849400174143...
0
2/1log
2
1)1(log
kk
k
k
k
k Prud. 5.5.1.17
.14476040944446771627...
15
21 )(
1
2
3,532)5(31
k k
1 .14479174504811776643... 1 1 1 2
2 221
1
1k k
k
kk
k
k
cos
( ) ( )
( )!
.14490138006499325868... 8
25 102 1
4 2
5
2
51 2
( )
loglog( )
= 1
8 51 k kk ( )
Prud. 5.1.7.23
.14493406684822643647...
2
3 22
1
16
3
2
11 2 1
k k
kk
k
k
( ) ( )
=H
k k
k kk
kk
k2
2 1
2 2 2
4
( ) ( )
=4
1 12 2 20
x
x edxx( ) ( )
= log( )1 2
0
1
xdx
x GR 4.29.1
1 .14493406684822643647...
2
26
1
21 1 2
( ) ( ) ( )k
k
k k
2 .14502939711102560008... 2 3/
.14540466392318244170... 381
2187
1
2
1 7
1
( )k
kk
k
.14541345786885905697... 2 112
21
log( )eke k
k
GR 1.513.3
.14544838683609430704... 1
25
3
58 25
8 5
5
1
25
7
51 2
21
( ) ( )
=1
5 2 21 ( )kk
.14552316993048935367... Likk
k3 3
1
1
7
1
73 0
1
7
, ,
1 .145624268262139279409... 2 3 2
12 32 18
3
2
1
3
3
162
2
3 4
7 3
4
GGlog
( )
= H
kk
k ( )4 3 21
.14573634739282131233...
0
2coslog442arctan85log3100
1
25
11dxxxex x
.14581354982799554765... ( )
!( )!
k
k kk I
k kk k
1
21
1
221
1
.145833333333333333333 =7
48
1
4
1
1 5
1
421 0
1
23
k k k k kk k
k
k( )( )
( )
= x x dx3
0
1
1log( )
= log 11
15
x
dx
x
.14583694321462489756... )2(56)4(140)6(56log24
53
=
1 4
1)2(
k k
k
.14585774315725853880... log ( )!!
( )!!
2
2
1
4
2 1
2
2 2
2
1
2 22
2
G k
k kk J385
.14588643502776689782...
1
21
)12(2
)1(
kk
k
kk
.14589803375031545539... 4
2
537
.14606785416078990650... 2 22
21
1
2
2 1
1
loglog ( )
k
k
k
H
k
3 .1462643699419723423... 32
.1463045386077697265... 44
2 21
2 1
2
21
log
( )k kk
GR 0.236.7
= log( ) log1 2
0
1
x xdx
.14630461084237077693... 2 21
2
3
22
1
2 1 2
1
1
arctan log( )
( )
k
kk k k
.14632994687169923278... 1
842 kk
.14641197161688952006... cos
/
x
edxx 2
0 1
.146443311828333259614...
4
1
12
1 2
2 21
cosh csch
sinh( )
( )
k
k
k
k
.1464466094067262378... 4
22
22
1
2
1
.1464539518270309601... 1
2
2 1 1
2 21
cos sin log sin log
e
x x
xdx
e
= xe xdxx sin0
1
1 .14649907252864280790...
1
0
2
12
log2
1
!
11},2,2,2{},1,1,1{ dxxe
kkPFQ x
k
.146581405847371179872...
1
1
1)1()1(2
)1(
16
9log1
kk
k
kk
2 .146592370069285194862... 6
52
3 20
2
cos
/ cos
x
xdx
.14665032755625354074... 24 13 1 20 11
2 2
1
1
cos sin( )
( )!( )
k
k
k
k k
1 .14673299472862373219... 1
121 ek
k/
Berndt 6.14.1
.146997759396759645699...
02 12
3
24
1xe
dxx
2 .147043391355541844907... 3 1 4
2 3132
3
8
1
1 4
( )
( / )
k
k
k
k
.14711677137965943279... )2,2()2,2(4
)2()2(4
)1()1( iii
iii
=
222
1
1
11)12()1(
kk
k
k
kkk
=
0 )1(
sin
2
1dx
ee
xxxx
.1471503722317380396... ( )
1
2 2
1
1
k
kk
.14718776495032468057...
0
6
3)cos(sin
10
1
40
3log9dx
x
xxx Prud. 5.2.29.24
.14722067695924125830...
loglog log
2 2
2
2
2 122 3
1
2
Li
= log log log2 3 21
2
1
42
2
Li
= log( )1
20
1
x
xdx
12 .147239059010648715201... 90
90
1
4 14
( )
.14726215563702155805... 3
643
0
1 x xdxarccos
.14729145183287003209... 11 1
111
Ei
e
H
kk k
k
( )( )
( )!
= dxxxex 1
0
1 log
.14747016658015036982... ( ) ( )( )
( )
1 3 11
11 2
1
3 3
3 32
k
k k
k kk k
k
.14758361765043327418...
2
1arctan
1
Plouffe’s constant, proved transcendental by Barbara Margolius
.14766767192387686404... 1
9 1
3
931 1k
k
kk
k
( )
.14775472229893261117... 58 1
2 30
e k
k
kk
( )
( )!
.14779357469631903702... 2
2
1
21
arcsinh
=( ) ( )!!
( )! ( )
1 2 1
2 4 1
1
21
k
k
k
k k
1 .14779357469631903702... 21log2
1
2
2
2
1arcsinh
2
2
= dxx 1
0
21
.147822241805417... ( ) log
1
122
k
k
k
k
.14791843300216453709... 3
23 1
1
9 31
log
k kk
J375
= ( )2 1
91
kk
k
.14792374993677554778... 4 31 4
1 4
0
ek
k kk
/
( )!
1 .14795487733873058... H5 23
/( )
.14797291388012184113... 2
42
1
4
1
22 21
coth
kk
1 .14812009937000665077... 1
7
2
8 2
1
4 4 721
tan
k kk
1 .148135649546457344478... ( )k
kkk 21
.148148148148148148148 =4
27
1 1
20
( ) ( )k
kk
k
3 .14821354898637004864... ( )
( !)
k
k kI
kkk
2 12
12 2 0
10
.14831179749879259272...
122 7
10,2,
7
1
7
1
kk k
Li
.14834567213507945656... 4 21
2
10
3
1
2 2 50
arctan( )
kk k
1 .14838061778888222872... 3
27
.14839396162690884587... 13
18
3
2
1
3
1
3
1
3
1
3 21
loghg
k kk
=( ) ( )
1
32
k
kk
k
=dx
x x x4 3 21
.14849853757254048108... 1
4
3
4
1 2
220
e k k
k k
k
( )
!( )
1 .1486983549970350068... 21 5/
1 .14874831559185321424... 16 7 4 31
16 1
2 2
0
( )( )
kk k
k
k
.14885277443216080376... 41
22
23
2
3 21
Lix
x xdx
log
.14900914406163310702... 21
2
1
3
1 2 1
2 3
1
1
log( ) ( )!!
( )!
k
kk
k
k k
.14901768840433479264... logk
k kk3 2
2
8 .14912752141674121819... 4 7
9
3
2
3 2
1
e k
k
k
k
( )!
.14918597297347943780... 93
2
7
2
1
2 1 2 30
log( )
( )( )( )
k
kk k k k
=
0 )2(3
1
2
1
kk k
.14919664819825141009... 8
19 42 2
0
2
ee e x xdxx
/ sin cos
.14936120510359182894...
1
13 !
3)1(
3
k
kk
k
k
e
6 .14968813538870263774...
112 )!12(
)!1(!
4
1,
2
3,3,22
k k
kkF
.149762131525267452517... 19
5 23 2
1
4 1
1 1
421
2
log
( )
( ) ( ( ) )
k k
k
k
k
kk
1 .14997961547450537807... 2 13
2
k
k
k ( )
.15000000000000000000 = 20
3
.150076728375356585727...
22 2
11log
2
1
)1(2
1)()1(
kkk
k
kkk
k
.15017325550213874759... 2
3
3
21
3
4 21
sin
kk
.150257112894949285675...
1
2
1
13 )1(
)1(
)2(
1
8
)3(
k
kk
k k
H
k
= dxx
xxdx
x
x
1
0
1
0
22
1
log)1log()1(log
= 1
0
1
0 1
)1log(dydx
xy
xy
1 .15050842410886528915... 24
)21log(228
5
8
1
16
1 2)1()1(
Berndt 3.3.2
= 12(21(2 22 iLiiLii
= 2
2 2 1
2
20
k
k
k
k k
( !)
( )!( )
= 22 2
0
2 x x
xdx
sin
cos
/
.15058433946987839463... 1
21 1
1
2 11
sin cos( )
( )!
k
k
k
k
= sin1
2 51 x
dx
x
.15068795010189670188... 3 3
4 15
2
42 2
2
4
4 2 23
4 ( ) loglog
log
6
1
2
21
43 24Li ( ) log
=log ( )
( )
3
20
1 1
1
x
x xdx
.1507282898071237098... 8 2 4 3 11
3 11
log log( )
( )
k
kk k k
.150770165868941637996... 1
4 2 6
2
3
1
3 2
1
21
csch
( )k
k k
.15081749528969631588... 1)1(1)(2
kkk
.15084550274572283253...
03 3
)1(
k
k
k
.1508916323731138546...
1
2
2 1010
10,2,
10
1
729
110
kk
kLi
.15097924480756398406...
12
2
)4)(2(36
113
3 k kk
k
1 .15098236809467638636... 3
10
3 3
10
2 2
5
1
6 5
1
1 6 121 0
log log
( )( )k k k kk k
= dxx
x
1
06
6 )1log(
.15114994701951815422...
022
112 )3(2
4
1
299
1
4
1
312 x
dx
k
kkk kk
.15116440861650701737...
7
627log
6
7log
6
7log
2
1
7
6
7
1)3( 233 LiLiLi
=
127k
kk
k
H
.1511911868771830003...
( )4 1
2 14 11
kk
k
81 .15123429216781268058... 8
.15128172040710543908...
1
01
3
)()1(
kk
k k
1 .15129254649702284201... 10log2
1
.15129773663885396750...
13
1
k kkF
.151632664928158355901...
1
21
2!)1(
4
1
kk
k
k
k
e
1 .15163948007840140449... ( ) ( )
( )
3 5
4
2 .15168401595131957998...
1 22/52/5 )1(
1
k k k
k
k
k
.151696880108669610301... 25
4
2csch
162coth
82
2
=
222
2
1
1
)14(
4
4
1)2()1(
kkk
k
k
kk
.15169744087717637782...
82 2 1 2
0
2
( log ) log(sin ) sin/
x xdx GR 4.384.9
.151822325947027200439... 1
2log
e
.15189472903279400784...
1
2
1
32
)1(
3
2log
69
4
k
k
kk
=4
12
11log
x
dx
x
1 .151912873455946838843... 1
16 5 4 3 20 k k k k k kk
.1519346306616288552... 5
6
6
51
51
1
log ( )
k kk
k
H
.152020846736913608124... log ( ) k
kk2
2
.15204470482002019445... 2 5 21
22 1
1
1
log1+ 5
2
( )
( )
k
k k
kk k
.152185252101046135100...
1
2)12(
61
2
3cos
5
1
k k
1 .15220558708386827494...
1
22
)12)(1(3
2log2
3
2log4
18 k
k
kk
H
.1523809523809523809 =)4)(3(4
12
105
16
0
kkk
kk
k
.152607305856597283596...
1 87
8log
7
8
kkkH
.15273597526733744319...
0
2
)7)(4(!
2
42
1
k
k
kkk
e
17 .152789708268945095760... 2)1(
1 .15281481352532739811...
1 3!)!12(
!)!2(
3
1arcsin
4
23
2
1
kkk
k
.15289600209100167943...
21
1
kkk
k
.15289756126777578495...
1
2
1
2
3
5
)1(
)1(dx
e
x
e
k
ee
eex
kk
.15300902999217927813...
13
234 )1(
11)3()2(3
kk kkkk
=
1
1)3(k
k
1 .15300902999217927813...
2
2)2()()3()2(4k
kk
.153030552913947039536...
3
1
3
1
2
1
3cot
322
1
=
113 3
)12()2(
3
1
kk
k
kk
kk
k
.15309938731499271977... )3(
)3(log
1 .15318817864647891814... 2 2
10
1
six
xdx
arcsin
arccos
1 . 15326598908047301786... 216
24
1
02
22
1
log
2dx
x
xx Borwein-Devlin, p. 54
dydxyx
yx
yxyx
yx
y
0
2
log)sinh(
)( Borwein-Devlin, p. 53
2 .153348094937162348268... )1(Im2)1()1(1coth iiii
=
1
11
k ikik
3 .15334809493716234827...
12 1
21coth
k k J947
4 .15334809493716234827... coth ( ) ( ) Im ( ) 1 2i i i i
.1534264097200273453... 1 2
2
1
2 1 2 2 1
1
1
log ( )
( ) ( )
k
k k k k GR 0.238.2, J237
= kk
kk
log2 1
2 11
1
J128
=
1 )48(
1
k kk
= 1
2 22
kk k k( )
= ( )!
( )!
2 3
22
k k
kk
= ( )
1
2 40
k
k k K Ex. 109b
= ( )
1
8 2
1
31
k
k k k
=
12
2
1 4
11log41
)12(4
)2(
kkk k
kk
k
= dx
x xxdx5 3
1
3
0
4
tan/
=
02
3
sinlogsin1
sindxx
x
x Prud. 2.6.34.44
= log
logx
xdx
x
x
dx
x21
2
2 1
=
1
02
12 )1(
)1log(
)1(
11log dx
x
x
x
dx
x GR 4.291.14
= ( ) arctan10
1
x xdx
= x
dx
xx log2
1
log2
1
1
11
02
GR 4.283.5
=
1
2
1 2
0 xe
e
x
dx
xx
x /
GR 3.438.1
= dx
e ex x2 20 1( )
1 .15344961551374677440...
0
2
4
0211 )!()1()2()2()2(3)2(2
k
k
k
kJJJJ
=
1
1 2
)!12(
)1(
k
k
k
k
k
.153486153486153486 = 13
2
.1535170865068480981...
3/1
3/13/2
3/1 7
112
7
1131
7
)7(731
76
1 ii
=
1
1
13 7
)3()1(
17
1
kk
k
k
k
k
.153540725195371548500... k
k kk
2
2 11 1
.15354517795933754758... 3600
5369)2()2()7( 6
)2()1( H
1 .15356499489510775346... 7/1e
1 .15383506784998943054... 8/1
.15386843320506566025... )1(360)2(6)3(42
2log
590
181
=
2 4
1)(
k k
k
.15409235403694969975...
1
22
2
1
2
)112(
12
12
)2(
32csc
3sin3
48 kkk k
kkk
.1541138063191885708...
23
3
1
)1(
2)1)()(2)(1()1(
4
9)3(2
kk
k
kkkk
= 3)2(
=
134
2log
xx
x
= x x
xdx
2
0
1
1
log
= x
e edxx x
2
20 1
.154142969550249868...
1
)3(
)1(4kk
k
k
H
.15415067982725830429...
11 7
1
7
16log7log
kk k
Li
=
0
12 )12(13
12
13
1arctanh2
kk k
K148
.15421256876702122842... 2
2 2164
1
2 1 4
(( ) )kk
J383
15 .15426224147926418976...
0 !k
ke
k
ee Not known to be transcendental
=
0
!/1
k
ke
.15443132980306572121... 2 12
1 2 11 2 1
12 2
k
kk
k
k
k kk k
( ) log( )
1 .15443132980306572121...
1
2log)(2
k k
kk
2 .15443469003188372176... 3 10
.154475989979288816350... ( ) log ( )
1 2 11
1
k
k
k
1 .1545763107479915715... 2/3)2(
2
39
40H
.15457893676444811... ( )
log
13
2
k
k k k
1 .1545804352581151025...
1 2 ),2(
1
k kkS
.15467960838455727096... 7
32 2
1
4 2 1
23
1
( )
( )
k
kk
k
k
k
k
.15468248282360638137...
13 1)13(
1
k k
=
6
35332
6
3534
3318
1
6
3log
3189
ii
i
i
.1547005383792515290... 2
31
1
2 1
21
0
( )
( )
k
kk
k
k
k
k
022 4)!(
)!2(
kkk
k
=( )!!
( )!
2 1
2 41
k
k kk
1 .15470053837925152902...
0
2
16
1
3csc
3
2
kk k
k AS 4.3.46, CFG B4
=
122!)!2(
!)!12(1
kkk
k J166
.15476190476190476190 =
0 )103)(13(
1
84
13
k kk K134
2 .15484548537713570608... 3 61 3
0
1
e e dxx /
7 .154845485377135706081...
1
4
)!1(13
k k
ke
8 .154845485377135706081...
1 !
)1(3
k k
kke
.15491933384829667541... 1
5
3
51
2
61
1
( )k
kk
k
k
k
.15494982830181068512... )(Re i
.154951797129101438393...
1 )14)(14(42log
8
123
k
k
kk
HG
.15519690003711989154...
3
2
.15535275300432320321...
122
2 11 2
k kkH
.15555973599994086446... )3()2(
)3()2(
.15562624502848612111... )3(2log723log54318216 2
=
1
1
134 6
)3()1(
6
1
kk
k
k
k
kk
1 .15572734979092171791... dtt
tdx
x
x
e
0222 1
cos2
)1(
cos
.155761215939713984946...
2 202
121
1
)!(
1)()1(
k k
k
kJ
kk
k
.15580498523890464859... 4 4 2 22
3
1
22
arctan
1
2log Li
=( )
( )
1
2 2 1
1
21
k
kk k k
.1559436947653744735... cos ( )( )!
2 12
20
kk
k k GR 1.411.3
.15609765541856722578... dx
x x x4 31
.1561951536544784133...
kk
kk
kk
k
kk
11log2
11log221
4
1
1
1)( 2
122
.15622977483540679645...
1 2
)(
kk
k kH
.156344287479823877804...
7
10 4 21
2 1
41
1
coth ( )
( )kk
k
k
=
222 2
1)(Re14
1
k
k
k
ik
k
.15641068822825414085...
dx
x
xdx
x
x
e 1
3cos
)1(
3cos2
0223
.15651764274966565182...
1 0
1
221
22 !
2
1
11
2
11coth
1
1
k k
kk
kk k
B
kee
1 .15651764274966565182...
022
022
2
1
11
2
1coth1
1 kkk kee
e
J951
.15652368328780337093...
1
1
)2()!2(
)1(
2
231sin101cos6
k
k
kk
1 .15654977606425149905...
1
22cos
!
sin
42)2cos(sin
2
122
k
ee
k
keeeee
ii
.156643842100387720666... log log log( )
log3 21
12
1
2 2 2 21
k
k kk
1 = Stieltjes const.
12 .1567207587610607447...
3 3
0
6 x xdxsin
1 .15688147304141093867... 2
11
2 2 20
log
log log
ex
e xdx
.156899682117108925297... 3
6
3
41
13
13
logx
dx
x
.15707963267948966192...
05520 xx ee
dx
.157187089473767855916... 1
33 e
3 .157465672184835229312...
0
2
2
1
)3log()31log( dx
x
x
1 .15753627711664634890...
13
133 )13(
1
)13(
11
27
)3(26
kk kkG
1 .15757868669705850021...
4
3
2 1
2
0
2
x dx
ex
.157660149167832330391... dxxx
xx
12
)1(2
1
log
3
1
6
2
9
1 GR 4.233.3
= 3/22
3/12
3/13/1
)1())1(()1()1(1
1
LiLi
.1577286052509934237... cos ( )( )!
3
0
2
11
41
3 3
2
k
k
k
k GR 1.412.4
.15779670004249836201...
2 21
kk
k
.15783266728161021181...
1
21
/12 !)1(
1
kk
k
k
k
e
9 .157839086795201393147... log8
.157894736842105263 = 19
3
.15803013970713941960... 1
4
1
4
14 9
1
e x
dx
xcosh
.1580762... powerinteger
trivialnonaω2)1(
1
.158151287891164991627...
12
21
02
2
)1(
log
)1(
log)2(
2
)3(3
xx
xdx
x
xx
.15822957412289865433...
1 )12(3kk
k
k
H
.158299797338769411097...
0 1 ),2(
!
k kkS
k
.15871527483457111423...
1
2
1
14
)1(
2csch
42
1
k
k
k
.158747447059348972898...
1
1
4
1)2()1(
22log
22log
kk
k
k
kii
.158806563699494307872...
1
2
1
3
)1(3csch
6
3
6
1
k
k
k
J125
.15886747797947540615...
14
12
2
1
log
)14(
1
216 x
dxx
k
G
k
.15888308335967185650...
1 )2)(1(242log6
kk kk
k
4 .15888308335967185650... 6 21
13
230
log( )
( )( )
k
k k k
12 .15888308335967185650...
1
2
22log68
kk
kHk
40 .15890107530112852685...
1
2
k kF
k
1 .1589416532550922853... B
k
k
kk
k
2
0 2
2
( )!
3 .15898367510013644053...
1 1 1 111 1 11 1
0
1
4
2
1
12
1
12
)(
2
)(
i j k liijkl
j kijk
i jkj
kk
kkt
7 .15899653680438509382... 3;1;)!(
3)32( 1
020 F
kI o
k
k
.15909335280743421091... 1
0
6 arcsin245
16
14dxxx
GR 4.523.1
.159105544771044128966... 525log515log255
1
=
11
1
)12(4)1(4 kk
kk
kk
kk
k
LF
kk
FF
.15911902250201529054... 5
2
1
21
1
2 2 1 2 1
1
2 11
arctan( )
( )( )
k
kk k k
.159137092586739587444...
1 )2)(12(
1
12
2log1613
k kkk
.1591484528128319195... ( ) log arctan
18
1
94
9
82 2
1
2 21 2
1
k kk
k
H
.159154943091895335769... 21
1 .15924845988271663064... )5(
)3(
.159343689859175861354...
1
2
2
)22()2(
kk
kk
.159436126875267470801... 2 2 1
132
5
12
2 2
3
2
3
1
3
log log ( )
( )
kk
k
H
k k
13 .159472534785811491779...
3
2
3
1
3
4 )1()1(2
.15956281008284001011...
1
2
41
2 cos)1(1
224 k
k
k
k
.15972222222222222222 =
1 )4)(1(
1
144
23
k kkk
1 .15973454203768787427...
0 )12(
1
4
1cothlog
2arccoth
kk ke
eee
.159868903742430971757... log log ( )
log ( )2
12
2 12
2
2 1
k
kk
k
k
k
k
k
2 .15990451200777555816... ( )1 5
6
.160000000000000000000 =
1
1
4
)1(
25
4
kk
k k
1 .160014413116984775294... dxx
x
03
3sin
3
1
4
3
.1600718432431485233... 15
4 5 141
( log log ) ( )( )
kk
k
k
2 .1601271206667850515...
12
)1log(log
k k
kk
.16019301354439554287...
112
1
2 76
)1(
6
1
kk
k
kk
k
k
H
kLi
.16043435647123872247...
1 )3(2
1
3
2log7
9
16
kk kk
.16044978576935580042... dxx
x
e
023 3
cos
32
.1605922055573114571... dxx
xkkk
1
0
4
12
11log
12112
1
60
72log24
.1606027941427883920... 25 1
30
2
0
1 2
21
e k kx e dx
x
xdx
k
k
x( )
!( )
log
=( )
( )! !
11 20
k
k k k
.160889766020364044721...
1 )12)(12(
2cos
2
1
4
1sin
k kk
k GR 1.444.7
.1610391299195894597... 22
22log
2
242log
2
2
=
12
1
4
)1(
k
k
kk
=2
14
1
0
4 11log)1log(
x
dx
xdxx
4 .1610391299195894597... 2
22
2
2
2 2
2 21
14
0
1
log log log
xdx
.16126033258846573748...
11
1
2
log
2
2log
2
1
2
1)1(
kk
kk
k kLi
k
.1613630697302135426... 1
2
1
21
1 2
22 2 121
2
1
coth( )!
k
B
kk
kk
k
.16137647245029392651...
222
2
1
)!2()!2(
1)2(00
1 kJ
kI
kk
k
k
.161439361571195633610...
1
2122
)!2(
2
2
1log1sinhlog
k
kk
kk
B
e
e
= 2
2
2 12
1
kk
k
B
k k
( )!
=
12
1 )2()1(11log
kk
k
k
kii
1 .161439361571195633610...
1
2
!
2)1(
2
1log1sinhlog1
k
kkk
kk
Be Berndt 5.8.5
.16149102437656156341... 3 2
20
1
192 1
arctan x dx
x
.161745915929683120084...
31
5
62arctan321217log
212
1 h
= dxxx
166
1
2 .16196647795827451054...
1 2k
k
k
kk
1 .162037037037037037037 = 3)3(
216
251H
1 .16204051536831577300...
2
1)12()(k
kk
.16208817230500686559...
1
1
3)!2(
)1(1
3
1cos
kk
k
k
8 .16209713905398032767... 3 3
2
11
65
60
( )( )k kk
.162162162162162162162 =
0
6log)12()1(6logcosh2
1
37
6
k
kk e J943
= dxe
xx
0
6sin
2 .1622134783048980717...
112
2
2
)2(
)12(
122log4
2 kk
k
kk
kk
3 .162277660168379331999... 10
1 .16231908520772565705...
1 21 )!(
11,0,
2
322)1(
k k
eerfe
.162336230032411940316...
1
3
)!3(2
7113
k k
ke
.162468300944839014604...
15
21
4
)(6
)5(93
k k
k
.162651779074113703372... 2/12
12
3
1 22212
1
!
)()1( kekk
kk
k k
kk
k
.16266128557107162607...
1
3 )12(2)12(27
)1(
812
k
k
kk
= dxx
xx
1
022 )1(
arcsin GR 4.512.7
.16268207245178092744...
2
1348
13log2log2
xx
dx
.16277007036916213982... 2 31
22
4
5
1
2 2 1 2 1
1
2 11
arctan log( )
( ) ( )
k
kk k k k
.162824214565173861783... 3
1
.162865005917789330363...
1
02
22
1
)1log(
8
2log
96
11dx
x
xx
1 .16292745855318123597...
3
2 2 2 23 3
256
1
1024
7
8
5
8
3
8
1
8
( ) ( ) ( ) ( ) ( )
= ( )( ) ( ) ( ) ( )
11
4 1
1
4 2
1
4 3
1
4 43 3 3 30
k
k k k k k
.16301233304332304576...
4
2
0ex e x x dxx
sin cos
.163048811670032322598...
( )log ( )
k
kk
k
1
2 1
.1630566660772681888... 7 3 12 1
2
1
e Eik
k kk
( )( )!( )
.163224793... ( )
( )
k
kk
1 2
2
.16329442747435331717... 8
9 62 1
4 2
3
2
31 2
( )
loglog( )
= 1
8 31 k kk ( )
Prud. 5.1.7.22
.163265306122448979592...
1 849
8
kk
k
.16345258536674495499... 2/
0
322
cos27
40
6
dxxx
.163763357369443580273...
13)3(
)3(3)2(8
17
k k
k
.16389714773777593436...
1
1
)!3(
)1(
32
3cos
9
2
9
1
k
k
k
ke
e
.163900632837673937287...
1 4
1)2(
2
5
2
3log
8
3log
kk k
k
.16395341373865284877... 3
1
12 2 1
41
e
e kk
2 .16395341373865284877...
0
2
)!2(1
1
2
1coth
k
k
k
B
e
e
4 .16395341373865284877... 3 1
1
12 2 1
40
e
e kk
.16402319032763527735... 1
12
2
3
4
3
3
4 9
1
6
1
31 1
21
( ) ( ) ( )
=
122 )19(
9
k k
k
2 .16408863181124968788...
22 3
)(
k k
k
.1642135623730950488... 25
4
2 1
2 22
( )!!
( )!
k
k kk
1 .16422971372530337364...
5
4
.16425953179193084498... loglog
224 4
2
2
1
2
1
2
2 2
2 2
Li
iLi
i
= log( )
( )
1
1
2
20
1
x
x xdx
1 .16429638328046168107...
13)!!(
1
k k
.164351151735278946663...
1
250 43
)1(
3
2log
331
xx
dx
kk
k
.164401953893165429653...
1
2
)1(2
)1(
2
3log
kk
kk
k
H
6 .16441400296897645025... 38
.16447904046849545588...
02
0
1
)2(!
)1(
)1()!1(
)1(1)1(
1
k
k
k
k
kkkk
kEi
e
= dxe
xxx
1
0
log
1 .16448105293002501181...
0
)2(
122
22
22
1
2
122log
6 kk
k
kk
H
kLi
.16449340668482264365... 2
5160
11
logx
dx
x
2 .16464646742227638303...
1
0
24 log)1log()4(2
45 x
xx
.164707267714841884974...
13
42
)12(1288
2log)2log1(
8
)3(7
k
k
k
kH
.16482268215827724019...
primepk p
p
k
k
1
loglog
)3(
1
)3(
)3(3
23
=
13
)(
k k
k
.164895120905594654651...
1
212
log)()6()31786050
)6(
)3(
k k
kk
1 .16494809158137192362... 1
4
.165122266041052985705... ( )( )
16 1
1
1
kk
k
k
.165129148016224359068... )5()3( 9 .1651513899116800132... 84 2 21
2 .16522134231822621051...
222
1)()1(
1
kk
k
kFkkk
k
= 2
5tan55
102
15
2
5
2
3
.16528053142278903778...
1
1
)!3(
)1(
2
3cos
3
2
3
11
k
k
ke
e
2 .1653822153269363594... e EiH
kk
k
( )!
11
.16542114370045092921... )1(
.16546878552437427...
23
123
log
)!1(
1
log
1)3(1
k
n
nk k
k
nkk
k
1 .16548809348122843455... 1
0
00 arcsin)1()1(2
dxxeeLI x
.1655219496203034616... 2 3 4 2 3 4 24 1
2 2
1
log log log log( )
H
kk
kk
1 .16557116154792148338...
0 36
)1(
16
)1(32log
6
3
4 k
kk
kk
= dx
x x2 21 1
.165672052444307559...
0
logcossin5log2arctan210
1dxxxxe x
3 .165673248894309178142...
12 22
112sincsc6
k kke
.16585831801671439398... )2(2
11
2
1
2
1
12
2log
12
2log1 33
32
LiLi
=log2
3 21 2
x dx
x x
.165873416657810296642... e
e
8
)1cos1(sin1sin1cos1sinh1cos1sin1cosh
4
1 2
=
1
1
)!4(
4)1(
k
kk
k
k
.16590906347585196071...
123
2
53
1
250
63
50
3
3050
3log9
k kk
16 .16596749219211504167...
1
8
1
42
2
3 2 1 21
( )/ /
log x dx
x x
.16599938905440206094...
14
2
)2(1)4(4)3(4)2(
k k
k
.166243616123275120553...
1
2
)12(
1
!)!2(
!)!12(1
4
k kk
kG
J385
1 .166243616123275120553...
0
2
)12(16
124
kk kk
kG
.166269974868850951116...
1
1
)!4(
4)1(1cosh1cos1
k
kk
k GR 1.413.2
.16651232599446473986... log3 2
2
.16652097674818765092... ( )
( )log2
3
23 2
1
4 2
2
4 31
k
k kk
= ( ) ( )k k
kk
2
22
.16658896190385933716... 2
2
.16662631182952538859...
4/
02cos)cos(sin
sin
8
2log
2
2log
4
dx
xxx
xx GR 3.811.5
.166666666666666666666 =1
6 1
11
4
32 2
12
3
k
k k kk k k( )!
=
1 4
1)2(
kk
k
=
123 )4)(3)(2(/1
1
kk kkk
k
kkk
=
0 )52)(12(
)1(
k
k
kk
=
2 1
1 2
)2(2
)1(
)2(!
1
k kk
k
k
k
kkk
=
32
41
k k
= k k
k k
k k
k kk k
( )
( )( )
( )
( )( )
4
2 2
6
5 11 1
J1061
= arccosh x
xdx
( )1 31
1 .166666666666666666666 =
1
4
)(2 2
)8(
)4(
6
7
k
k
k
HW Thm. 301
=
primep p
p4
4
1
1 Berndt Ch. 5
.16686510441795247511... sinh sin sin
( )!
1 1
2 4
1
4
1
2
1
4 30
e
e kk
.16699172896087386807...
2 20
11
2
!!
1)(
k k kkI
kk
k
1 .16706362568697266872...
0
4/14/1
)!4(
4)1(cosh)1(cos2cosh2cos
2
1
k
k
k
22 .167168296791950681691...
0
2
!
)1(22
k
k
k
ke GR 1.212
1 .16723171987003124525... 8
arccos
2 .16736062588226195190... 22
2cos2cosh
=
142
4/34/1 11
)1(sin)1(sin
k k
.167407484127008886452...
6
5
12
11
13
13log32log
=
12
1
6
)1(
k
k
kk
.16745771316555894862...
1 )!1(2
11
k k
5 .16771278004997002925... 6
3, volume of the unit sphere in R6
.16782559481552120796...
loglog ( )
( )
22 1
22
kk
k
k
k Dingle 3.37
= 1
2
2 1
21 k
k
kk
log
.16791444558408471119... 9
2
3
2
1
3
12 2 1
91
cotkk
.16805750730968383857...
1 13
1
3
)(
k primeppk
k
1 .16805831337591852552...
0 )!3(
1
2
3cos
2
3
1
k kee J803
.16807806319774282939...
1
3
)!3(
))!1((
k k
k
.16809262741929253132... ( )
( )
3 1
3
1 .168156588029466646... log
( )
2
2 2 1
k
k kk
1 .16823412933514314651...
1
4 1)3(k
k
.16823611831060646525...
0
12 )12(6
1
6
1arctanh
kk k
AS 4.5.64, J941
.16838922476583426925...
6
2
32
5 3
4
3 3 3
1
6
5 3
4
3 3 3
log
i
i
i
i
=1
2 1
3
831 1( )
( )
k
k
kk
k
1 .168437795256228934...
1
1
23
1)13(2)1(2
2
k
kk
k
kk
k
.16846317173057183421...
144 6
1
6
1
kk k
Li
1 .1685237539993828436...
3
1
2
1
322log2
2
3log3
=
1
1
02
2
2
2
1
1log
1
1log dx
x
xxdx
xx
x
.168547888329363395939... 4 16
1
8 2 1 2 2 1
2
4 21
( ) ( )k kk
.168609905834157724449... 1)1()2)(1(
2)1(7216log2
4
1
1
1
kkkk
kk
.168655097090202230721... 4 2 2 14
2 2 1 8 27 3
16
2
log log( )
=1
8 2 1 2 2 14 31 ( ) ( )k kk
.16870806500348277326...
1
21
1
2 2
1
21
4 2
4
2 1
81
( )kk
k
= 1
8 31 k kk
.16905087983986064158...
2
1)(
kkk
k
1 .16936160473579879151...
5/4
1
5/1
1
.16944667603360116353...
23 3
)1(
k
k
k
.16945266811965260313... 1
0
2 arccosarcsin927
14dxxxx
.16945271999473895451...
32
3cos
2
9
1 e
e
.16948029848741747267... ( )k
kk
4
42
= 1
1
1 1 14
24 4
212 4
220 4
2k k k k k k kk k k k
= ( ) ( ) ( ) ( )4 1 8 4 1 16 4 1 16 4 11 1 1 1
k k k kk k k k
.16951227828754808073... 21
2
12 2 1
41
cotkk
=
112
1tan
2
1
kkk
Berndt ch. 31
3 .1699250014423123629... 9log2
11 .1699273161019477314... 12 22 6 2 23 2
0
x xdx
ex
log
1 .17001912860314990390...
0 )12(6
12
2
3arcsin
2
3
kk kk
k
.17005969112949392838... k
k
kk
1log
1
22
.17032355274830491796...
133 6
1
6
1
kk k
Li
.170557349502438204366...
1
0
2
2
1
1)1log(
121
xe
dxx
eLie
2 .17063941571244840982...
2
1)(k
k
1 .17063957497359386223... 4/34/34/5
2coth2cot214
23
=
12
41)4(8
8
8
k
k
k
kk
.1706661896898962359...
1
1
12 9
)2()1(
19
1
2
1
3coth
6 kk
k
k
k
k
.1707701846682093606...
12
)1(
)1(3
16
3
2
9
1
12
5
362
3log
k kk
k
=
1 3
)1)((
kk
kk
.17083144025442980463... kkF
8
3
.1710822479183635679... ( )k
kk 41
.1711785303644720138...
1)(
1)13(
8
1
k
k
2 .17132436809105815141... Re ( ) ( ) i i i
1 .1713480439548652889...
0 )!2(3
1
3
1cosh
kk k
.17138335479471051464... 41 1)1()1( kk
1 .17139260811917011128...
12
1 1( ) ( )k kk
1 .1714235822309350626...
1
428
)()4(8100 k
kkd Titchmarsh 1.2.1
.171458091265465835370...
1
2
5
41
21
log coscosk
kkk
.1715728752538099024...
10 )1(2)!2(
!)!12(
)1(8
12223
kk
kk kk
k
kk
k
=( ) ( )!!
( )! ( )
( )
( )
1 2 1
2 1
2 1
4 1
1 1
11
k k
kkk
k
k k
k
k k
=
0
)2()!2(
)1(
2
1
k
k
kk
.17166142139812197607... ( )
( )
2 1
1
112
1
22 2
2
k
kk Li
kk k
.17172241473708489791... 7
)3(
.17173815835609831453...
0 )2()!2(
)1(
2
11sin51cos36
k
k
kk
= e
dxx
xx
1
3 logcoslog
.17186598552400983788... 11
22 2( ) ( )i i
= 1)1()1(2
11)()(
2
1 iiii
= 11 2 1 13
2
1
1k kk
k
k
k
( ) ( )
=
1
)14()14(k
kk
= 1
10
cosx
e edxx x
.17190806304093687756... dx
x x x4 21
1 .1719536193447294453...
0
3/22
3/12 1
)1()1(3 xx ee
dxxLiLi
i GR 3.418.1
= 1
3
1
3
2
3 11
2
20
1
( ) log
x
x xdx GR 4.23.2
5 .172029132381426797... dxx
x
1
03
32
)1(
log
4
)3(9
1 .1720419066693079021...
1
3)1(4
11
2sinh
5
8
k k
1 .1720663568851251981...
22 2
log
k k
k
.17210289868366378217...
1
2/1)1(4
tanh2
1
2
1
k
kk e J944
3 .1722189581254505277... 2e
.17224705723145852332...
1
2
)!3(k k
k
.17229635526084760319... ( ) ( )
( )
( )2 3
4
4 1 31
H
k kk
k
.17240583062364046923... 1)2(49
4
16 1
2
kk
kk
.17250070367941164573... 11
2
1
2
12 2 1
21
cotkk
.172504053920943245495... dxxx
li )log3sin(1
23log3
10
1 1
0
GR 6.213.1
.17254456941296968757...
1
1
223
3
1)3()1()1( k
k
k
kkk
k
2 .17258661797937013053...
13
1
k kF
.17259944116823072530...
1
2
)2(32
2
3log4
2
3log3
2
3
kk
k
k
H
.17260374626909167851...
1
21
12 2)!2(
4)1()2(1sinlog
k
kk
k
kk kk
B
k
k
AS 4.3.71, Berndt Ch. 5
2 .172632057285762871628... ( ( ))2
.17264325482630846776...
12
23
11)()1(
kk
k
kkLi
k
k
.17273053919054494131...
1
1
)3(!
)1(
3
55
k
k
kke
.1728274509745820502... log ( )2
2
1
2 1
1
0
GH
k
kk
k
Adamchik (24)
= dxx
x
1
02
2
1
)1log( GR 4.295.5
.172855673055088999098...
16/
4cos4
22
x
xe
.17289680030426476...
2 2
log
kk k
k
1 .17303552576131315974...
1 2
cot22
11)2()2(
k k kkkk
=
1 122
1 22 1
1)2(
m kk nk kmn
k
=
2
2
2
cot)1(2
3
4
3
k kkk
k
.17305861469432215834...
1
1
)!1(
)1()1(2)1(
2
k
kk
k
kH
e
EiEi
e
.17311810051999912821...
2
21)( dxx
2 .17325431251955413824...
n
mn
nmhn 1
)(1
lim)3(
)2/3(
h(m) = lowest exponsent in prime factorization of m.
.173280780307794293319... 11
22 2 e ei i
= ( ) cosk kk
1 11
.17328679513998632735...
1 )14)(24)(34(
1
4
2log
k kkk J253
=
0 44
)1(
k
k
k
=
1 12)14(
1
r srs J1124
= 1
8 2 1 2 2 131 ( ) ( )k kk
[Ramanujan] Berndt Ch. 2
=
15 xx
dx
= dxx
xx
1
022 )1(
log GR 4.234.2
1 .1733466294835716297...
12!!
1
k kk
.17338224470333561885...
2 !
1)(
k kk
k
3 . 17343648530607134219... 3
232log4
1
1
1
1221 yx
dydx Borwein-Devlin, p. 53
.17352520779833572367...
03 32
)1(
k
k
k
2 .173532954993580700771... log ( )2
0
1 1
x
xdx
1 .17356302722472693495...
1 !
)1()1(
k k
keEi
.173749320202926737911...
1 )2(
)2(
k kEulerE
kBernoulliB
.17378563457299202316... kk)6(
1
.17402964843834081341...
14)1(
)5(2)3()2(4
)4(
k
k
k
kH
.17412827332774280224...
1 )13(3
1
kkk k
.17417956274165000788...
1
1
122 5
)1(
6
1
6
1
kk
kk
kk k
H
kLi
.17426680583299550083... 2
221log
2
11arcsinh
=
1
02/32
2
)1(x
dxx
.17440510968851802738...
2
3 216
3 3 9 3 6 4 361
6
log logk kk
= ( )( )
12
61
1
kk
k
k
2 .17454635174140...
1
/11)1(
k
kk
k
e
.174644258731592437126...
12 2/1
11
k k
.17472077024896732623...
1
41)(k
k
.17476263929944353642... )3(log)(1
13
kk
k
p kprimep
.17488106576263372626... 24
3 2
2
1
8 21
log
( )k kk
.1750000000000000000 = 40
7
1 .1752011936438014569... iiee
sin2
1sinh
2
1cosh2
21sinh
1
=1
2 10 ( )!kk
AS 4.5.62, GR 1.411.2
= 112 2
1
kk
GR 1.431.2
= cosh1
12x
dx
x
.175311881551605975687...
1 4
1)1(
3
1
22log3
kk
k
.17542341873682316960... 2,2,1,12,2,1,14
11212
iiFiiiF
i
= dxe
xx
0 2
cos
.175497994056966649393...
1 )!1(
log
k k
k
.175510690849480992305... 1
121 e
k
k
.175639364649935926576...
1 22 2!
log
!
1
2!
1
21
n kk
n
kk k
k
nk
ke
.17575073121372975946... 83
8 20
1
e
x x
edxx
sinh
.175781250000000000000 = 45
256 9
2
1
kk
k
.175886808246818344795...
)3(
3
2
3
1
2
12
4
9log
3
23log
2
3log 3332
2 LiLiLiLi
=
1
)2(1
)1(2)1(
kk
kk
k
H
.17592265828206878168...
0
102
)!2(!
)1(;3;
2
1)32(
k
kk
kk
eeF
e
J
.17593361191311075521...
22 2
11log
2
1
)1(2
1)(
kkk kkk
k
1 .1759460993017423069... i i i x x
xdx
4
1
2
1
21 1
0
( ) ( ) sin
sinh
.17606065374881092046...
1
1
13 6
)3()1(
16
1
kk
k
k
k
k
.176081477370773117343...
1 3)!2(3
1sinh
32
1
kkk
k
12 .1761363792503046546... 4 3
20
3
08 1
log
sinh
xdx
x
x dx
x
1 .17624173838258275887... )(
1 .176270818259917... smallest known Salem number 1 .1762808182599175065... larger real root of Lehmer’s polynomial
134567910 zzzzzzzz Borwein-Devlin. p. 35
.176394350735484192865... 20
1
20
2
10 61
log arctan x
xdx
1 .176434685750034331624...
12
1
2
)1(1
k
k
k
.176522118686148379106... 3
2 2
8
21
1 1
21
1
log( )
( )k
k
k
k
= 1
21
11
2 kk
kk
log
.176528539860746230765... 12log2/11
2/11log
24
3
=
0 )3)(2)(1(2
1
kk kkk
J279
.17659040885772532883...
1
1
13
)1(
kk
k
.1767766952966368811... 1
4 2
1
4
2
0
( )k
kk
k k
k
.176849761028064425527... /1e
.17709857491700906705... 5 1 3 11
2 1 2 31
cos sin( )
( )!( )
k
k k k
=log sin log
sin3
15
1
1x x
xdx
x
dx
x
e
1 .177286727908419879284... 1
2
2 1)(
k
k
k
.17729989403903630843...
0
1
)53)(13(
)1(
8
1
36 k
k
kk
1 .17730832347023263216... 27
43 9 2
1
2 1 2 1 1 920
log log( )[( ) / ]
k kk
.17733145610396423590...
12
12
)1(12)3(92log24
12 k
kk
kk
HH
1 .17743789377685370624...
0102 2)!2(!
12
2
1;3;)2(4
kkkk
FI
.177532966575886781764...
12
22
2
)2(
1)1(
121
kk
k
kkk
=
1
22
22
2
2cos)1(
2
1
k
kii
k
keLieLi
=
02
12
1
0 )1(
loglog)1log(
xx ee
dxxdx
xx
xdxxxx GR 3.411.10
= x
e edxx x
10
=
x x
xdx
log
10
1
=
1
0
1
0 1dydx
xy
yx
.17756041552698560584... 1
82 3 2 3 3
2 11
log( )cos
k
kk
J513
2 .1775860903036021305... 2log
= dxx
x
1
121
)1log(
=
02
2
02
2
4
)4log(
1
)1log(dx
x
xdx
x
x
= dxx
x
02
2
1
)1log( GR 4.295.15
= 2
0
2
1
1log
x
dx
x
x
GR 4.298.9
=
0
tan dxxx
= 2/
02
2
sin
dxx
x GR 3.837.1
=
0
2
1
1
arctanarcsindx
x
xdx
x
x
=
2/
00
)tan4log(2
sinlog
dxxdxx
1 .177662037037037037 = 4)3(
1728
2035H
1 .17766403002319739668... 7
.17778407880661290134...
012 )12(2)!12(
12log
2
1
kk kk
ci
.17785231624655055289...
1
3
)!3(k k
k
.177860854725449691202...
2
3
3 6
2 3
1( ) ( )
( )
n , where n has an odd number of prime factors
[Ramanujan] Berndt Ch. 5
.17802570195060925877... 4/
02
22
cos
tan
4162
2log dx
x
xx GR 3.839.3
.17804799789057856097...
12 1
1
kke
Berndt 6.14.4
3 .17805383034794561965... !4log
1 .17809724509617246442... 21arctan8
3
= ( )
( )sin
1
2 1
6 3
2
1
21
k
k k
k J523
= ( )
( )( )
k
k k
k k
k kk k
1 4 8 4
4 8 3
2
12
321
2
21
= dx
x( )2 30 1
= x
xdx
3 2
0
1
1
/
GR 3.226.2
= sin3
0
x
xdx
GR 3.827.4
.1781397802902698674... 12 24
2 2
20
log sinx x
xdx
.17814029797808795269...
126k
kk
k
H
.1781631171987300031... ( ) ( )
( )( )
2
2 4
1
2
1
1 21
k
k k kk
2 .1781835566085708640... cosh( )!
22
2
2
2 2
0
e e
k
k
k
GR 1.411.2
10 .17822221602772843362... k
kk
2
2 !!
11 .17822221602772843362...
1
2
!!2
2
1
233
k k
kerf
ee
.1782494456033578066... 1
2 42 4 2
1
2
arctan
dx
x x
.17836449302910417474...
0
4
242
)22(
)12(
4
)3(
14424 k k
k
.178403258246176718301... k k k
kk
! ! !
( )!31
.17843151565841244881... 1
162
12 1 1
2
1
log ( )k
kk
k
.178487847956197627252...
0 )5)(2(!
13
3
25
k kkke
.1785148410513678046... 45
54
14
1
1
log ( )
k kk
k
H
.1787967688915270398...
4 3
3
4
1
3 3 1 3 21
log
( )( )k k kk
J250
= dx
x x4 21 1
.17881507892743738987... 21
21
1
2
20
1
arctantan sin
cos
x
xdx
.17888543819998317571... 2
5 5
1 21
21
( ) ( )!
( !)
k
k
k k
k
2 .17894802393254433150...
2
21)(2k
k k
.178953692032834648497... 1
2
1
2
1
12 21
cot
kk GKP eq. 6.88
=
1
2121
12 )!2(
2)1()2(
k
kkk
kk k
Bk
.17895690327368864585... 8
1 11
41
3erfi erfx
dx
x
sinh
1 .178979744472167270232...
0
sin2)(2dx
x
xsi
.17905419982836822413... 5 2 61
1 2
1 12
1
( ) ( )!
( )!
k
kk
k
k
.17922493693603655502...
0 )4(2
)1(
2
3log16
3
20
kk
k
k
.1792683857302641886... 2 3
4
1
2
1
41
2
20
1
log
log( )e e
dx
e ex x
.17928754997141122017...
2 1)(
1)()1(
k
k
k
k
.179374078734017181962... ee
ke
k k
k
11
1
1
1
( )
( )!
.17938430648086754802... 1 1 11
11
( ) log( ) log( )
k
k k k J149
376 .179434614259650062029... 145739620510
387420489 10
10
1
kk
k
.1795075337635633909...
0
2
2222 log
4
2log
2
2log
2
2log
2424 xe
xdxx
5 .179610631848751409867... e
1 .17963464938133757842... 1
6 4 3
3
2
1
4 321
cot
kk
.17974050277429023935... )32(log2
3
4
3
3
1arctanh
2
3
4
3
=
0 )12(3kk k
k
.1797480451435278353...
1
1
)13()!2(
!)!12(2log2
6
5
3
13
k kkk
k
.179842459798468021675... 7
6
7
6 71
log
Hkk
k
.180000000000000000000 = 50
9
1 .18011660505096216475... 5 3
16
2
0
1 ( ) arcsin arccos
x x
xdx
.180267715063095577924... ( )
! ( )
1
4 2 1
1
1
k
kk k k
Titchmarsh 14.32.3
. 18028136579451194155...
02
2
)1(16
2124
kk k
k
k
k
1 .18034059901609622604...
0
2
2 32
12
4
3 kkk
k
.1804080208620997292... 23 1
1 20
e
k
k
k
kk
( )
( )!
.18064575946380647538... 50
10 5 51
201 5 2 5 2 ( ) arccsch arcsinh
log5 1
32
= ( )
1
5 1
1
1
k
k k
5 .180668317897115748417... )2(e
.18067126259065494279...
1
222
22
)58(
1
)38(
1
8
3csc
642216 k kk
= dxxx
xx
1
042
2
)1)(1(
log GR 4.234.5, Prud. 2.6.8.7
.1809494085014393275...
8 3
3
24 831
log dx
x
.18102687827174792616... cosh
sinh1 1
3
13
14
x
dx
x
.181097823860873992248...
2 12
21
kk
.18121985982797669470...
21 2
11log2
1
)1(2
1)1(
kkk kkk
k
3 .18127370927465812677...
1
1
2
)!12(
1)2(2
k k
k
kI
.18132295573711532536... 6)1(
1 .18136041286564598031... e1 6/
1 .1813918433423787507... dx
x!1
1 .18143677605944287322...
012
3/1
)13(2
11,
3
4,
3
1,
3
12
kk k
F
4 .18156333442222918673...
1
152/)55(
!51532
515
1
k
kk
k
FFee
e
1 .18156494901025691257... primep
p 31 Berndt 5.28
=
1
3
)(
)6(
)3(
k k
k
Titchmarsh 1.2.7
=
squarefreeq
q 3
1 .18163590060367735153...
0
4
22 )(sin
3
2dx
x
x GR 3.852.3
.18174808646696599422... 2 2
124 2 1 2 2
2
24 arcsinh log
log
=1
2 2 121
kk k k( )
.1817655842134115276...
3
5
2
3log3
322
3
.181780776652076229520... 5
cot2
25)3(10log25
6
5125
2
5
2sinlog
5
4cos
5sinlog
5
2cos50
=
k
k
k
k
kk 5
)3()1(
5
1 1
134
.181783302972344801352...
123
2
34
1
27
16
9
2
4
2log3
18 k kk
.18181818181818181818 = 11
2
.1819778713379401005... 612
6 2 6 33 1
3 1
1
6
2
3 21
log log( )k
k k k
.18199538670263388783... ( )k
kk 31
1 .18211814116655673118... HypPFQk
kk
1
211
1
4
3
4
1
16
14
20
, , , , ,
.18214511576348082181... 2 2
1
1122
2
21
1
loglog
( )( )k
k
k
k
.182153502605238066761...
212 3
)3(
)13(3
1
13
1
kk
k
kkk
kk
.182321556793954626212...
11 6
10,1,
6
1
6
1
5
6log
kk k
Li
=
0
12 )12(11
1211arctanh2
kk k
J248
.182377638546170138737...
1
1
12
22
)1(
8
1,
2
3,2,2
4
1
k k
k
k
kk
F
.182482081626744776419... 11
6 4
2
23 1 3 2
1
32
log( ) ( )
( )k
kk
.182487830587610500802... ( )
1
3
1
31
k
k k
.18254472613276952213... ( ) ( ) ( )( ) ( )3 11
42 22 2 i i
= )22,3()22,3(2
11)3( ii
=x x
e edxx x
2 2
0 1
cos
( )
.182770451872025159608...
14
12
2 log
)3(
1
9
)2(
54 xx
x
kk
27 .182818284590452353603... e10
.18316463254842946724... 1
41 2 1
2
1
log( cos )sin cos
k k
kk
.183203987462730633284... 5
2 22 2
2
27 41 4 1 4
42
/
/ /cot cothkk
= 2 4 11
k
k
k ( )
.18331367237300160091...
)3(36962log824
1
4
3
512
1 23)3()3( G
= dxx
x1
02
4arctan
.18334824604139334840... ( ) ( )
arctan
1 1
2 1
1 1 1
2 2
k
k k
k
k k k k
.183392750954659037507...
0
)1(
22
1
2
1
k
k
k
1 .18345229451243828094...
2
1
2
1
4
1
2
3
4
1
2 10
, ,( )
k
k k
= dx
xlog cot
/
0
4
GR 3.511
2 .183488127407812202908... G
2
.1835034190722739673... 12
3
1 2 1
2 2
1
1
( ) ( )!!
( )!
k
kk
k
k
.183547497347006560022...
2 2
)()1(
kk
k k
.18356105041486997256... dxee
xxx
0 1
sin1
2
coth
.183578490978106856292...
12
)()1(
2
1
14
)2(
12
)2(
11k
k
kk
kk
kkk
.18359374976716935640... ( )k
k
k 221
.18373345259830798076...
488 .183816543814241755771...
3
1)3(
23 .183906551043041255504... e e 2cosh
.1839397205857211608...
1 )!12(2
1
k k
k
e
=
1
2xe
dxx
= 3
15
12
1sinh
2
11sinh
x
dx
xx
dx
x
.183949745276838704899...
12 644
1
6
1
2
5tanh
20
5
k kk
1 .184000000000000000000 =
1
2
4125
148
kk
k kF
2 .184009470267851952895...
12
2/12/312/)12
1
kk
kkk
3 .184009470267851952895...
32
2/31)2/(
1
kk
kkk
.1840341753914914215...
primep p 31
1log)3(log HW Sec. 17.7
=
23 log
)(
k kk
k Titchmarsh 1.1.9
3 .18416722563191043326...
121k k
k
.18424139854155154160...
1
16 443
5log
xx
dx
.18430873967674565649... 3
4
3 3
42
6 1
12
121
log ( )!
( )!( )
k
k kk
.18432034259551909517...
012
1
)4)(1(
)1(
3
)1(
18
5
3
2log2
k
kk
kkkk
= 4
1
1
0
2 11log)1log(
x
dx
xdxxx
.184471213280670619168... 381
4
9
)3(8 3
= ))1(())1(()1()1(1
2 3/23
3/13
3/13/1
LiLi
=
1
02
2
1
log
xx
dxxx
.18453467186138662977... 3
32
20
27
2 1
4
4 4 1
4 1
2 2
1
2 2
2 32
k k k k
kkk k
( ( ) ) ( )
( )
1 .18459307293865315132...
00
4/1
2!)!12(
1
)!12(
!
2
1
kk
k kk
kerfe
= dxe
ex
x
1
0
2
1 .184626477442252628916...
21
121 2sin
1
)!12(
)1)2((2)1(
kk
kk
kkk
k
.184713841175145145685... )6()3( 1 .184860269128474... 1 6( )
.184904842471782293011...
2
/1
1
12
11
!
)1)12(()1(
k
k
k
k
ekk
k
.184914711864907675754...
1 1
2
2
/11
1
kxk
dxe
x
e
k
eee
.185066349816248591534... 8
491
8
7 81
logkHk
kk
2 .18528545178748245... 1 3 2( / )
.18533014860121959977... sin1
31
3x
dx
x
8 .1853527718724499700... 67
24 .18541655363135590105...
1 !
)(1k
kk
e
k
HeeEie
.18544423087144965373...
1
164
2
48
2
12
1
16
1
43
2 3
3Li Li( )log log
=log2
31 4
x
x x
1 .18559746638186798649...
111112 2
)(
4
1
22
1
14
2
12
1
kk
kkk
kk
k
kk
k
1 .185662037037037037037... 256103
2610003
5 H ( )
1 .1856670077645066173...
32 22 2 2
2 12
2
2 21
csc sin( )
k
kk
.185784535800659241215... 4/
0
tan8
2log
2
dxxx
G GR 3.747.6
.18595599358672051939...
22
1
1 11log
11)12()1(
kk
k
kkk
k
2 .186048141120867362993... 1
22 1 3 2 2
221 1
1
log log( )
Hk
k
4 .18605292651171959669...
2
/3
2
1!
)1)((3
k
k
k
k
ek
k
.186055863117145362029...
1
22
)32)(1(2log42log88
3
1
6 k
k
kkk
H
.18618296110159529376... ( )
1
2 20
k
kk
.18620063576578214941... 23
3
12
2 11
k
kk k
k( )
= dxx
x
1
031
1log
.186232282520322204432...
0
4
3 )!34(2
sinh
k
k
k
.1864467513294867736... H
kk
kk
( )2
1 6
.186454141432592057975...
2
1arcsinh26log
831
32log
4
= arcsinxx
xdx
1 2 20
1
GR 4.521.3
.18646310636181362051...
( )3
41
kk
k
.18647806660946676075...
1
12
2
!
2)1(
)2(2
k
k
k
k
k
k
e
I
.18650334233862388596... 1
11 3 13
2
1
1kk
k
k
k
( ) ( )
=
3
5
6
1
61 3
1 3
21 3
1 3
2
i
ii
i
1 .18650334233862388596... 1
11 1 3 15 4 3 2
0
1
1k k k k kk
k
k
k
( ) ( )
=
23 12
1
k k
k
=
2
3131
2
3131
6
1
23
ii
ii
= 1
63 2 2 1 1 2 1 2 1 2 12 3 1 3 2 3 2 3 ( ) ( ) ( ) ( )/ / / /
1 .1866007335148928206... k
ekk
11
.1866931403397504774... )2(2
1)2(
3
2
2
14,3,
2
1F1
2
1120211 I
eI
e
=
k
k
kk
k 2
)!2(
1)1(
1
1
.186744429183676803404... 2
12sin
4)!(
)!2()1(
4
1sin
)2/1cos(2
1
02
k
k
k
kk
k
.186775363945712090196...
1156
2log
36 xx
dx
1 .18682733772005388216...
12 2
1
2
7tanh
7 k kk
1 .18683727525826858588...
0
3
3)!1(
1133
kkk
.186945348380258025266...
1
3
12
)23(
)1(2736)3(6
16
1
k
k
kk
.18704390591656489823... k
k
k k
k
k
3
2)1(
7
3
7
2
0
.18716578302492737369...
2 )1)((2
1)1(
kk k
k
1 .18723136493137661467...
1 23
1
32
1arcsin11
121
12
11
1
k k
k
k
1 .18741041172372594879... 7
K Ex. 108d
.187426422828231080265... 2/
0
22
sinlog16
)3(3
24
2log dxxx
.187500000000000000000 =
1
1
3)1(
16
3
kk
k k
=
21
1
)2)(2(
)1(
kk
k
kk GR 0.237.5
=
1 116
4)(
kk
kk
1 .187538169020838240502... 12 24
22
2
22
0
1
i i i
Li e x x dxilog ( ) log( cos )
.188016647161420393823...
2 )1)(2(
1)(
k kk
k
.18806319451591876232... 9
1
1 .188163855465169281367... dxx
xG
1
21
)1log(
8
2log
= 4/
0
)cot1log(
dxx GR 4.227.13
1 .18834174768091853673...
1/1
2
11
)!1(
)()1(
kk
k
k
kekk
k
.1883873799298847433... 21log22
1
2
1
= 1
42 2 1
1
21
1
2
1
2
arcsinh arctanh
= 1
2 2 1 2 11k
k k k( )( )
1 .18839510577812121626...
0
2
)!2(
42)1(1csc
k
kkk
k
B [Ramanujan] Berndt Ch. 5
.188399266485106896861...
12
2
)2)(4(6
11
k kk
k
.18842230702002753786...
24 10
1
k k
.18872300160567799280... 2
)21(82log4)21log(2
=
8
3sinlog
8sinlog22log4
8cot
28
=
k
k
k
k
kk 8
)1()1(
8
1 1
12
Prud. 5.1.5.21
= dxx 1
0
6 )1log(
.1887270467095280645... 4
91
16
9
1
41
log( )k Hk
kk
.1887703343907271768... 1
2 1( ) e J147
4 .188790204786390984617... 3
4 , volume of the unit sphere
3 .1889178875489624223... sinh
2
3 21
22
2
kk
.188948447698738204055...
12 )!2(!
1
2
1)2(
k kkI
.189069783783671236044...
1 )1(3
1
3
2log21
kk kk
J149
=
1
1
)1(2
)1(
kk
k
k
1 .18920711500272106672... 2 11
4 34
0
( )k
k k
.18930064124495395454... )1()1()()( iiii
1 .18930064124495395454... )2()2( ii
.1893037484509927148... 1 4
32
0
1
x xdxsin
1 .18934087784832795825...
12
/1
2
12
)!2(
)()1(
k
k
k
k
kke
k
k
.1893415020203205152..
1
1
2
)1(
kk
kkk HH
.189472345820492351902... dxexe
erf x
1
2 2
2
1
4
1
.189503010523246254898...
1
1 2
)12(2
)1(
2
2arcsinh21
kk
k
k
k
k
=
12)1(2k
kk
k
H
2 .18950347926669754176...
2
22 1)(k
kk
.18950600846025541144... dxxx
xdx
xx
x
2
12
21
0
23 log
)1(
)1(log
3
2log
4
)3(
.189727372555663311541...
1
221 )18(
8
4
)12(
2
11
22
11
28
1
kkk k
kkk
406 .189869040564760332...
0
2
!
cosh
24
1 22
k
ee
k
keee
.1898789722210629262... 1
2
1
1
1
21
csch
2
( )k
k k
.189957907718062725272...
2 5
517
20
1
5
1
23
csc( )
k
k k
.18995863340718094647...
12
1 9
11log
9
)2(
33
2log
kkk kk
k
.190028484997676054144...
22
1 11log
1)1()(
)1(
kk
k
kk
kkk
k
.190086499075236586883...
1 )24(3
1
13
13log
kk k
1 .19020822799902279358... 1
1 32
kk
1 .190291666666666666666 = 6)3(
24000
28567H
.19042323918287231446...
1
21
21
2)!1()!1(
)!()!(124
2
7
kkkk
kkE
.190598923241496942... 15 8 3
6
1
6 2
2
1
kk k
k
k( )
.190632130643561206769...
2
1)(12
)1(
kk
k
k
1 .19063934875899894829...
3
7
.1906471826647054863...
12 1
2 1
2
20
1
arctan tan sin
sin
x
xdx
1 .190661478044776833140... 11
220
( )k
kk
.1906692472681567121...
1 )!2(
1
2
)1(
24
9
k kk
Eie
=1
1 33 ( )! !k kk
.190800137777535619037...
12
22
66
126log6log5log25log
2
1
kk
k
k
HLi
=
12
1
5
)1(
kk
k
k
.190856278271211720658... i i i i i
16
1
2
2
2
2
2
1
21 1 1 1 ( ) ( ) ( ) ( )
= ( )
( )
1
1
1
2 21
k
k
k
k
.190985931710274402923... 53
.1910642658527378865...
22
22
111)(
kk kkLi
k
k
.19123512945396982066...
12 544
1
5
1
8
tanh
k kk
.1913909914437681436...
12
1
2
)1(
k
k
k
3 .191538243211461423520... 2
8
.19166717873220686446...
( )2
51
kk
k
.19178804830118728496...
2
143
3log2
3
2log4
xx
dx
3 .19184376659888928138... ( )
( )
2 1
3 1
.191899085506264827981...
11
1 1sin
1
)!12(
)12()1(
kk
k
kkk
k
1 .19190510063271858384... G2
.192046874791755813246... 1ee
e J152
.19209280551924242045... log k
ekk
1
.192127760367036391091...
0
2
2
2
12cschsinh
2
1
k k
k
.1923076923076923076 =
0
5sin
26
5dx
e
xx
=
0
)12/()5(log)1(5logcosh2
1
k
kk e J943
.192315516821184589663... 3
.192405221633844286869... 3
.192406280693940317174...
1
3
8kkkH
.19245008972987525484...
0
2
2
)1(
33
1
kk
k
k
kk
.19247498022682262243... 2 6 2 3 62 1
2 31
log log( )!!
( )!
k
k kkk
.192547577875765302901...
1
2
2 )2(2sin2
8
1csc
kk
kk
1 .192549103088454622134... 26
3
.192694724646388148682...
1 2 11 2
0
e Eix
e xdxx( )
( )
.192901316796912429363...
4
3
4 4
1 2 1
2 1
1
0
loglog ( ) log( )
k
k
k
k
[Ramanujan] Berndt Ch. 8
1 .193057650962194448202...
129
11
3sinh
3
k k
.193147180559945309417...
222 )12(2
1
24
1
2
12log
kk kkkk
=
1 )12(2)12(
1
k kkk J236
=1
8 231 k kk
[Ramanujan] Berndt Ch2, 0.1
=
15
1
0 4
)1(
3
)1(
k
k
k
k
kkk K Ex. 107f
=
2 2
1
kk k
=
0 )3)(2)(1(2
1
kk kkk
J279
=
112
2 2
)12(
2
1)(
kk
kk
kk
=
134
223 xx
dx
xx
dx
=dx
x x x
dx
x( )log
11
12
12
13
=
02 sinh)1( xx
dxx
GR 3.522
= dx
e ex x20 1
1 .193207118561710398445... 7)3(
8232000
9822481H
2 .193245422464301915297... 2
9
32
2
21
k
k k
kk
2 .19328005073801545656... e i i / /4 2
.193497800269717788082... 4/
0
tanlog16
)3(7
4
dxxx
G
.193509206599196935107...
0
3 3/1
6xe
dx
1 .193546496554454968807...
1
2/1
1
1)12(!
1
kkk
k
ek
41 .193555674716123563188...
1
1
)!1(k
ke
k
ee
.19370762577767145856... )7()3(
.193710100392006734616... 1log1
!
1)3()1(
3
21
1
k
kk
k ekk
k
.193998857662600941460...
log log log2
1 3
2
1 3
2
i i
=
3/23/1 )1()1(2
1log
= log ( )( )
11
13 1
32
1
1
k
k
kk
k
k
.194035667163966533285...
0
2 32
)1(
2
3csch
3
6
6
1
k
k
k
.19403879384913797074... ( ) ( )
log
1 1
1
1
21
1 1
2
1
422
2
2
k
k k
k
k
k
k k k
.194169602671133044121...
1
)3(1
4)1(
kk
kk H
.194173022150715234759... 2
1)1(1)1(1
44/14/1 i
4/34/3 )1(1)1(14
i
=
1
1
24
1)14()1(1 k
k
k
kk
k
.194182706187011851678...
122 2
1
22
2cot
4
1
k k
.194444444444444444444 =
11 )3)(1(
10,1,
7
1
736
7
kkk kkk
k
.19449226482417135531... 2
1
2 .194528049465325113615... e k
k
k
k
2
1
3
2
2
2
( )!
3 .194528049465325113615... e
k
k
k
2
0
1
2
2
1
( )!
4 .194528049465325113615... e k
k
k
k
2
1
1
2
2
1
( )!
57 .194577266401621023664... 2 2 22 1
12
0 11
e I Ik
k kk
( ) ( )( )!
.194700195767851217061... 1
41
41 41
1e
k
kk
kk
/ ( )!
5 .194940984113795745459... 9
1 .194957661910227628163... 2
1
2
1
2 2 10
erfik kk
k
! ( )
.195090322016128267848... 16
7cos
16sin
.195262145875634983733... 4/
04
3
cos
sin
3
22
dxx
x
.195289701596312945328...
12
24
2
3
)12()2(
3
11
3cot
3)2(
kk
k
kk
kk
k
.19541428041665277483...
1
1)4(2
k
kk
k
.19545006939687776935... dxxx
1
1)12(
.195527927242921852033...
1 )!1(k
k
k
kB
.195686394433340358651... dxe
xHypPFQ
x
1
222 log
1},2,2,2{},1,1,1{26
1 .195726097034987200905... 2
cot2
.19573254621533971315...
1
123
3
72
3
72
1
12
1
33
2 3
3Li Li( )log log
=log2
31 3
x
x xdx
.19582440721127770263....
2
2)1)()1(
k
k
k
k
1 .195901614544554380728... ek
k kk
(cos )/ cossin cos
!1 2
0
1
2 2
GR 1.463.1
= 2/2/
2
1 ii ee ee
2 .196152422706631880582...
0
2
6
1133
kk k
k
5 .196152422706631880582... 27
.19621530110394897370....
1
2
1
13
)1(
3csch
322
1
k
k
k
.196218053911705438694...
1
2
92
2log3
8
3log9
kk
kH
1 .19630930268377512457...
21
2
0
tanh sinsinh
x dx
x
.19634954084936207740...
16
=
04 4)12()12(
)1(
k
k
kk K Ex. 107e
= ( )
( )sin
1
2 1
2 1
4
1
21
k
k k
k J523
= sin cos sin cos3
1
3 2
1
k k
k
k k
kk k
=
22
2
032
2
)4()1( x
dxx
x
dxx
=
044 xx ee
dx
1 .1964255215679256229... H kkk
21
2 1( ( ) )
1 .1964612764365364055... 44
2 2
20
1
Gxdx
x arcsin
= dxx
xx
2/
02
2
sin
cos
GR 3.837.5
= dxxx
x
022
3
1
arctan GR 4.534
4 .19650915062661921078... 14},1,1{,2
12
)!(
1
12
HypPFQk
k
kk
.196516308968177764468...
3
1
6
3log
318327
2
9
)3(13 )1(3
=
12)13(k
k
k
H
.196548095270468200041...
0
2
10
!
2)1(22
k
kk
kJ
.196611933241481852537... e
e
k
e
k
kk( )
( )
1
12
1
1
.1967948799868928... ( )( )
15 1
1
1
kk
k
k
1 .19682684120429803382... 2
3
.1968464433591154...
2 log)12(
)1(
k
k
kkk
.196914645260608571900... 23 3
2 1
2
0
( )
)
x
e edxx x
=
023
2
123 xx ee
dxx
xx
dx
.196939316767279558589... 9
.196963839431033284292... ( )
! ( )
1
2
k
k k k
2 .19710775308636668118... 25 1
32
27 1
32
13
16
1
32 2
5
1
cosh sinh
( )!
e
e
k
kk
2 .19722457733621938279... 3log2
.19729351159865257571... 12
2
1 4
2 1 2 1
1
1
si
k k
k k
k
( ) ( )
( )!( )
1 .19732915450711073927...
4
5,21682 G
17 .19732915450711073927...
12
)1(2
)4/1(
1
4
18
k kG
.19739555984988075837...
0
12 )12(5
)1(
5
1arctan
kk
k
k
=
1
2)2(2
1arctan
k k [Ramanujan] Berndt Ch. 2, Eq. 7.6
.197395559849880758370...
0
12 )12(5
)1(
5
1arctan
kk
k
k
1 .19746703342411321823...
1
2
)2)12()2((128
3
k
kkk
.19749121692001552262... 2 13 1
21
2 11
2
1
cossin
( )( )!( )
k
k
k
k k
1 .197629012645252763574...
12 116
21
4coth
4 k k
.197700105960963691568...
11 9
)2(
)13)(13(
1
362
1
kk
k
k
kk
.197786228974959953484... 1
254 3 2 2 5 4 2 1
41 2
1
arccot log log ( )k kk
k
kH
.197799773901004822074... 60 222 101
ek
k kk !( )
.19783868142621756223... 2 2 2 2
04
2
2
2
4 48
log log log sinx x
xdx
3 .19784701747655329989... 4
)3(39
36
5
6
7
16
1
8
)3(1327
312
19 3)2(
3
=
13
1
)3/1(
)1(
k
k
k
.19809855862344313982... 11 5 2 13 2
2
1
1k kk
k
k
k
( ) ( )
.19812624288563685333...
( )log
33
1
k
kk
1 .19814023473559220744...
4
5
4
3
2)1()1(2 1
KE
= dxxdxx
x
2/
002
2
sinsin1
sin
.198286366972179591534...
1
2)12()2(k
kk
.198457336201944398935...
0
21 )32()!)!12((
)1(2)1(
k
k
kkH
.198533563970020508153...
1
21
2
)1)1(()1(
kk
k k
41 .198612524858179308583...
1
4
)!1(
4
4
13
k
k
k
ke
.198751655234615030482...
0 23
)1(
kk
k
.198773136847982861838...
4 32 3
17
66
1
1224
csc( )k
k k
.19912432950340708603... ( )
! !!
1
0
k
k k k
.199240645990717318999... 1
0
2 log)1(2223 dxxxeEie x
.199270073726951610183... ( )( )
log ( )
1 2 11
1
k
k
k
kk
.199376013451697954050...
1 2log
k
k
kk
.199445241410995064698...
1 )74(
1
7
8log
14147
40
k kk
.19947114020071633897... 8
1
1 .19948511645445344506...
3
2
3
13log3log2log
6 222
2
LiLi
2 .199500340589232933513... 2
22
1
2
2
0sintan
kk
k
k
Berndt ch. 31
.199510494297091923445... 4
21
22 2
1
21 1cos ( ) sin ( ) cos CosIntegral SinIntegral
= dx
e xx 20 4
.199577494388453983241...
13
1
2
)1(
k
k
k
k
1 .199678640257733... root of ( ) ( )x e x ex x 1 1
5 .199788433763263639553... 3
.20000000000000000000 =
11
2
15
)(
6
1
7
1arctansin
5
1
kk
kk
k
= e
x x
dxx
e
dxx
1
4
0
log2cos
= k k
k kk
( )
( )( )
5
4 11
J1061
.20007862897291959697...
2
34)31()2()2(
6
1 iiii
2
34)31(
2
34)31(
ii
ii
2
34)31(
ii
= k
kk
k
k
k
3
62
1
111 6 3 1
( ) ( ( ) )
.2000937253541084694... 3
2 2 31
3
1
21
log( )k
k k k
=2
13
1
0
3 11log)1log(
x
dx
xdxx
3 .2000937253541084694... dxx
1
03
11log
6
1
3
12log2
3
.2002462199990232558... 2 2 2 1 2 21
2 2 11
log( ) log( )k
k k k
= 2 2 2 1 2 arcsinh log
.2003428171932657142... ( )3
6
1 .20038385709894438665...
3
212
2
18
64
729log3log22 22
2 LiLi
=
3
1
)3,(1
)1(
k
k
kStirlingS
1 .2004217548761414261... 1 1 1
¼ ¾ ¼ ! ¾ !
.2006138946204895637... 1 2
2
1
81
21
2
3
2 2
3
2 2
log ii
i i ii
i
= 1
8
1
2 2
1
2 21
21
2 2
i i i itanh
3
4
2
2 8 2
logcoth
= ( )
11
1
3 21
k
k k k k
.2006514648222101678... Ei Ei( ) ( ) e e dxe x
10
1
.201224008552110501897...
0
2
)1()!2(
4)1(1sin2sin
k
kk
kk
6 .201255336059964035095... 3
5
.2013679876336687216... 9
8
2
3
2
0
e k
k
k
k ( )!
.20143689688535348132... ( )( ( ) )
!
11 2
2
k
k
k
k
.2015139603997051899... 1
6
6
3
3
2
1
2 320
csch ( )k
k k
.201657524110619408067...
0
2
2
cos
2dx
x
ex x
1 .20172660598984245326... 1
3 21k
k
.201811276083898057964...
3
23
2
123log2log33log3
2 222
2
LiLi
=
1
1
)1(2
)1(
kk
kk
kk
H
.20183043811808978382... e
e
k
kk8
3
8 2 1
2
1
( )!
.201948227658012869935... e
e e
k
ek
kk
sin
cos( )
sin1
1 2 112
1
1
.20196906525816894893... 16 4 3 5 8 12
12
( ) ( )
i i
=1
41
2 5
47 51
1
1k k
k
k
kk
k
( )( )
1 .20202249176161131715... 1
2 1G
.2020569031595942854... ( )log
3 11 1
2 132
2
0
1
k
x x
xdx
k
GR 4.26.12
= 6 2
1 1
3
2 30
2
x x
x
dx
e x
( ) Henrici, p. 274
=
1
0
1
0
1
0 1dzdydx
xyz
xyz
1 .2020569031595942854...
1
32
2
13 )1(
1331)3(
kk kk
kk
k
= )1,2(MHS
= H
kk
k ( + )1 21
Berndt 9.9.4
=
122 )1(
)12(
k
k
kk
kH
=
1
)2(
)1(k
k
kk
H
= 5
2
12
5
2
1
2
1
31
1 2
31
( ) ( ) ( !)
( )!
k
k
k
kk
kk
k
k k Croft/Guy F17
Originally Apéry, Asterisque, Soc. Math. de France 61 (1979) 11-13
= ( )( !) ( )
(( )!)
1205 250 77
64 2 11
10 2
5k k k k
k Elec. J. Comb. 4(2) (1997)
=
1
23
3
123
3 )(2
180
7
1
12
180
7
kk
kk e
k
ek
Grosswald
Nachr. der Akad. Wiss. Göttingen, Math. Phys. Kl. II (1970) 9-13
=
2
12
)(4
45
2 22
123
3
kke
k
kk
Terras
Acta Arith. XXIX (1976) 181-189
= 4
7
2
4 2 2
2
72
7
2
0
2 2 ( )log
k
kkk
Yue 1993
= 4
9
2
4 2 3
2
92
2
27
2
0
2 2 ( )log
k
kkk
Yue 1993
=
22
4 2 2 2 32
0
( )
)(
k
k kkk
Yue 1993
=
4
7
2
4 2 1 2 2
2
0
( )
( )( )
k
k kkk
Ewell, AMM 97 (1990) 209-210
=
8
9
2
4 2 1 2 3
2
0
( )
( )( )
k
k kkk
Yue 1993
=
0
2
)32)(22)(12(4
)2()52(
3 kk kkk
kk Cvijović 1997
=
0
22
)32)(22)(12(4
))2(1)(52(
32
3log31
3
4
kk kkk
kk
Cvijović 1997
=
0
122
)22)(12(16
)2()12(4
4log
2
3
14 kk
k
kkk
k
Ewell, Rocky Mtn. J. Math. 25, 3(1995) 1003-1012
= primep
p 41)6(
=
primep pp
ppp21
321
1
1)4(
= 5
41 1 1 1
3
22 2
1
4HypPFQ[{ , , , },{ , , }, ]
= dxx
x
1
0
2
1
log
2
1 GR 4.26.12
= dxexdxxx
x x
01
2
2
1loglog
= 1
0
2 )1(logdx
x
x
= log( ) log10
1
x xdx
x GR 4.315.3
= 4
32
0
4
log(cos ) tan/
x xdx
GR 4.391.4
= dxx
xx
4/
0
22
tan1
)tan(logsec
= 16
32
0
1
arctan log (log )x x x dx GR 4.594
=
0 2
0
2
12
1
12
1dx
e
ex
e
dxxx
x
x
= x dx
ex
5
0
2
1
= 2
3 1
3e
edx
x
e x
GR 3.333.2
= 1
2 1
3e
edx
x
e x
GR 3.333.1
=
0
2 dxeLi x
2 .2020569031595942854... ( ) ( ( ) ( ))( )
3 1 22 1
12
2
32
k k kk k
k kk k
.20205881140141513388... )1)1316((1
12133
kk
kkk
.20206072056927833979... )1)1215((1
12143
kk
kkk
.2020626180169407781... ( )log ( )
12
k
k
k
k
.2020645408229235151... )1)1114((1
12113
kk
kkk
.20207218728189006223... )1)1013((1
12103
kk
kkk
.20208749884843252958... )1)912((1
1293
kk
kkk
.20211818111271553567... )1)811((1
1283
kk
kkk
.20217973584362803956... )1)710((1
1273
kk
kkk
.20218183904523934454... ( )( )
!
12
12
2
2
1
kk
k
k
k
k
ke
k
.20230346757903910324... )1)69((1
1263
kk
kkk
.2025530074563600768... 4/34/1 )1(cot)1(cot82416
7
4
ii
4/34/1 )1()1(4
)2()2(8
1 i
ii
= 1
8 5 13 5
2 1k kk
k k
( ( ) )
.2026055828608337918...
144 5
1
5
10,4,
5
1
kk k
Li
.20261336470055239403... 3 1
81
2
3
1
cos( )
( )!
k
k
k
k
.20264236728467554289... 2 2
22 21
( )
( )
k
kk
= x xdx x xdxcos cos 0
12
0
1
=
0
2/1xe
dx
.20273255405408219099... log log3 2
2
1
5
arctanh AS 4.5.64, J941, K148
= 1
5 2 1
1
2
1
3
1
22 10 1
1
11
kk
kk
k
kkk k k
( )
( )
=
1 1
)(
1
1
32
1
3 n kk
kn
kkk HH
= log
( )
xdx
x2 1 21
= ( ) sin sinx x x
xdx
5
0
1
GR 1.513.7
.20278149888418782515... 2/12/12/12/1 22224
eeeei
=
1
sin1)12(k
kk
.2030591755625688533... 1
7 4 13 4
2 1k kk
k k
( ( ) )
.2030714163944594964... ( )
log( )6 3 1
11
3 6
2
k
kk k
k k
3 .203171468376931... root of ( )x 1
2 .2032805943696585702... tan(log )
i ii i
i i
.2032809514312953715... log3
4
.2034004007029468657... 1 11
1
1
1
1
21
1
2
Eik k k k
k
k
k
k
( )( )
!( )
( )
( )! !
.2034098247917199633... ( ) ( ) ( )2 3 3 2 4
.2034498410585544626... 3 3
12
1
2 3 20
( )
( )( )
k
k k k
.20351594808527823401... log ( ) 31
kk
.2037037037037037037 = 11
54
1
3 91
k kk ( )
.20378322349200082864... ( ) ( )( )
2 2 326
27 2 32
k
kk
2 .20385659643...
1
2
2
2 )1(
11
6)1(11
primepprimep pppp
p
.20387066303430163678... 1024 12832
3
2
45768 2 16 3 5
2 4
log ( ) ( )
=1
41
5
46 51
1
1k k
k
k
kk
k
( )( )
61 .20393695705629065255... 5
5
.2040552253015036112...
2
53log5205log5
4
552553
6
=
1
1
123 5
)2()1(
5
1
kk
k
k
k
kk
.2040955659954144905... ( ) ( ) coth4 4 2 4
28
=1
41
2 4
46 41
1
1k k
k
k
kk
k
( )( )
.20409636872394546218...
3
1
12
1
6
3 3
2
3 3
2
i i
=1
6 3 13 3
2 1k kk
k k
( ( ) )
.204137464513048262896... ( )
log( )5 2 1
11
2 5
2
k
kk k
k k
.20420112391462942986... ( )4 1 1 1
21
22
4
k
kk Li
kk k
.20438826329796590155... H k
kk
( )3
1 6
2 .204389986991009810573... loglog
2
2
[Ramanujan] Berndt ch. 22
1 .20450727367468051428... e
x e dxx4
2 1
0
2
5 .204869916193387826218...
0
2
2
1
2csc2sinh2
k k
kh
.20487993766538552923... 2 1 2 2 2 2 2 32 1
4 1
1
1
log log( )
( )
k
k k k
k
kk
.205074501291913924132...
1
1)1()1(2
2log2log4k
kk
k
k
.205324195733310319...
133 5
1
5
10,3,
5
1
kk k
Li
.205616758356028305...
2 1
2 21148
2
8
1
42
( ) ( ) cosk
kk k
k
k
= )()(2
122 iLiiLi
=
1 )1(k
k
kk
H
=log xdx
x x31
=
x xdx
x
log2
0
1
1
= log 11
41
x
dx
x
.20569925553652392085... ( )
12 1
1
1
k
kk
1 .2057408843136237278... 11
61
kk
.2057996486783263402... 7
180
3
31
coth k
kk
Ramanujan
.205904604818110652966...
0
33/5
sin3
4
2
13 3
dxxe x
.20602208432520564821...
2
3
1
3 32
2
20
1
Lix
xdx
log
( )
.2061001222524821839... 2
2136
49
720
1
6
k kk ( )
.2061649072311420998... ( )kk
2
2
1 1
.206183195886038352118...
1 611
5821log
5
13coth6
55
1
kk
kkk
k
FFFarc
.206260493108447712133...
1
1)1()1( )12(
4
)1(
21
21
8 kk
k
kkiii
.2062609130638129393... 15 2 13 2
2 12k k
k ik k
( ( ) ) Re ( ) Li
.20627022192421495449... ( )3 2
6
1
615 2
1
k
k kkk k
.2063431723054151902... 2 2 2 4 2 3 2 3 41
32 2
2 log log log log Li
=( )( )
12 2 2
0
k
kk k
.20640000777321563927... ( )( )
13
5
1
5 11
13
1
kk
k k
k
k
.20640432108289179934... ( )
log( )4 1 1
11
4
2
k
kk k
k k
.2064267754028415969... 3
6 3
3
4
1
3 122
coth
kk
= ( )( )
12 13
1
1
kk
k
k
.2064919501357334736... H kk k
k
kkk k
( ( ) ) log
3 11
113 2
2
1 .206541445572400796162... 1
2
3
21
1 3
22
log ( )( )
k
k
k
k
Srivastava
J. Math. Anal. & Applics. 134 (1988) 129-140
.2066325441561950286... ( )3 1 1
21
2 32
k
kLi
kk k
.2070019212239866979... I
e
k
k kk
k
k
04
0
41
2 2( )( )
!
1 .20702166335531798225... 1 11
2 11
e erfik kk
( )!( )
=e
xdx
x 2
12
0
1 GR 3.466.3
5 .2070620835894383024... e
e
e
k
e k
k
1
10 ( )!
3 .20718624102621137466...
1
42
)4(1728
3023
288
25
2
)3(
k
kk
kk
HH
4 .2071991610585799989... 7 3
2
1
1 1
2
0
2
1
0 ( )
sinh
log ( )
( ) sinh(log( ))
x dx
x
x
x xdx
1 .207248417124939789...
1
4
1
2 10
eerfi e e erfi
e
k
k kk
( )sinh
!( )
.2073855510286739853... ( )5
5
.20747371684548153956... ( )( )
( )
12 1
2 11
1
k
k
k
k k
.20749259869231265718... 10
8
754
0
1
x xdxarcsin GR 4.523.1
.207546365655439172084... 1
22 2 1
12
log ( )( )
( )
k
k
k
k k
2 .2077177285358922614...
2
23 ))()((k
kk
.2078795763507619085... iii eei log2/ )valueprincipal(
=
2sinh
2cosh)sin(log)cos(log
iii
.20788622497735456602...
1
2
3 .207897324931068989... e e e
e
k
k
e e
k
2 1 2
0
1
2 1
/ cosh
( )!
1 .208150778890... 1
42 ( )kk
.2082601377261734183... dxxxx
dx
2/
0
3532
cossin)2(232
3
.2082642730272670724... 256)3(42log192)4(3
832
2
=1
41
4
45 41
1
1k k
k
k
kk
k
( )( )
.20833333333333333333 =5
24
15
1
log( )x
xdx
.2084273952291510642... dxe
x
eEiEie
x
1
0
)1log(2log))1()2((
.20853542049582140151... dxxx
xLi
13
2
3 2
log2
8
1
.2087121525220799967... 5 21
2
1
7 1
2
1
kk k
k
k( )
= 11
3 1
21
0
( )
( )
k
kk k
k
k
1 .2087325662825996745... e ek e
ek
k
1 1
0
1
1
/
( )!
.20873967247130024435...
( ) Re( )
( )3 4 2 1
22 1
2
3
21
i i k
i kk
=1
41
2 3
45 31
1
1k k
k
k
kk
k
( )( )
6 .208758035711110196... dxeStruveLI x
0
sin00 )1()1(
.2087613945440038371... 2 1
2112
2 2 21
1
log( )
( )
k
k k k J365
= ( )
Re( )
( )
1 3
23 22 1
k
kk
kk k
k
i
= log log( )x x dx10
1
GR 4.221.2
1 .2087613945440038371... 2
0
1
122 2 1
1 1
log
( ) log( )x x
xdx
2 .2087613945440038371... 2
2
1122 2
1
log( )
H
k kk
k
=
log log1
1
0
1
xxdx
= log( )log1
21
x x
xdx
1 .20888446534786304666... ( ( ) ) k k
k
1 2
2
.208891435466871385503... ii ee 2log2log2
11cos)1(
=
2
1)(cos
k
kk
k
.20901547528998000945... 6
5
1
6
1
5
1
5
1
6 1 62 21
2
1
Lik
Hk
k
k
kk
( )
( )
.2091083022560555467... ( )( )
14 1
4
1
415
1
kk
k k
k
k k
=
1
2
1
2
3
2
3
2
i i
.2091146336814109663... 2
8 2
1
3
1
4 4 321
tanh
k kk
.2091867376129094839... ( )
( )!sinh
2 1
2 1
1 1
1 1
k
k k kk k
.2091995761561452337... 2
3 31
1
2
2
0
2
( sin )
sin
/ x
xdx
1 .2091995761561452337... 2
3 3 2 1 2 2 10
2
0
k
k
k
kkk k
!
( )!!
( !)
( )! CFG D11, J261, J265
= 1
22 1
0 k
kk
k
( )
= 9
3 1 3 1
2
1
k
k kk ( )( )
=dx
x
dx
e e
dx
xx x30 0 01 1 2
sin
=xdx
x30 1
GR 2.145.3
= dx
x x
dx
x x20
211 1
.209200400411942123073...
5046
5495514311log
3
4log5
89
5413arctan52
555
1h
2
2)1)((k
kk
1 .20930198120402108873... 81
2
9 3
2
3
2
1
1 9
2 1
2 21
( )
( / )
k
k k
1 .20934501087260716028... = 2 2
20
2
4 2 2 1
log log
xdx
x
.20943951023931954923...
6
2)cos(sin
15 x
xxx Prud. 2.5.29.24
.20947938452318754957...
2 12 2
1
2
)(
2
12log
k jkj
kk j
kk
k
.2095238095238095238 = 22
105
1
2 1 2 9
1
4 2 4
2
0 0
( )( ) ( )( )k k k k
k
kkk
k
.209657040241010873498... log( )csc ( )
1 2 1
1
k
kkk
.2099125364431486880...
1
2
2 88
10,2,
8
1
343
72
kk
kLi
.2099703838980124359... ( )log
!
11
k
k
k
k
2 .2100595293751996419...
0 03
2
3
21
02
23
1
log
1
log
1
log
381
10
x
dxxx
x
dxx
xx
dxx GR 4.261.2
= log2
2 10
x
x xdx
1 .2100728623371011609...
2
1
2
1
28
2log
248
13
2
1,2,
2
1
8
1
8
22
222
Li
=
1
02 12
log
x
dxx
.21007463698428630555...
1
31
)()3(log
)()3(
kk k
kk
k
k Titchmarsh 1.6.2
.21010491829804817083... 4
13
6
1
4121
( )!
( )!
k
k kk
Dingle p. 70
.2103026500837349292... 1
51 ( )k kk
.21036774620197412666... sin ( )
( )!
( )
( )!( )
1
4
1
4 2 1
1
2 1 2 10
2
1
k
k
k
kk
k
k k
2 .2104061805415171122... ( ) ( )3 7
.210526315789473684 =4
19
.2105684264267640696... 6 3 2209
12 1 40
(log log )( )
( )( )
k
kk k k
8 .2105966657212352544...
2
3( )
.21065725122580698811...
122
616
12 1 2 40
log ( )( )( )
k
k k k
= x xdx2
0
1
arctan
.21070685294285255947... 1
3
1
3
13 7
1
e x
dx
xcosh
1 .21070728769546454499... log( )10
e kk
k
1 .21072413030105918014... 11
21
( )k
k k
46 .21079108380376900112... 17e
.2109329927620049189... 1
82 2 4 1 ( ) ( )i i
=
2
3
8
1
42 2
142
( ) ( )i ik
kk
= )()(4
1
8
1
2ii
= ( ( ) ) 4 1 11
kk
.210957913030417776977... 2 3 11
2
20
e Eix
e xdxx( )
( )
.21100377542970477...
1
1
122 4
)1(
5
1)
5
1(Li0,2,
5
1
kk
kk
kk k
H
k
= ( ) log log log2 4 5 54
52
2
Li
2 .211047296015498988... 2 2 1
1
1log log ( ) /
k k
k
1 .2110560275684595248... E(¾)
.21107368019578121657... ( )
!/3 1
11
1
2
3k
ke
k
k
k
7 .2111025509279785862... 52 2 13
1 .21117389623631662052...
11
2 1
21
( )!!
( )!!
k
k kk
J166
.2113553412158423477...
( )3
21
kk
k
.2113921675492335697... 12
1( ) Berndt 7.14.2
.2114662504455634405... ( )log
3 1 11
13
21
3
2
k
k kmm
kk
mk m
=
log cosh log logcosh
1
3
3
23
3
2
=
2
35log
2
35log
ii
.2116554125653740016...
e ke
kk2 1
sin
.2116629762657094129... 14
coth
8 .21168165538361560419... Ei e( )
.211724802568414922304... 2 4 32
3
23
3
162 2
4
7 3
4
2 3 2
GGlog
( )
= H
kk
k ( )4 1 21
.21180346509178047528... ( )
!
!
1
23
k
k k
.21184411114622914200...
10 )3(2
1
)82(2
1
3
162log8
kk
kk kk
.211845504171653293581... 1
42 2
0
1
log ( ) log sin cosci x x x dx
.2118668325155665206... 2 1 11
11
e e ee k kk
k
( ) log( )( )
J149
.21192220832860146172... 3 1
318
41
1 2
Gk
k
k
k
( )
( / )
1 .21218980184258542413... 1
3 1 30k
k
/
.21220659078919378103... 2
3
0
3 23
0
1
/
sinx x dx
.21231792754821907256... log log31
22 2 3 2
1
2
dx
x x
7 .2123414189575657124... 6 32
3
0
( )
x dx
e ex x
=
0 13/1xe
dx
.2123457762393784225... e
.21236601338915846716... ( )2 1 1 1 1
21
22
k
k kLi
kkk
k
3 .21241741387051884997...
2 )1(
log
k
k
kk
kH
.21243489082496738756...
( )k kk
k 2 11
.2125841657938186422...
2
2
1 42 20
2e
xdx
x
xdx
x
cos cos
AS 4.3.146, Seaborn Ex. 8.3
= sin( tan ) tan/
20
2
x xdx
GR 3.716.6
= cos( tan ) sin20
x xdx
x
GR 3.881.4
.2125900290236663752... logk
kk3
1 1
.2126941666417438848... log log2 21
2 42
31
H
k kk
k
.21271556360953137436... 10 3 24 2
1081
3
2 1
3 21
log ( )k
k k k
.2127399592398526553... Ik kk
3 0 10
216
4 11
3( ) (; ; )
!( )!
F LY 6.114
.21299012788134175955... 2 2 6 4 21 2 1
2 2
1
1
log log( ) ( )!!
( )!
k
kk
k
k k
=
1
1 2
8
)1(
kk
k
k
k
k
.2130254214972640800... ( ( ) ( )) k kk
1 2
2
.2130613194252668472... 2
11
2 1
1
1e k
k
kk
( )
( )!
.2131391994087528955...
02 )22)(12(4
)2(
4
)3(7
kk kk
k
Yue & Williams, Rocky Mtn. J. Math. 24, 4(1993)
.2131713636468552304...
8
1
2
3
29 42 2
0
2
ee erfi erfi e x xdxx
/ sin cos
.2132354226771896621... hgk k
k
k
kk
k
( ) ( )( )1
71
71
172
1
1
1
.21324361862292308081... 17
1 .21339030483058352767... 1
22
1
2 111
1k
kk
k
½
3 .21339030483058352767...
111
0
2
)(
2
)(
kk
kk
kDk,
1
0 )()(k
knD
.21339538425779600797...
12 64
1
3
2log
9
4
k kk
= 1
0
2 log1 dxxxx GR 4.241.10
= log(sin ) sin cos/
x x xdx2
0
2
GR 4.384.11
.2134900423278414108...
sin ( )
!( )
4
2
1 41
120
k k
k k k
.21355314682832332797... 2
3
1879
7560
1
3 3
1
1
log ( )
( / )
k
k k k
.21359992335049570832... 1
82 2 2 2
1 2
2 1
1 2
1
( cos sin )( )
( )!
k k
k
k
k
=
1
211
)!2(
2)1(
k
kk
k
k
.2136245483189690209... 16
32 3
132 2
1
cotkk
.2136285768839217172...
111 )92(!
1,2
13,
2
11
11
1
k kk
kF
511
32
945
641
eerfi
.2137995823051770829... 1
9
8
27
1
2 2
1
22
1
1
arcsinh( )k
kk k
k
.21395716453136275079... 1
6
3 1
641 1k k
k
kk
k
( )
.2140332924621754206... 5 5 15
515
2
k
k k
kk
( ( ) )
1 .2140948960494351644...
233/23/1
11
))1(())1((2
1
2
3cosh
2
1
k k
.2140972656978841028... 1
1
1 1
1 1e e e
dx
ekk
x
1 .214143334168883841559...
( ) ( ) ( )k k kk
12
.214252808872002637721...
2&,1#1#48649
7
9
4
81
1 22)1()1( Root
=
133 1
log
xx
dxx
.2142857142857142857 =3
14
1 .2143665571615205518... ( )logk
kk
1
2
.214390956724532164999...
122
2
1
2
)19(
9
9
)2(
36
3
27 kkk k
kkk
.21446148570677665005...
12
1)2(1)(dx
x
xdx
x
x
.2146010386577196351...
1 2
2
1
8
3
2
1
2
3
2
1
2
1
2
1
2
log
=( )
123
2
k
k k k
.21460183660255169038... 14
1
2 30
( )k
k k
=( )!!
( )! ( )
2 1
2 4 121
k
k kk
J387
=dx
x x
xdx
x
x
xdx4 2
1
2
20
4
20
1
1
sin
cos
arcsin
( )
/
2 .21473404859284429825... 3
14
1 .2147753992090018159... 4
3
)(log33log32log dxx GR 6.441.1
.2148028473931230436... 22
csclog2
2log3log
= ( )
log2
81
1
812
1
k
k kkk k
.21485158948632195387...
0 )42(4
)1(25log84log8
kk
k
k
.21506864959361451798... ( ( ) )
! k
kk
1 2
2
.215232513719719453473... ( )
1
2 1
1k
pp prime
, where p is the kth prime
.2153925084666954317... ( )( ( ) )k kk
2
2 1
.2154822031355754126... dxx
xdxxx
4/
02
34/
0
2
tan
sincossin
26
1
3
1
12
24
.2156418779165612598... 1
0
222
arccosarcsin2
624 dxxx
.21565337159490150711... 2
32
7 3
4
4
27
4
2 1
( )
( )
k
kk
= ( ) ( )( )
1 11
22
kk
k
k kk
.2157399188112040541... 7 331 5
4 6 1 2
4 2
51
( )
( )( / )
k
kk
.2157615543388356956... 34
43
13
1
1
log ( )
k kk
k
H
1 .2158542037080532573... 12 2
22
( )
2 .21587501645954320319...
( )
( )
k
kk
1
1 12
2
.21590570178361070... 1
2
1
2 2
1
0
e e
k e
k
k
( )
1 .21615887964567006081... ( ) ( )k kk
1 11
.216166179190846827...
1
12
)!2(
2)1(
4
1sinh
4
1
k
kk
ke
e
1 .2163163809651677137... cos sin sin( )
( )
1
4
1
41
1
21
1
2 10
k
k
=( )
( )!!( )
( )!!
14 4
14 2 2 40 0
k
kk
k
kkk k
.21634854253072016398...
1
2132
4
)1()1(
2
)3(7
8424
kk
k kkG
13 .2163690571131573733... k kk
4 2
2
1( ( ) )
.2163953243244931459... 3 3 3 2 11
3 11
log log( )
kk k
GR 1.513.5
= ( )
( )
12 11
k
kk k k
= log 120
1
x
2 .21643971276034212127... k
k kk k
2
21
21
3
21
1
2
.2167432468349777594... 2
323 2 2
1
8 7 8 31
log( )( )( )k kk
J266
1 .2167459561582441825... log( )( )
26
1
4 3
1
1 4 121 0
k k k kk k
.216823416815804965275... 1
2
2
2
1
8
1
2
1
2
log i i i i
= cos
)
2
0 1
x
e edxx x
.216872352716516197139... 5
4log
5
4
2
5log5log2log25log2log2)3( 2
322
Li
5
4
5
133 LiLi
= H
kk
kk 5 2
1
.2169542943774763694...
sin
2
21
22
2 k
48 .2171406141667015744... = 93 5
2
4
0
( )
sinh
x dx
x
.21740110027233965471... 2
21
94 1 3
k
k kk ( )( )
1 .2174856795003257177... 24 2 5 1 241
44 0 10
Ik kk
( ) (; ; )!( )!
F
.217542160680600158911... 1
41
1
12 2
1
2 21
cosh2 csch
sinh cosh( )
( )
k
k k
.2178566498449504... ( )
143
2
k
k k
.218086336152111299701... 1
2 242
1
2
2 1
1
log
( )
( )
kk
k
H
k k
.218116796249920922292...
1
22 1)1(cossin224 k
ii kkkeei
2 .2181595437576882231... 2
5
1 .21816975871035137... j g2 22 J311
.2182818284590452354... ek k k kk k
52
12
12 41 0( )! !( )( )
1 .218282905017277621... arctan( )
( )e
e
k
k k
k
1
2 1
2 1
0
.2185147401709677798... ( ) log3 48 22
38 64
14
2
4 31
k kk
= ( )( )
13
41
1
kk
k
k
37 .2185600000000000000 =116308
3125 4
2
1
F kkk
k
3
1 .2186792518257414537... 2 2
2 2
sinh
2 .21869835298393806209...
2
1
1arcsin
kk
.218709374152227347877...
12
11log
1arctan
21
)12(
)()1(
kk
k
kkkkkk
k
.2187858681527455514... Hk
kk 6
6
5
6
51
log
3 .2188758248682007492... 2 5log
.218977308611084812924... 136
625
4 2
625 51
2
4
1
arccsch( )k
k
kk
k
.21917134772771973069... cosh( ) ( ) sinh
1
2 81 1
12
14
erf erfix
dx
x
.219235991877121... ( )
1 1
31
k
k k k
1 .21925158248756821... 8 2 4 2 2 32 1
22
1
log log ( )( )
H
k kk
k
.2193839343955202737...
Eik k
k
k
( )( )
!1
1
1
LY 6.423
= logdx
e
e
xdxx
x
1 0
1 1
= dx
ee x
0
= xe dxx ex
0
2 .2193999651440434251... ( ) ( )3 6
.2194513564886152673...
12 2
1
2
11
kkLi
k
9 .2195444572928873100... 85
.21955691692893092525... 2 22
1
4 4 21
( ) coth
k kk
= ( )( )
12 2
41
1
kk
k
k
=
Re( )
( )
k
i kk
2
21
.21972115565045840685... e
e x
dx
x x
dx
x4
5
4
1 1
2
12 7
14
1
cosh cosh
.2197492662513467923... ( )( )!
1 1 0
1
k
k
k
k
.2198859521312955567... 1 1 10
1
ci x x xdx( ) cos log cos
.219897978495778103... ( ( ) ) kk
2
1
2 1
2265 .21992083594050635655... 34
7
.2199376456133308016... ( ) ( )
1
4
1 3
1
kk
kk
H
k
.21995972094619680320... 3 2 2 2 2 3
6 24
2
2
2
24
2
2
2
6
3
3
log log log log ( )
= log sin2 2
0
x x
xdx
.22002529287246675798... 1
21
1
41
1
21
Jk
k
k
kk
( )( )
( !)
.220026086659111379573... log21
41
2 31
2 3
i i
1
4
3 3
6
3 3
6
i i
= ( )
1
3
1
31
k
k k k
.2200507449966154983... 2 24 2 1
2
1
2
2
1
log( )!!
( )!!
G k
k kk J385
.220087400543110224086... ( )
log2 1
21
1
2
1
222
1
k
kk
k kkk k
.2203297599887572963... 2 4 2
12 4 2
12
coth coth tanh
=1
4 4 2
2
221 1k k
k
ikk
k
Re( )
( )
= ( )( )
14
4
1
4 114
1
kk
k k
k
k
.22037068922591731767... )2()2(16
)2()2(8
1
64
7
4)1()1( ii
iii
=
2
243 )1(
1
k kk
1 .2204070660909404377... ( ) ( )3 4
.2204359059283144034... 27
4
3
1
6
1 2)1(
= 3/22
3/12
3/22
)1(3
1)1(
2
)1(
108
5
LiLi
3/22
3/2
)1(4
)1(
Li
= 1
36
1
3
5
6
1
3 2 20
( )
( )
k
k k
=
02
1
03
13 1
log
1
logxx ee
dxxdx
x
xx
x
dxx
.220496639095139468548...
02 12
3
28
33/2xe
dx
.22056218167484024814... 1
22
1
22 1
1
2
1
1eEi
kH
k
kk
kk
log( )
!
.2205840407496980887... cos Re
2
2 i
.220608143827399102241... 3 3
24 2
36 1
22
0
1 ( )log log( ) log x xdx
6 .220608143827399102241...
( ) log( )
log log2 4 23 3
21
1
0
1
2
x
xdx
.2206356001526515934... log
tan tan2 1
20
1
0
1
x xdx x xdx
2 .220671308254797711926... arctanarctan log
log2
12 24 4 2 3 3
2 32 2 2
2
20
x
x xdx
1 .2210113690345504761... 11
22
( )kk
.2211818066979397521... 1
1627 3 3 2 27 3 8 32( log ( ))
=1
3 231 k kk ( )
.2213229557371153254...
21
6205144
5 1 2 5 2 5 ( ) ( ) ( , , ) ( , )
1 .2214027581601698339... e5
2 .22144146907918312350...
12 4x
dx Marsden p. 231
=
2/
04
2
tan1
dxxx
dxx
=d
1 20 sin
Marsden p. 259
= log cosh
cosh
x
xdx
xdx
x2 12
2
2
00
= log 11
2 40
xdx
.22148156265041945009...
122
2
1)(11
1
kkk kk
k
.22155673136318950341...
8
.221565357507408085262...
51cot22coth53
4
51arctan537
5310
1arcarc
=
1
11
)12(4
)1(
kk
kkk
k
FF
6 .221566530860219558... 6 51 3
0
1
( )log( ) log
x x
xdx
.22156908018641904867... ( ) ( ) ( )( ) ( )3 11
42 22 2 i i
=x x
e edxx x
2 2
0 1
sin
( )
.22157359027997265471... log ( )!
( )!
2
2
1
8
2 4
2
2
2
k k
kk
= ( )!!
( )! ( )
2 1
2 122
k
k kk
.221689395109267039... 1
13 1 3 1
13
2 16 3
2kk
k kk k k
( ( ) ) ( )
=1
31
1 3
2
5 3
2
1 3
2
5 3
2
i i i i
=
1
3
1
61 3
3 3
21 3
3 3
2( ) ( )i
ii
i
=
3
2
3 3 3
5 3
2
1
3
2
3 3 3
5 3
2
i
i
i
i
= 1
31 1 2 1 1 2 12 3 1 3 2 3 2 3 ( ) ( ) ( ) ( )/ / / /
.2217458866641557648... 16
3 4 22
41
(log log )( )
kk
k
.22181330818986560703... 2 2 26 2 1
22
20
log log( )
k
kkk
1 .2218797259653540443... log
k
kkk
112
2
1 .221879945319880138519...
1
4
26
)3(222680
97
k
kk
k
HH
.221976971159108564... 1
1142 kk
.22200030493349457641... ( )
!
/2 1 1 1
1
1
2
2
k
k
e
k kk
k
k
.22222222222222222222 =2
91
21 2
0
1
1
( ) arccoskk
k
kx xdx
=
1 19
3)(
kk
kk
.222495365887751723582...
1 )!1(k
k
k
B
.2225623991043403355... 1
6
2
3
2
3
1
21
21 2 1
1
log arctan ( )
k kk
k
H
.222640410967813249186...
1
2
23 )2(91sin123cos
32
1csc
kk
kk
2 .2226510919300050873... 34
12
12
2
0
e
e
k
kk
( )!
.2227305519627176711... 21 3
4 16
12
2 1
4
41
( )
( )
k
kk
= ( )( )( ) ( )
11 2
21
3
kk
k
k k k k
.22281841361727370564... G log2
2 .22289441688521111339...
8 4 1
1
2 21
2
4
=8
8
2 1
231
11k k
k
kk
k
( )
8 .22290568860880978... 2 1 1
1
2
1
k
k
k
k
k
k
e
k
( )
!
/
2 .2230562526551668354... 1
29 3 3
3
3 321
22
( log )( )
k k
k
kk
k
=1
1 230 ( )( )k kk
.2231435513142097558...
1
1
11 4
)1(
5
1
5
1Li0,1,
5
14log5log
kk
k
kk kk
= 21
2 1 92
1
9 4 12 10 0( )k
dx
ekk
x
arctanh K148
.2231443845875105813... H
kk
kk
( )
( )
3
1 3 1
.2232442754839327307... 2 1 1 21
2 1
1
2 1 2 2
1
1
1
1
sin cos( )
( )!( )
( )
( )!( )
k
k
k
k
k
k k k k
= x xdx2
0
1
sin
=log sin log
sin2
21
41
1x x
xdx
x
dx
x
e
.22334260293720331337... ( ) ( )
( )!2 2 1
21
k k
kk
.2234226145863756302... ( )
!
110
k
k k
.22360872607835041217... ( ( ) ) k
kk
1 2
2
98 .22381079275099866005... 267 1 9
0e
k
k
k
k
( )!
.223867833455760676... 1
4 2
1 1
4 22
2 2
e e
csch
=( )
1
2
1
21
k
k k J125
.2238907791412356681... Jk
k
k
k
k
k
k0 0 1 2
0
1 2
21
2 1 11 1
( ) (; ; )( )( !)
( )( !)
F LY 6.115
= ( )( )!
( )( )!
11
2
21
2
2
0
12
0
k
k
k
kk
k
k
k
k
k
k
1 .223917650911311696437... 1)3( e
.22393085952708464047...
2
2
12223
)1log(1)12(
)1(
4
kkk k
k
k
k
kk
3 .2241045595332964672... ( )( )
! /
11 3 13
2
2
1 32
k
kk
k
k k
k k
k k
e k
.224171427529236102395...
3 82 1
2 11
log( )
( )
k
k kk
.224397225169609588214... ( )( )
12 1 1 1
22
2 21
k
k k
k
k kkLi
k
.2244181877441891147... 7
361
7
6 71
( log )
k Hkk
k
.2244806244272477796... )1(22
2log
33
5
= ( )
logk
k kk k
kk k
1 1
3
1
3
1
21
1
1
2
2
.22451725198323206267... e
.2247448713915890491... 3
21
2 1
2 31
( )!!
( )!
k
k kk
1 .2247448713915890491... 3
2
1
12
2
0
kk
k
k
= 11
6 30
( )k
k k
.22475370091249333740... 8 1
51e
ex
dx
x
sinh
1 .2247643131279778005... 1
183
61
92 21
cotkk
.2247951016770520918... 1
2 4 2 2 2
1
8 1
2
821 1
cot
( )
k
k
kk
k
.22482218262595674557...
2 3 3
13 1
2 132
2 1
cot( )
k
k
kk
k
.224829516649410894... 181
32 3 24 2 27 31
2 32
31
( log ( ))( )
k kk
.225000000000000000 = 40
9
.22507207156031203903... 1
0
3arctan2
2log
324dxx
.225079079039276517...
4
3
4
1
2
1 11
.225141424845734406... 22
2 11
2 20
1
G E kdk
k
log ( ) GR 6.149.1
.2251915709941994337...
1
1
!)!1(
2)1(
k
k
k
.2253856693424239285... )()(14
1
16
)3(3333 iLiiLiLi
=
1
02
2
13
2
1
loglog
x
dxxx
xx
dxx
=x dx
e x
2
20 1
1 .225399673560564079... 2
2 11
20
e
e e kk
( )
1 .22541670246517764513...
4
12
4
3 1
.22542240623577277232...
11 1
)(
)1(
)(
kk k
k
kk
k
1 .22548919991903600533...
2
2
1
1
coth(log )
.225527196397265246307... 1
184 2 2 2 3 2 2 1
21 2
1
arccot log log ( )k kk
k
kH
.2256037836084357503... H k
kk
2
1 8
1
74
8
72
4 2
4 2
log log
1 .2257047051284974095... ( )31
kk
.2257913526447274324...
2112
log)1(2
)2(
2log
k
k
kk
kk
k
K ex. 207
.226032415032057488141... 8
1
6 51
arctan x
xdx
3 .22606699662600489292... ( ) ( ( ) )
1 2 11 3
1
k
k
k
8 .22631388275361511237...
42
4
2 10
erfik k
k
k
!( )
.2265234857049204362... e
12
2 .2265625000000000000 = 285
128
1
54 0
1
5 54
4
1
, , Lik
kk
.226589292402242872682...
2222
51
12
4
151
8
5353
515
1LiLiLi
=
12
11
4
)1(
kk
kkk
k
FF
.226600619263869269496... 1
4
5
44
0
4
x e dxx
.226724920529277231...
12 53636
1
38 k kk
.22679965131342898... 2874
1532
2772
22 1 2 3
23
1
log log( )( )( )
k
k k kk
kk
H
.226801406646162522... 22
22log22log4
8
7cot
2
=1
8 21 k kk
.2268159262423682842... 1
972 2 168 2 149
2 42
1
log log( )
H kk
k k
1 .2269368084163... root of ( )x 5
.22728072574031781053... 2
272 3
2
3 2
2
31
, ,log x
xdx
.227350210732224060137...
11 3
11log
1
3
1)1(
kkk kkk
k
.22740742820168557... Ai(-2)
.22741127776021876... 3 4 21
2 2 3
1
10
1
2 21
log( )( )
( )
( )kk
k
kk k k k
= ( )!!
( )! ( )
2 1
2 1 21
k
k kk
1 .22741127776021876... 4 4 21
4 1
22
0
log( )k
k k
k
k
=
22
12
1
1 2
)()1(
4/1
)1(
)½(
1
kk
k
k
k
k
k
kkkk
=( )!
( )!
k
k kk
1
2
121
1 .2274299244886647836... 23
2
1 32
1
21
arcsinh
( )k k
kk
k
k
4 .2274535333762165408...
4
1
)4/1(
)4/1(2log3
2
1 .2274801249330395378... 3 5 66
372
12 22 1
1 1 12 42
12 3 4
( ) ( ) log
H
k k k kk
k
1 .227558761008398655271...
2 1
1)()(
k k
kk
1 .22774325454718653743... H
k kk
k ! 21
1 .2278740270406456302... 1
11 k kk ! ( )
1 .227947177299515679...
0,1,2
1)22log(
.228062398169704735632... ( )( )
arctan
12 1
1 3 1
32 1
k
k k
k
kk
k
k
k
2 .2281692032865347008... 7
16 .2284953398496993218... e erfk
k
k
2
0
12
22
!!
.2285714285714 = 8
35
1
1
2 2
2 4
p
p pp prime
2 .22869694419946472103...
2
)(
2 2 2
)(2
)1(
221
)2(
)(
k
k
k s ks
k
kkkk
k
.22881039760335375977... )1,2,2(1
)2,3(2
)5(11)3(
2 123
2
MHSjk
MHSjk
.22886074213595258118... dxxx
x
1
34
222 )1log(
48
7
24
2log3
.2289913985443047538... G
k
k
k4
1
4 2 20
( )
( )
=1
8 6
1
8 22 21 ( ) ( )k kk
=xdx
e e
xdx
x xx x2 20
3 11
log
=x x
xdx
log
1 41
.2290126440163087258... e
ee
H
e
kk
kk
11 1
1 1
1
(log( ) )( )
.22931201355922031619... H
kk
kk
(2)
51
.22933275684053178149... dxxx
x
12
22
)1()1(
log
128
)3(7
.229337572693540946...
1
21
1 1
21
loge e kk
k
.2293956465190967515... I e ee
k
k
k0 0 1 2
0
2 1( ) (; ; )( !)
F
.22953715453852182998... 3
22 1
11
2
log ( ) ( )k
k
k
kk
= log 11 1
12
k kk
.22955420488734733958... 1
8
1
2
1
4
1
2
1
2 1 4
1
1
cos sin( )
( )!
k
kk
k
k
.229637154538521829976... 1)(1
)1(2log2
3
2
kk
k
k
k
.22968134246639210945... 1
2 2
1
2
1
2 2
1
1
sin( )
( )!
k
kk
k
k
.2298488470659301413... sincos ( )
( )!2
1
1
1
2
1 1
2
1
2 2
k
k k GR 1.412.1
= sin1
21
3x
dx
x
2 .2299046338314288180... ( )
( )!cosh
2
2 2
1 1
12
1
k
k k kk k
.2299524263050534905...
1 )54(
1
105
2log3
25
24
k kk
.2300377961276525287...
1
022 )1(
arccos
2
12
4 x
dxxx GR 4.512.8
.2300810595330690538...
1
2)12(
51
2
5cos
4
1
k k
.23017214130974243... 1
2 2
1
2 2
12
21 8
1
1eerfi
kk
k
k
kk/
( )
!
.230202847471530986301... 1626 1
2 30
e k k
k
kk
( )
! ( )
.2302585092994045684... log10
10
3 .23027781623234310862... 12
22
14 12
0
csch
( )k
k k
.23027799638651103535... 1
4 21k
k k( )
.23030742859234663119...
1
2 3102k j s
jks
.23032766854168419192... 5 5
121
1
2
21
1
( )k
k k
k
k
.23032943298089031951... 16
.230389497291991018199... 1
4
1
2
1
2 2 2
i i i i
= cos
)
x
e edx
x x
10
.230636794909050220202...
4/
0
)1()1(2
cossin8
3
8
1
32
1
82
xx
dxxG
.2306420746215602059...
log
cos2
2
1
1 2 20 x
xdx
x
.230769230769230769 = 3
13
5
0
sin xdx
ex
9 .2309553591249981798... k
kkk
4
1
.2310490601866484365... log ( ) ( )2
3
1
3 3
12
20
1
1
k
k
k
kkk k
kk
J292
= dx
x x
dx
x x
dx
e x41 0
203 1 3 2 3
( )( )
1 .23106491129307991381... cosh cosh sinh sinh cosh sinh 2 2 2 2
=2/
ei eei
.231085022619546563... ( )!k
k kk2
2
1 .231141930209041686814...
1
3
)3(26
2
)3(
1890 k
k
k
H
.231274970101998413257... log ( ) 2 11
kk
.2312995750804091368... k
kk
!!( )!
30
1 .23145031894093929237...
2
1122 2 1 2
1
2
log ( log )( )k H
kk
kk
.23153140126682770631...
1
1
/1 !
)1(1
kk
k
k
k
e
.2316936064808334898...
2
1Ai
.23172006924572925... k
kk
kLi
kk k
11 1
1 12
22
2
( ) log
2 .23180901896315164115... 1
21
2
( )kk
.2318630313168248976... 2 2 21
20
log( )
kk
k
1 .2318630313168248976... 2 2 2 121
log( )
k kk
k
2 .2318630313168248976... 2 2 2 21
20
log( ) k k
kk
1 .23192438217929238150... ( )( ) ( )
12 1
1 1
1
kk
k
k k
.2319647590645935870...
0 )82(!12
2
9
k kk
ke
.23225904734361889414... 6
2
91 2 2
0
1
K x x dx( ) arcsin( )
.2322682280657035591... 351 14 3 126 3
5881
3 71
log( )k kk
.2323905146006299... ( )log( )
112
k
k
k
k k
.232609077617500793647...
03
1
)3(63/1xe
dx
1 .2329104505535085664...
02/
2
118)3(16
xx ee
dxx
3 .2329104505535085664... 16 3 11
2
0
1
( ( ) )log
xdx
x GR 4.512.8
=
02/2/
2
116)3(16
xx ee
dxx
.23300810595330690538...
1
2)12(
51
2
5cos
4
1
k k
4 .23309653956994697203...
2
1
2
3
2
122 )2/1(i
.2331107569002249049... ( )3
32
3
.233207814761447350348... 4 48
2
2
2
0
1
log
arctan logx xdx GR 4.593.1
.23333714989369979863...
( )2
41
kk
k
1 .2333471496549337876... )(Imcsccoth1 )2(23 ih
1 .2334031175112170571... arcsinh2
1 .233527691640341023900...
0
)1()1(k
kk JI
.2337005501361698274...
222
2
2
)1)()(1(
)12(
11
8 kk
k
kk
k
=( )!( )!
( )!
k k
k
k
k
1 1 2
21
= dxx
xx
1
02
2
1
log
=log x
x xdx4 2
1
1 .2337005501361698274... 2 2 2
2 32 08
11 2 1 1
k k
k
k
k kk k
( )( )
!( )!!( )
J276
= iLiiLi 11 22
= ( )( )
( )( ( ) )2
3 24
12 1
2 120
2
1
kk k
k k
= 1
4 1
1
4 32 21 ( ) ( )k kk
= 2
221
k
k k
kk
=
1 1 2
)(
j kjk
jk
= log xdx
x20
1
1 GR 4.231.13
= log xdx
x40 1
= arctan xdx
x1 20
= arctan x dx
x
2
20 1
GR 4.538.1
= arcsin xdx
x1 20
= K kdk
k( )
10
1
GR 6.144
= E k dk( )0
1
GR 6.148.2
.2337719376128496402... ( )
!
( )
( )!!/k
k
k
ke
kkk k
k
k2 21
1
22 1
1 2
1
.23384524559381660957... 2
13
1sin
4
1,
3
5,
2
3,
3
2
4
1
x
dx
xPFQ
.23405882321255763058... 1
2 2 2 2
1
21
1
21
1
2
cot
=k
k k
k k
kk
k
1
2
2 2 1
231 1
( ) ( )
.2341491301348092065... 1
6 1 61
0
1k
kk
k
k
( )
.234163311975561677586...
1
)1()1(
)12(2
3log33
2
3
1
9
2
k
k
kk
kH
.23416739426215481494... ( )3
6
1
36 616 3
1
k
k kkk k
.23437505960464477539... ( )
!
120
k
kk
.23448787651535460351...
21
11
arctanh12
1)2(2log
2
3
kk kk
k
k
.23460816051643033475... logk
k kk3
2
.23474219154782710282...
( )3
6
1
21
1
61
1
6
=1
36 6
2 1
65 31 1k k
k
kk
k
( )
.2348023134420334486... 1 11
40
1
21
Jk
k
kk
( )( )( !)
.2348485056670728727...
4
12
4116
16 43
2
1
,( )kk
.234895471785399754782...
18sin32
18cos16
9
7
9
4
729
1 3)2()2(
=
133
2
1
log
xx
dxx
.23500181462286777683...
2
133
5log
2
2log3
xx
dx
.23500807155578588862... 2 1
5 4 11
21
1
sin
cos( )
sin
kk
k
k
.235028714066989... 1
3612 3 2 3
3
2
3
22
sin csc sec
=( )
134 2
2
k
k k k
.2352941176470588 =4
17
4
0
sin xdx
ex
=1
2 41 2 1 2 2
0cosh(log )( ) ( ) log
k k
k
e J943
1 .2354882677465134772... 33
2 1
3 13
32 1
sech
k
k
k
kk k
exp( )
= exp logk
kk
3
32 1
=
13
)(
k k
kz
= 3 1 15 3
2
5 3
21 3 2 3 ( ) ( )/ /
i i
.23549337339183675461... 3 3
42 2
133
20
1 ( )log
log ( )
x
xdx
.23550021965155579081... 1
2 2 1
1
2 11
1
11
k kk
k
kk( )
( )
.2357588823428846432... 2 1
2
1
2
1
1
1
1
1
1 2
1
1e k k
k
k k k
k
k
k
k
k
k
( )
!( )
( )
( )!
( )
( )! !
.23584952830141509528... 89
40 CFG D1
.235900297686263453821...
5
15log5log2log42log4
2
1
4
12
222 LiLi
=H
k kk
kk
k
kk5
1
41
1
21
( )
=
1
0 4
logdx
x
x
.2359352947271664591... ( )( )!
/11
1
1
e e ek
k ee
kk
5 .23598775598298873077... 5
3 1 6 50
dx
x /
.23605508940591339349...
4/
0
)log(sinsin22
2log2)12(2log1
dxxx
.236067977499789696... 5 21
3
1 .236067977499789696... 5 11
2 1
21
1
( )k
k k
k
k
=
( )
( ) !8
3 1
2 20
k
k k kk
Berndt Ch. 3, Eq. 15.6
2 .236067977499789696... 5 2 1 21
5
23
0
kk
k
k
4 .236067977499789696... 3 2 1
.2361484197761438437... 12 2 9 3 3 36
1
6
2
3 21
log log
k kk
2 .236204051641727403...
( ) ( )2 5
22
224
277 3
1 .2363225572368495083... 14 2 6
23
13 22
1
cotkk
.236352644292107892846...
113
2
2
log
2
1,3,2
32
1
xx
x
.23645341864792755189... 3 2 1 2 11
2 1
1
1
cos sin( )
( )!( )
k
k k k
1 .2365409530250961101... 3
4 2 2
2
0
2
0
e k
k
k
kkk k
! ( )!!
1 .236583454845086541925... csc ( )3
6
1
18
1
9
1
2 20
k
k k
.236585624515848405511...
1
122
)12(
)1(
2
2log2log
122
k
kk
kk
HG
.2368775568501150593...
3
1
6
5
216
1
18
)3(13
381
5 )2()2(3
=
02
2
03)23(
)1(2
xxk
k
ee
xdxx
k
=log2
31 1
x
xdx
.2369260389502474311... 25
641
2
1 6
0e
k
k
k
kk
( )!
2 .2369962865801349333...
1 424
k
k
kk
k
.23722693183674748744... 9 3
4 1 1
2 3
31
3
30
1 ( ) log
( )
log
( )
x
x
x x
x
.2374007861516191461... log( )3 3
.237462993461563285898... 2
43
15 20
( )/
k
k k
=
2/
02
2
)sin1(
sin
dxx
x
.23755990127916081475... 252
15log
4
3
24 22
2
Berndt Ch. 9
=
1 2
1
)24(2
)1(
4
1,
2
3,
2
3,1,1,
2
1
4
1
k
k
kkk
kPFQ
9 .2376043070340122321... 16 3
3 CFG D11
17 .2376296213703045782... k
kk
3
1 1( )!!
.23765348003631195396... 2
2130
1371500
15
k kk ( )
.2376943865253026097... ( ( ) ) jkkj
3 111
98 .23785717469155101614... 24 212 1
22 3
30
log( )
e x dx
e
x
x GR 3.455.2
.2378794283541037605... 1
144
1
6
2
31 1
3 31
( ) ( ) log
xdx
x x
.2379275450005479544... ( ) ( ) ( )
( )
( )2 3 2 2
1
1 21
k
k kk
.23799610019862130199... dsskkk
32
31)(
log
1
.2380351360576801492... ee
erfk
k
k
2
1
2
1
0
( )
!!
.238270508910354881... 8 212
14615
12 2 70
arctanh
kk k( )
6 .238324625039507785... 9 2log
.2384058440442351119... 2
12e
9 .23851916587804425989... = 2 2 2 3
20
2 3 2
16 2 2 1
( log ) log
xdx
x
1 .2386215874332069455...
1
/1
1
1 2
1!
)2()1(
k
k
k
k ek
k
.2386897959988965136... ( )
112
0
k
k k k
.23873241463784300365... 3
4
.238859029709734187498...
1
212
4
)2()1(
2csc3
2coth
2coth2
32 kk
k kkh
.2388700094983357625... 1
163
2 1
3
1
( )( )!
ee
k
kk
2 .2389074401461054137... k H k
kk
( )3
1 2
2 .23898465830296421173... ( ) ( )3 5
.2391336269283829281... 2 1 11
2 2 31
2 2200
cos sin( )
( )!( )( )
( )( )!
k
k
kk k k
k
k
=log coslog2
1
x x
xdx
e
.2392912109915616173... Ei ekek( ) log( )
/ 1 1 1
1
.23930966947430038807... ( )k
kk
3
32
= 1
1
1 13
26 3
212 3
2k k k k kk k k
= ( ) ( ) ( )3 1 9 3 1 9 3 11 1 1
k k kk k k
.2394143885900714409... 1
8
2
4
1
42 21
cot
kk
2 .2394673388969121878... 2 3 2 4 ( ) ( )
.239560747340741949878...
2log2
4
)33(cot
4
)33(cot2
6
1 ii
=
13
2
1
)1(
k
k
k
k
.2396049490072432625... log( )
e
e2 1 J157
.23965400369905365247... 12 3
3
1
2
1 22
2 1
1
1
arcsinh( )
( )
k k
k k
kk
.2396880802440039427... 1
831 kk
.2397127693021015001... 12
12
14 2 1
1
1
sin( )( )!
k
kk k
1 .239719538146227411965...
1 )(
1
k k k
.2397469173053871842...
( )log
32
31
k
kk
.2397554029243698487... 21
2 21
2 22
1
1
sin( )( )!
k
kk k
.23976404107550535142... 12
2 212
3 21 2
22 0 10
Jk k
k k
k
( ) (; ; )( )!( )!
F
.2398117420005647259... ci x xdx( ) log sin10
1
GR 4.381.1
= ( )
( !)
1
2 21
k
k k k
.239888629449880576494... 2
)3(3log
4
7)2(12log
4
7
=
1 2
1)2(
k k
k
=
kk
kk
42
2
2
11 1
2log
1 .23994279716959971181... dxx
x
1
06
22
)1(
log
6
2log5
303
1
.2399635244956309553... 1
2
1
4,
.24000000000000000000 =
1 60,1,
6
1
25
6
kk
k
.24022650695910071233...
11
1
12
)1(21)1(2log
2
1
kk
k
k
kk
k
H
k
H
=
1 2
1
22
)1(
k k
k
kk
k
= dxx
x
1
0 1
)1log( GR 4.791.6
=
0
2
1
1
0 1
loglog
1
)1log(xe
xdx
x
xdx
x
x
.2402290139165550493... 2 1 2 2 11
10
1
log( ) log e
e
edx
x
x
1 .24025106721199280702...
4
3
4
1
100
1
25)2()2(
3
.2404113806319188571... ( )3
5
.2404882110038654635... ( )( )
13
13
1
k k
kk
H
1 .2404900146990321114... 1
1 3
1
2 3/ /
.24060591252980172375...
k
k
kkk
k 2
)12(4
)1(
4
5824log2log
2
1arcsinh
2
1
012
.2408202605290378657... 1 2 2 32 3
1
( ) ( )( )
k
kk
= ( ) ( ( ) )
1 1 11 2
1
k
k
k k Berndt 5.8.5
2 .240903291691171216051... 1
1 21 ( )! ( )k kk
Titchmarsh 14.32.1
.241018750885055611796...
1 3
1)1(
2
1
6
3
3
3log2
kk
k
.241029960180470772184... 2 24
23 3
16
22
0
1
log( )
arctan logx xdx
.241172868732950643445... 1
0
logarcsinh1arcsinh22log222 dxxx
.24120041818608921979... 1
6
2
3
1
1 2
1 12
1
( ) ( )!
( )!
k
kk
k
k
13 .2412927053104041794... 257
32 2
5
1
e k
k kk
!
.241325348385993028026... 2 2 3
2112
2
2
2
6
3
8 2
log log ( ) ( )k
kkk
.2414339756999316368... 5 2
458 2 12
2
log( )
k
kkk
.24145300700522385466... 1
1
.241564475270490445... log( )
log
4 1
1
2
20
1
x
x
dx
x GR 4.267.2
=x x x
x xx dx
log
loglog( )
112
0
1
GR 4.314.3
.24160256000655360000...
( )
( )
2
5 1
1
512
1
kk k
k k
1 .24200357980307846214... 15 33
6
Associated with a sailing curve. Mel1 p.153
1 .24203498624037301976...
2
20
1
1
( )
( )sin/
e
ee xdxx
.24203742082351294482... ( ( ))22
kk
1 .2420620948124149458... 1
2 1
2
2 11 2
21
1
11( )log log( )
( )
`k
k
k
kk
kk
kkk
k
=
1
1
2
)(
kk k
k
.242080571455143661373... logsin/3
32
27 8 8
50
x
xdx
.24209589858259884178...
2
061
2
6 1
( ) logx x
edxx
.24214918728729944461... 112 9 24 2
271
2
0
1
loglog( ) logx x xdx
.242156981956213712584... ( )( )
14 1
1
1
kk
k
k
.2422365365648423451... 3 2
2 20
2
3 11128
x dx
e e
xdx
x xx x
log
.24230006367431704... ( )
1
230
k
k k
.242345452227360949... c231
63 2 3 ( ( ) ( ) ) Patterson Ex. A4.2
.24241455349784382316... ( )
!
1 1
1
k ek
k
H
k
3 .24260941092524821060... 1
2 (log )k kk
.24264068711928514641... 3 2 48 1
2
1
k
k
k
kkk ( )
4 .24264068711928514641... 2318
.24270570106525549741...
1
11
kk k
k
k
kLi
2 .24286466001463290864... 8 13 38 3 3 1
12
4 3 2
3 31
( )( )
k k k k
k kk
= ( ) ( ( ) ( ))
1 23
1
k
k
k k k
.24288389527403994851... 2 3 3( ) ( )
.243003037439321231363...
1
21
21
)!12(
)!()!(
23 k k
kk
= 1
0
32
2
)arctan(1
dxxx
x
.24314542372036488714... 1126
9 7 5
2 2
9
1
2 921
log
( )k kk
2 .2432472337755517756... 128
1
48
7
120
24
= k kk k k
kk k
3
1
2 4 2
2 42
2 11
1( ( ) )
( )
( )
.2432798195308605298... e
e
k
kk
k
21
12
0( )( )
!B
.24346227069171501162... ( ) ( )
!/
/k k
ke
e
k kk
kk
k
1 11
2
11
21
.24360635350064073424... 5 82 21
ek
k kkk
! ( )
8 .2436903475949711355... 9G
.2437208648653150558...
0 )3(2
)1(3,1,
2
13
2
3log8
kk
k
k
.2437477471996805242...
1
1
04
2
40 1134
)1(
22
)21log(
24 x
dxx
x
dx
kk
k
=dx
e ex x30
.243831369344327582636... 7
2
1
2
3
2
3
2
3
22 1
1
cot ( )k
k
k
.24393229000971240854...
k kk kk k
( ( ) ) 3 11
113 3 2
2
.2440377410938141863... )1(
5 .24411510858423962093...
4
1
2
1 2
, related to the lemniscate constant
.2441332351736490543...
1
2
1
22 4
)1(2
2
2csch
4
1
4
1
k
k
kee
=
sin2
10
xdx
ex
.24413606414846882031... 9
3log2
9
9log J137
.24433865716447323985... k
kk
1
2
21 2( )
.24470170753009738037... 2 2 22 1
2 2 11
log( )!!
( )! ( )
k
k k kk
.244795427738853473585... 4/
0
2
sinlog128
)3(35
32
2log
8
dxxx
G
.24483487621925456777... 41
4
1
4 2 12
1
sin( )
kk k k
.2449186624037091293... e
e
11 J148
.24497866312686415417... arctan( )( )
14
14 2 12 1
0
k
kk k
6 .24499799839839820585... 39
.245010117125076991397...
1 3
13
22
)1(
6
2log
4
)3(
k k
k
kk
k
.2450881595266180682...
126
1
6
12log2
2
3log3
2
36
k kkhg
= ( )( )
11
61
1
kk
k
k
= dxx 1
0
6 )1log(
.24528120346676643... 2 2
20
4
16 42 log
cos
/
Gx dx
x GR 3.857.3
= arctan2
0
1
xdx
.2454369260617025968... 5
643
0
1 x xdxarcsin
2 .245667095372459229031...
1
3122
2
)5(7)3(1212
72
1
k
kk
k
HH
.24572594114998849102...
22 1
sin1cos22log1sin
2
1
4
1sin
k k
k
1 .245730939615517326... 35
2 .245762562305203038...
0
2
2 )1()!2(
42sinh
2
2cosh
2
1
44
3
2
1
k
k
kk
e
e
.2458370070002374305... 1
2
1 1
2 0
1
sin cos sin
e
xdx
ex
=sin log x
xdx
e
21
.24585657984734640615... dx
e xx ( log )11
.2458855407471566264...
1234
12log12162)2(
k kk
= ( )( )
12
41
1
kk
k
k
2 .2459952948352307... root of ( ) ( )1 12
x
.2460937497671693563...
( )2
221
kk
k
1 .24612203201303871066...
4
3
4
12log)2log23(log2
6 22
2
LiLi
8 .2462112512353210996... 68 2 17
.2462727887607290644...
14
1
4
)1(0,4,
4
1
kk
k
k
.24644094118152729128... 1
14 3 21 k k k kk
1 .24645048028046102679... 2 1 2 2 1 2log( ) arcsinh arctanh 12
=1
2
1 2
2 1log
= 1
2 2 1 20 1k
k
k
kkk
H
( )
O
= 1 12
2 31
( )( )!!
( )!!k
k
k
k J141
=( ) ( )!!
( )!!
1 2
2 1
1
1
k
k
k
k k
=dx
x x( )
1 1 2
0
=
1
0 1)1( xx
dx
=
2/
0 sincos
xx
dx
.24656042443351259912...
1
22
2
1 )18(
8
8
)2(
2sin2
32 kkk k
kkk
.2465775113946629846...
( ) ( )22 1
2 14 11
1 1
k kk
kk
k
.2466029308397481445... 3
2
3
2 3 12
1
log( )
H
kk
kk
.24664774656563557283... dxxx
x
133
2)2()2(
3 log
6
1
3
2
1728
1
144
)3(13
3648
5
1 .2468083128715153704... e e ee kk
k
log( )( )
11
10
1 .2468502198629158993... arcsec
.2469158097729950883... 1
2 2
1 1e
dx
ekk
x
3 .24696970113341457455... dzdydxxyz
xyz
1
0
1
0
1
0
4
1
log
30
.2470062502950185373... log ( )3
2 6 3
1
9 3 321 2
3 21
k k
k dx
x x xkk
k
3 .2472180759044068717... 3 46
4096 30 387072014 6
61
( )
( )( )
a kk
where is the nearest integer toa k k( ) 3 . AMM 101, 6, p. 579
.24734587314487935615... 1
44 4
0
1
( log ) log ( ) sin x xdx GR 6.443.1
.24739755275181306469... 3/13/13/23/13/2
2222)1(231)2(23126
1
ii
=
23 2
1
k k
.2474039592545229296...
00
1212/1
4!)!24(2
)1(
4)!12(
)1()1(Im
4
1sin
kk
k
kk
k
kk
.2474724535468611637... 7
4
4
3 23 2
log
.24777401584773147...
2
)(
12
2
kk
k
1 .24789678253165704516... 33
2
1 12
4 12 21
Gk
k
k
k
( ) ( )
( )
2 .24789678253165704516...
1
32
3
)4/14(
)1(
2
13
k
k
k
kG
.24792943928640702505... ( ) ( )2 12
k kk
.24803846578347406657... ( )
logk
k kk
k jkk
1
1
1
2
1
42
1
21
12
22
.24805021344239856140... 3
125
.24813666619074161415...
12 16
1
2
1
6coth
62 k k
.24832930767207351026...
222
1
21
21
2
1 iH
iH
ii
=
134
1
k kk
= ( )( )
Re( )
( )
12 1
4
1
21
11
kk k
kk
k k
i
.24843082992504495613... 84
1
1 2
3 1
31
( )
( / )
k
k k
.24873117470336123904... 2
2 2
1 2 2 120
1
log arcsin
x
xxdx GR 4.521.3
.248754477033784262547... 1log A , the log of Glaisher’s constant
= 22
)2(2log
12
1
.2488053033594882285... ( ) ( ) ( ) ( )( )
2 3 4 3 3 51
3
51
k
kk
2 .248887103715128928098...
124
242
12
1212csc12cscsinh
k kk
kkh
.24913159214626539868... 7
360 6
4 2 4
40
x
xdx
cosh
1 .249367050523975265... sinh3
.24944135064668743718... log x
edxx
11
.2494607332457750217...
3/13/2
3/1
3/2212)2(2
2
131
2
1131
212
1 ii
=
1
21
13 2
)3()1(
14
1
kk
k
k
k
k
.2498263975004615315... sin sinh( )
( )!
1
2
1
2
1
4 2 2
1
2 11
k
kk k
GR 1.413.1
.2499045736175649326... dxe x
3
1)( 1
.250000000000000000000 = 1
4
1
3 6
1
4 421
21
k k k kk k
= 1
1 12 ( ) ( )k k kk
J268, K153
= 1
2 132 1k k
kk k
( ( ) )
= k
kk 4 141
=
3
1
222
)1)()(1()1()1(
12
k
k
k
kkkk
k
= ( ) ( )
( )( )( ) ( )
11
11 3
12
122
2 0
1
21
1
23
k
k
k
k
k
k
k
kk k k k k k k
= ( )( )
14 1
4 4 114
1
kk
k k
k k
k
= ( ) ( ) ( )
1
3 4 1
2
4 1
1
1 1 1
k
kk
kk
k
kk
k k
= dx
x
x dx
x
xdx
x
xdx
x51
3
50
31
2
311
( )
log log
= x x xdxarcsin arccos0
1
= x x dxlog0
1
= log(sin )sin cos logsin sin cos/ /
x x xdx x x x dx0
22
0
2
= dxx
xxx
05
2)cos(sin Prud. 2.5.29.24
= log 11
13
x
dx
x
=
44
7
0
3
xx e
dxx
e
dxx
1 .250000000000000000000 = 5
4 22 H ( )
2 .250000000000000000000 =
1 1
11 334
9
k kkkk
k
FFk
68 .25000000000000000000 =
1
5
30,5,
3
1
4
273
kk
k
.250204424109600389291... )1(
.25088497033571728323... ( )k
kk
12 12
1 .25164759779046301759... 12primesmallest;2,lglim 1)(
npnn
n
nppp
Bertrand’s number b such that all floor(2^2^…^b) are prime.
1 .25173804073865146774... 1
22 2
1
2 20 00
I Jk kk
( ) ( )( )!( )!
.2517516771329061417... 1
41111
3
2
3
22 2
1
16
1
2
3
21
HypPFQ , , , ,{ , , , },(( )!)
(( )!)
k
kk
2 .2517525890667211077... ( )10
.2518661728632815153...
12 54
1
10
1
2
5coth
54 k k
.251997590741547646964... 3
1
11
)(
)4(
1
)3(
3
)2(
3
kk
k
k
6 .25218484381226192605... 7 3 4 22
2 2 1
2 1
2
31
( ) log( )
( )
k
k kk
=k k
kk
2
1
22
( )
.252235065786835203... 11
2 11
10
1
e e
dx
e ex xlog log( )
.2523132522020160048... 15
4 .2523508795026238253... Ik
k
k0 0 1 2
0
2 2 1 22
( ) (; ; )( !)
F
1 .252504033125214762308...
2
1
2
1
2sech
ii
.25257519204461276085...
1
2
2 )(
6)3(
k k
k
.2526802514207865349... arccsc4
1 .2527629684953679957... Li15
7
.25311355311355311355 =691
2730 6 B
.25314917581260433564... 2 8 3 ( )
2 .25317537822610367757... k
kk
!!
( )!!21
.2531758671875729746... = 11
2
1
2 1
1
1
erf
k k
k
k
( ) ( )
!( )
.253207692986502289614...
7
1 .25322219586794307564... 1
2 3 31
( )( )f k
kk
Titchmarsh 1.2.15
.2532310965879049271...
123
2
23
1
128
393log9
k kk
.25326501580752209885... 2/
0
223
cos848
dxxx
1 .25331413731550025121... 2
i i ilog
.25347801172011391904... 1
42 2
2 21 10
I Jk
k kk
( ) ( )( )!( )!
1 .25349875569995347164... 1
12 kk !
2 .25356105078342759598...
0
2
2
3
)3log(
32
12log
x
x
.25358069953485240581... 130 ( )! !k kk
.25363533035541760758...
0 21
1
11 2)!(
)1(
2
1,
2
5,2
3
21
2
121
kk
k
k
kFerfi
e
2 .25374537232763811712... 24 81 4
0
1
e e dxx /
.25375156554939945296...
023
11410
3
5
4xx ee
dx
xx
dx
.2537816629587442171... 656 45
3 3 51
1
2 4
5 21
( ) ( )( )k kk
2 .2539064884185793236... 2
2
2
21
k
k
k
.25391006493009715683... 1
42
1k
k
.2539169614342191961...
12 2
1
2
11
1log
2
1
kk
kk
Likk
k
.25392774317526456667... 1
2 21 ( )k k
k
.254023674895814278282...
1 log)12(
1
k kkk
.2541161907463435341... 1
44 0
1
4
1
44 41
, ,
Likk
k
.254162992999762569536...
1
sindx
e
xx
.2543330950302498178... 2
4 2 6 CFG D1
.25435288196373948719...
2
13 16
3log
3
2log
36 x
dx
.25455829718791131122...
011 )72(!
11,
2
11,
2
9
9
122)1(21
32
5
k kkFeerfi
.25457360529167341532...
2
csch2
coth216
2
=
1
11
121
122
2
2
)2()1(
)2(
1
)12( kk
k
kk
kk
kkk
k
.2545892393290283910...
0
42/2/
16)!4(41
2cosh
2
1
kk
k
k
ee
1 .2545892393290283910...
1
4
12 16)!4()12(
11
2cosh
2
1
kk
k
k kk
= sin sinh/
x x dx0
2
23 .2547075102248651316... 34
3
7 .2551974569368714024... dxx
03 1
261log
3
4
11 .2551974569368714024...
0 2
3
314
k
k
k
k
2 .25525193041276157045... 2
1cosh2
1
e
e
.25541281188299534160...
0
12 )12(4
1
4
1arctanh
4
5log
2
1
kk k
AS 4.5.64, J941
4 .25548971297818640064... 72
422 ( )
.25555555555555555555 = 2390
12 1 2 70
( )( )k kk
K133
.25567284722879676889... 125
15 8 512
1 12 1
1
1
( )( ) !( )!
( )!
arcsinhk
k
k k
k
1 .25589486222017979491... 120 ( )! !k kk
.25611766843180047273... ( )4 156
.2561953953354862697... 4 2 212
622
2
0
1
log log arcsin log
x xdx
.2562113149146646868... 2
csch4
2
2
1
=( )
12 1
1
21
k
k k
.2564289918675422335... 7
33
1 3
1 3 2
1
e k
k kk
/
( )!
.25651051462943384518...
1
0
222
)1(
)1log(
4
2log3
16dx
xx
x
60 .25661076956300322279... 333
0
ek
k
k
k
!
1 .2566370614359172954... 25
.2566552446666378985... H k
kk
( )3
1 5
2 .2567583341910251478... 4
1 .25676579620140483035... csc2
.25696181539935394992... 1
82 2
2 20 0
2
0
I Jk
k kk
( ) ( )( )!( )!
1 .2570142442930860605... HypPFQ { },{ , },
( )!
11
2
1
2
1
16
1
220
k
k
kk
2 .25702155562579161794... 16 253
6 23
1
log
Hkk
k
.25712309488167048602... 4
85
4
3
4 1
2
20
sin
sinlogsin
x
xx dx
1 .25727411566918505938... 5
.2574778574166709171...
1
2
)1(2
)1(2log2log2
kk k
k
.25766368334716467709... dxxx
xLi
1
0
2
22
)2)(2(
log
2
1
4
1
8
2log
48
.25769993632568293344... ( )
12
1
31
k
k k
1 .25774688694436963001...
( )log log
kk
k k
k
kk km
km22
2 21
=1 1
11
1 11 2 1k
k
k
k
kH k
k
k
kk
k
log ( )
( )( ( ) )
.257766634383410892217... H
kk
kk
( )
( )
3
1 2 2 1
.2578491922439319876...
k
k
k
kII
e kk
k 2
2!
)1(2,2,
2
3F)1()1(
1
01110
.25791302898862685560...
1
125
2
52
2
2 30
( ) x dx
e ex x
.25794569478485536661... ( ) ( )
( )4 2 3
23
51
1
2
4 21
k kk
2 .25824371511171325474... 2 21 1 2 1
0
sinh /( )!
k
k
404 .25826782999151312012... 1
4
2 2
0
e ek k
ke e
k
sinh cosh
!
.25839365571170619449...
( )
( )k
kk
1 1
2
.25846139579657330529... =
133 4
10,3,
4
1
4
1
kk k
Li
=log( / ) log1 4
0
1
x x
xdx
.2586397057283358176... 2
12
226
2 21
2
1
2 1
log( ) ( )
( )
k k k
k kkk k
.258742781782167056... 3
42
7
12 1
2
0
log
xe dx
e
x
x GR 3.42.0
3 .2587558259772099036...
22
2
2
062 2 2 2
log loglog
xdx
ex
.25881904510252076235... 26
1
2
32
CFG C1
= )13(4
2
12sin
AS 4.3.46, CFG D1
.25916049726579360505... 2 2 21
2
1
2 2 30
arctan( )
( )
k
kk k
.259259259259259259259 =
1
2
70,2,
7
1
27
7
kk
k
.25938244878246085531...
1 21
k
k
kk
1 .2599210498948731647... 2 11
3 23
0
( )k
k k
.2599301927099794910... log8
8 J137
.2602019392137596558... 2
)3(7
42
2
=3
12
1
21
2
11
2
11
2
1
2
)1()1(
kkk
kk
kkk
k
1 .26020571070524171077... ( )2 11
kk
.26022951451104364006...
ie e e
k k
ke e
k
i i
4
1
22
2 2 2
1
cos sin(sin )sin cos
!
.26039505099275673875...
1 2!
)1(12logk
kkk
kk
Be Berndt 5.8.5
.260442806300988445...
2
23
41
4
1
10
1
2log loglog
x
dx
x GR 4.325.4
=log xdx
e ex x
0
.2605008067101075324... 11
51
kk
.26051672044433666221... ( ) ( ( ) )
1 1 3
2
k
k
k
.26054765274687368081... 1
2
1
2
1
2 1 41
sinh( )!
k kk
1 .260591836521356119... cosh( )!
12
12 20
k kk
.2606009559118338926... 2
2 2112 3 4 3 3
12
13 1
csc cot( )
kk
J840
.2606482340867767481... 81 3
24
32
9 32
811
3
2
5 41
log( )
k kk
=( ) ( )
1 4
3
1
1
k
kk
k
.26069875393561620639... ( )k
kk 50
.2607962396687288864...
0 1
2
),2(!
),2(
k kkSk
kkS
.2609764038171073234... 2
31
3 3
24
2 1
2
( )
( )
k
kk
.26116848088744543358...
212
12 1
21
22
1
log(cos ) ( )( )
( )!k
kk
k
B
k k AS 4.3.72
.26117239648121182407... 21
2 2
1 1
2 22
1
arcsin( )!( )!
( )!
k k
k kk
1 .26136274645318147699... 1
3 21 k k kk
.26145030431082465654... ( )3 33
21
31
3
i i
.261497212847642783755...
primep pp
111log Mertens constant
=
nprimepn pn
1logloglim
4 .2614996683698700833... 742343 3
1 35
1
ek
k kk
/
!
.26162407188227391826... 4
3log
2
1arctanh
4
33log
=
12 )12(4
1
2
1
2
1
kk
kkk kk
LiLi
1 .261758473394486609961... eG
1 .2617986109848068386... 11
2 2 11
kk k( )
= arccosh x
xdx
dx
e ex x41
3 30
.26179938779914943654... 12
1
6 3
1
4 2 1
2
02 1
0
( )
( )
k
kk
kk k
k
k
=x dx
x
5
120 1
=
07
333 cossindx
x
xxx
.26182548658308238562... 11 2
122
2
2
2
6
7 3
8
2 2 3
( log )
loglog log ( )
= 1 21
2
1
2
1
2 12 3 31
log( )
Li Lik kk
k
1 .26185950714291487420... log3 4
.26199914181620333205... ( )4 3
4 11
kk
k
.2623178579609159738... dxx
xx
1
0
22
1
log
13500
256103)3(16
.262326598980425730577...
1
22
1)1()1(216
log4 k
kk
k
k
1 .2624343094110320122... e
e iiiik
k
/
/
2
201
11
.2624491973164381282... | ( )|( ) k k
kk
1
3
1
1
.2625896447527351375...
( ) ( )k kk
k
0
1 2
.262600198699885171294...
1
)3(42
)2)(1(2
1
2
)3(
18012 k
k
kkk
H
1 .2626272556789116834... arctan
.2627243708971271954... log
k
kkk
12
2
.262745097806385042621... ( )k
k
k 2 121
.26277559179844674016... 2
228
25
18
4 1
2 2 1
log( )
k
k kk
= ( )( )
11
22
kk
k
kk
.2629441382869882525...
2 !!
11
k k
.26294994756616124993...
022
21
04
2
1
log
32
)3(7tt ee
dttdx
x
xx
.2631123379580001512... 21211,21,24
1
2
1
2
1iiiEiiEi
e
= )1,0,2()1,0,2(224
1iii
=( )
!( )
1
2
1
21
k
k k k
.263157894736842105 =5
19
.263189450695716229836...
2
75
1
1
2 2
2 2
p
pp prime
.2632337919139127016... ( )( )
!!
11
2
k
k
k
k
.263300183272010131698... e dxex
2
0
.26360014128128492209...
112
)12(24
1,
2
5,2,2
6
1
27
32
3
2
k kk
k
kF
3 .26365763667448738854...
1
22
)3(12
5)3(
6 k
kk
kk
HH
.26375471929963096576... H
kLi
k
kk
kk
( )
( )
2
12
025
1
4
1
5 1
5
4
1
5
.26378417660568080153... 2
3 2118
49
21627
12 3
logk kk
5 .263789013914324596712... 815
25
, the volume of the unit sphere inR
.263820220500669832561...
1
2
2 )!1(
)1(1
6)(
k
kk
kk
BieLi
[Ramanujan] Berndt Ch. 9
.2639435073548419286... 2
2 21
2
1
21
log( )k
k k k
= ( )!
( )!( )
k
k kk
1
2
121 4 1
=2
02
1
0
2 11log)1log(
x
dx
xdxx
= arcsin logx xdx0
1
GR 4.591
2 .2639435073548419286...
4
1
2
12log
2
= log log( )
11
12
220
1
0
xdx
x xdx
.26415921800062670474...
2 2
)()1(
kk
k kk
1 .2641811503891615965... k
kk ( !)31
.26424111765711535681... 12 1
2
1
1
1
10
1
1
4
0
e k k
k
k
k
k
k
k
k
k
k
k
( )
!( )
( )
( )!
( )
( )!
=1
2 11
11 0( )!( )( )
( )! !k k k kk
k
k
= ( )! !
( )log
!
11
12
1 1 2
k k
k n
kn
k
k H
k n
k
k
= xe dxx
xdxx
e
log
210
1
.264247851441806613... ( ) log ( )( )
3 116
3 14
3 12
12 3
4 1
2 1
0
kk
k k
.264261279935529928...
2
25
7
5 1
2
5 20
csc
( )
x
x
1 .2644997803484442092... 1
2 1
1
2 10 1k
k
k
kk
k
( ) ( )
.2645364561314071182... 9121
511 60
ek k kk
!( )( )
.26516504294495532165... 3
8 2
1
4 2 1
21 2
1
( )
( )
k
kk
k
k
k
k
4 .26531129233226693864... ( )k
Fkk
12
.26549889859872509761...
1 4
sin
1cos817
1sin4
kk
k
1 .26551212348464539649... log log21
2
4744 .2655508035287003564... 8
2
1 .2656250596046447754... 1
20k
k!
.2656288146972656805...
( )
( )
3
4 1
1
413 3
1
kk
k k
k
1 .2656974916168336867... 1
21 4 4 1 2 2 1 12 2
ee e e e si ci( ) cosh ( )( ( ) ( ))
( ) log ( ) ( ) ( ) ( )1 2 1 2 1 22 2 2e e ci e si =
2log
)2()1(22log)2(
2
1)1(1sinh2
1e
e
EiEieEieEi
ee
=H
kk
k ( )!2 11
.26580222883407969212... sech21
22
2 2 cosh e e
.265827997639478656858...
2
511
2
511
2
151
2
151
52
1HHiHiH
=
1
1)12(k
k kF
.2658649582793069827... 23 12
2 3G
log( ) Berndt 9.18
.2658700952308663684... 1
2 11
k kk
( )
4 .2660590852787117341...
1
1,,11
kk
e
e
ke
ee
1 .2660658777520083356... Io ( )1
=
02
21
0 )!(
2)!(1
!
)1(
k
k
k
k
k
k
ek
J
= e dxxcos
0
1
.26607684664517036909... 9
4
1
4
5
6
1
3
1
2 31 1
21
( ) ( ) ( )
( / )
k
k k
118 .2661309556922124918... 31
252
465 6
4 1
65
0
1
( )
(log )xdx
x GR 4.264.1
x dx
ex
5
0 1
.2662553420414154886... cos( ) 2
1 .2663099938221493948... ( )
1 2
1
k
kk
k
F
5 .26636677634645847824... 2
2
( )kk
2 .26653450769984883507... dx
x( )1
23 .266574141643676834793...
1
02
45
1
log
3243
32dx
xx
x
.26665285034506621240... )2,3()2,3(cschcoth2
1 23 iii
=
0
2
1
sindx
ee
xxxx
.26666285196940009798... | ( )| 2
41
kk
k
.266666666666666666666 =
0 115
44/1xe
dx
1 .266741049904...
1 )2)(1(
11
k kkk
1 .26677747056299951115... 8 2 2 2 21 2 1
0
log log( )
(k +1)8k
k
k
k
.26693825063518343798... dxex
x
e
e
1
02)1(
log
1
)1log(1
2 .266991747212676201917... 11
8 2
3
1
e k
k kk
!
.2673999983697851853... 1
22 2log
= log cos sin log 8 8
1
22 2 2 2
=1
41 k
k
k
cos
=
1
4 41 2 1 1 1 11 4 1 4 3 4i
( ) log( ( ) ) log( ( ) )/ / /
.26765263908273260692...
4
32log2log23log2
64
12
2
2 LiLi
=( ) log( / )
1
3
1 41
1 0
1kk
kk
H
k
x
xdx
5 .267778605597073091897... 2 2 7 31
22
0
log ( )sin
cos
x xdx
x
.267793401721690887137... ( )
( )! ( )
1 2
1 21
k k
k k k Titchmarsh 14.32.1
.26794919243112270647...
k
k
kkk
2
)1(6
132
1
1 .26794919243112270647...
k
k
kkk
2
)1(6
133
0
4 .2681148649088081629... 160
27
8
27
64
272
35 2
22
0
log ( )/x Li x dx
.268253843893107859...
6 4 23 3
21216 12 2
1 2 1
( ( ) )( )f k
kk
Titchmarsh 1.2.14
.268941421369995120749... 1
1
1 1
1e e
k
kk
( )
.26903975345638420957... erfi
ee x xdxx1
4
2
0
sin cos
.269205039384214394948... ( )
11
k
kk
k
F
.2696105027080089818... ( ) cos cos(log )
1
1 1 1
1
21 0 0
1k
kx
k
k
x
edx
x
xdx
=1
41
21
2
1
2 2
1
2 2
i i i i
=
2
1
2
1
224
1 iiii
4 .2698671113367835...
0
cos )sin(sin2 x
dxxxe
e x GR 3.973.4
.270034797849637221...
022 2
cos
22 x
dxx
e
.27007167349302089432... 3
22)log(122log2log2
)3(32
Glaisher
=
1
1)1(4k
kk
k
.27008820585226910892... 16
105
.27015115293406985870... cos ( )
( )!
( )
( )!( )
1
2
1
2 2
1
2 1 2 10 1
k
k
k
kk
k
k k
.27020975013529207772...
2
14 14
1arctan
2
1
8 x
dxx
.270222741100238847013...
1
21)2(
24
11log2
)3(6
k
kk
k
.27027668478811225897... 2
2116
2
2 1 1
log log
( ) ( )
x
x xdx
.270310072072109588...
1
k1
2)1(
2
3log
3
2
k
kk H J 137
.2703628454614781700... log( )
( ( ) )2 11
12
k
k kk
=1 1
1 12 k kk
log/
K 124h
1 .2703628454614781700... log 2
2
1
)( dxxH
=1
1 20
xx
dx
xcos
.2704365764717983238... 14 2 2
14 4 12
1
tan
k kk
4 .27050983124842272307... 5
23 5 5
1
5
2 2
0
( )
kk
k
k
.27051653806731302836... dxx
x
1
061
1log
32
32log
2
32log5
.270580808427784548...
1
3
4
)1,3()1(4
)4(
360 k
k MHSk
H
.270670566473225383788...
1
3
1
2
1
1
2 !
2)1(
!
2)1(
)!1(
2)1(2
k
kk
k
kk
k
kk
k
k
k
k
ke
3 .27072717208067888495... 2 212
2
3
3 3
2
3 3 4
40
2
loglog ( )
sin
/
x
xdx
1 .27074704126839914207...
0 !2
)1(
)12 kk
kk
k
B
e
e
1 .2709398238358032492...
1
2
4
5,1
24
2 kk
kk
G
.27101495139941834789... cosh
1
2
1
2i i
=
1
21 )12(2)1(k
kk k
.27103467023440152719... ( )3 1
6 2 3 2 3 1 1 1
4
120
6
120
x
xdx
x
xdx
.2711198115330838692...
21
21
22
1
22
1
8
1
2
2log iiii
=( )
1
2
1
31
k
k k k
.27122707202423443856... e
e
k
k kk16
1
16
1
4
1
2 2 1 41
sinh( )!
1 .27128710490414662707...
011
4/3
)14(!
11,
4
5,
4
1F1,
4
1
4
54
4
)1(
k kk
.271360410318151750467... 1
2 2
1
2 2 21
sinh( )!
k
k kk
.27154031740762188924... 3
12
2 1sinh
2
1sinh
2
11cosh
x
dx
x
.2716916193741975876... 212
2 2 22
2 2
0
2
log log logsin sin/
x xdx
1 .2717234563121371107... 0 10
21
22 2
1
2 1F1 ; ; ( )
!( )!
Ik kk
k
.2717962498229888776... 63 5 3 45 3
1501
3 51
log( )k kk
.2718281828549045235... e
10
1 .27201964595140689642...
.27202905498213316295... sinh
11
11
2
121
22k kk k
=
( ) ( ) ( ) ( )i ii i
i i
3 3
101 1
= )2()2(10)2()2(2
1iiii
1 .2721655269759086776... 6
9
.27219826128795026631...
2
1
2
1
28
2log22
iLi
iLi
iG
= dxx
x
1
021
)1log( GR 4.291.8
= log( )x
xdx
1
1 21
GR 4.291.11
=
log x dx
x1 40
1
GR 4.243
=x dx
e ex x 2 20
2log
GR 3.418.3
=
log(sin )sin
sin
/
xx dx
x1 20
2
GR 4.386.1
= log1
1 20
1
x
x
dx
x GR 4.297.3
=x dx
x x x(cos sin )cos
/
0
4
= x x x dxarccos log0
1
=arctan x
xdx
10
1
= log( tan )/
10
4
x dx
GR 4.227.9
= log((cot ) )/
x dx 10
4
GR 4.227.14
.272204279827698971... 1
1242 kk
15 .272386255090690069056...
1
2
k kk
k
FF
.2725887222397812377... 4 25
2
1
2 21
log( )
kk k
6 .2726176910987667721... 8 21
4
12 2 1
161
cotkk
.27270887912132973503... 3 2 21
2 1 21
e Eik
k kkk
log ( )( )!
.27272727272727272727 = 3
11
.27292258601186271514...
( )( )( )
12
1
11
kk
1 .27312531205531787229...
1
22
24
2
1)2(132
5tan
521
kk
k
kFkk
k
.273167869100517867... 2
7
1
2 2
1
221
arcsin
kk k
kk
1 .2732395447351626862... 4 2
3 2
1
1 2
5
4
2 3
22
2
2
( )
( / ) /
( )!!
( )!!
k
kk
J274
=
0 )1(16
12
kk kk
k
= 12 20
2kk
k
tan
K ex. 105
= ( )
( )
2 1
2 2 2
2
1
k
k kk
.2733618190819584304... 1
2 121
kk
.27351246536656649216...
3
2
27
1
381
4
27
)3(26 )2(3
= dxee
xdx
x
xxx
12
2
13
2
1
log
9 .2736184954957037525... 86
1 .27380620491960053093... log log( )3
21 3
12
20
1
xdx
x GR 4.295.38
.2738423195924228646... 1
28 2 14 2 13
2 32
1
log log( )
H
kk
kk
.27389166654145381... 1
254 3 3 27 3 2 32 log ( )
=1
31
334 3
1
1
1k k
k
k
kk
k
( )( )
.2738982192085186132...
8sin
8cosarctanh2
8cos
8sinarctanh2
8
1
=dx
x x4 41
.273957549172835462688... ( )k
k k k kk k
1
2 1
1 1 1
22
2
arctanh
.27413352840916617145...
2 3
31
3
2 21
sin
( )
kk
.2741556778080377394...
12
1 2
23
cos
22
1
36 kk k
k
kk
k
= dxx
x
x
dx
x
1
0
6
13
)1log(11log
2 .2741699796952078083... 32 5 2 71
8
2 3
0
( )
kk
k
k
1 .2743205342359344929...
12 )2/1(
11
2coth
2 k k
J274
=
1
11
2
)2()1(
kk
k k
28 .2743338230813914616... 9
.27443271527712032311...
4
1,
2
3,2,2
2
1
125
log54
625
612
2
F
=
1
1
2)1(
k
k
k
k
k
.2745343040797586252... 83711
304920
1
111
k kk ( )
.27454031009933117475... k
Likk 5
1
5
1
5
1
20
11 2
/ ( , , )
.2746530721670274228... log
( )
3
4
1
2
1
2
1
4 2 11
arctanhk
k k
=dx
e x20 2
.2747164641260063488... ( ) arctan log
14
1
52 2 2
5
41 2
1
k kk
k
H
7452 .274833361552916627... 9
4
.27489603948279800810... 49 6
21
4 31
log
( )k kk
1 .27498151558114209935... 1
3 21k k
k
.275000000000000000000 = 40
11
.27509195393450010741... sin
( )( ) ( )
k
k
ie e e e
k
i i i i
1 221
22
2Li Li
3 .27523219111171830436... G
3
.27538770702495781532... 2 2
2122
2
21 2
1
21
log
loglog Li
=1
2 121
kk k k( )
.27539847458111520468... ( )
! ( )
1
2
1
1
k
k k k
.27543695159824347151... ( )
113 3
0
k
k k k k
=1
42
2
2
1
81
21
2
1
2 2
1
2 2
csch
log( ) ( ) ( ) ( )
i i i i
.27557534443399966272... )1(log)1(log22
iii
=1 1 1 2 1
2 11
1
1k k
k
kk
k
k
arctan( ) ( )
( )
.27568727380043716389... log tanx x dx0
1
.27572056477178320776... cschB
21 1
22 2 1
22 2
2 2 12
1
e e k
k kk
k
( )( )!
J132, AS 4.5.65
.27574430680044962269... loglog(cos ) log( cos )
( )cos
21
2
2
81 1
1
4
k
k
k
k
.275806840982172055143... )(2
1)1log(
162
)3(
4 3
2
iLiiG
= 4/
0
22 cot)3(352log21664
1
dxxxG
3 .275822918721811... Levy constant
4 .2758373284623804537... 8
5
2
5 5
4
5
4
5 1 45 20
csc /
xdx
x
.27591672059822730077...
22 )1(
)(
)(
11)1(
kk
k
kk
k
k
.27613216654423932362... 1
4 2 22
1
2 3
1
21
csch
( )k
k k k
.27614994701951815422...
12 49
1
3128
1
k k
.27625744117189995523... 5
324 3
3 2
3 30
log x
x xdx
1 .27627201552085350313...
0
5
33 sin
32
13dx
x
xx GR 3.787.3
.276326390168236933...
012 )12)(4(!
)1(2
4
1
kk
k
kkErf
.2763586311608143417...
( ) ( )
( )
3 1
6
1 13 3 3
1
k kk
.27648051389327864275... log( )
25
12
2 1
4 21
k
k k kkk
1 .27650248066272008091... 43
81
1 4
2 1
2 21
( )
( / )
k
k k k
.27657738541362436498... ( )
!
11
1
1
k
k k
2 .27660348762875491939... 2
2 3 222 0 1
0eI e e
e
k k
k
k
( ) (; ; )!( )!
F
1 .276631920058325092360...
0
2/)1(
02/3 2
)1(
2
1
2)14(
1
4
1,
2
3
8
1
kkk
kk
k
[Ramanujan] Berndt IV p. 405
.276837783997013443598... ( )
13
1
1
k
k k
k
.276877988824579262196...
1
22
)12)(1(62log4
k
k
kkk
H
.27700075399818548102... 43
234
9 13
23413
csc
= ( )
1132
4
k
k k
.2770211831173581170...
2 1 12
12
1 1
1
( ) ( )! ( )
( )!
k k
kk
k
k
.27704798770564582706...
0
2sinlog565log32arctan8100
1
25
14dxxxex x
.27708998545725868233... 1
4 12k
k k( ( ) )
.2771173658778438152... ( ( ) ) kk
1 3
2
4 .27714550058094978113...
2
1,2,
2
1
.27723885095508739363...
4
3
3
4log2log42log3log43log2log)3( 2
22 Li
4
3
4
133 LiLi
=H
k
H
kk
kk
kk
kk4
1
321
1
1
( ) (2)
.27732500588274005705... log2 2
.27734304784012952697...
3
1
1 .27739019782838851219... x x
edxx
log( )1
10
1 .277409057559636731195...
12 23
1
4
3log3
12
3
k kk
= dxx
x
1
03
3)1log(
.27750463411224827642...
1
2 )!1(
)1(1)1(
k
kk
k
BeLi
1 .27750463411224827642...
0
2 )!1(
)1()1(
k
kk
k
BeLi
.2776801836348978904...
8 2 2 12 20
2
4 20
dx
x
x dx
x( ) ( )
192 .27783871506088740604... 6
5
1 .27792255262726960230... 2 2 2 2sin log ( ) i i i
.2779871641507590436... log7
7
1 .2779927307151103419...
222 log)(
1
k
k
kkk
.278114315443049617352... G46
2log2log28
)3(5 22
=( )
( )
1
2 1
1
21
kk
k
H
k k
.27818226241059482875... 12
2
1
2 1
21
1
arcsinh ( )k
k k
k
k
2 .27820645668385604647... ( )
!( )!
2
1
1 2
1 11
k
k k kI
kk k
7 .278557014709636999...
1
02
2
4/1
log
2
1,3,4
x
x
=
3
23 34
2 22 2
log i Li
iLi
i
.27865247955551829632...
11 22
1
2log2
11
xk
k
dx
17 .27875959474386281154... 11
2
.2788055852806619765...
1
1,2
1)11(
xe
dxerf
x
.27883951715812842167... e
e x
dx
x2
5
22
1
14
sinh
.27892943914276219471...
1
11 5
1
4
1
54
5log
4
5
kk
kkk
k
H
2 .27899152293407918396...
2
)(logk
kk
.27918783603518461481... ( ) ( )
logk k
k
k
kk
kk k
1 11 1
1
22
1
.27930095364867322411... 33
2
1
6 1 6
1
3 2 320 0
k kk
k
kk/
( )
( )
.27937855536096096724... S k k
k kk
22
1
22( , )
( )
.27937884849256930308... 1
2
1
21
1
2
3
2
1
2 2
1
1
, ,( )
k
k k
=1
22 2 2
1
2
( )
.2794002624059601442... 1
4 11
4 11
1
1k
k
k
kk
( )
.27956707220948995874... 1
22
1
2 2
20
1 2
0
Jk
k
k
k
k
kk
( )( )
( )!
2 .2795853023360672674...
0
2
2
020 )!()!(
1)2(
kk k
k
kI LY 6.112
=
deF
2
0
cos210 2
1)1;1(; Marsden p. 203
= 1
2
2
2
2 1 2
0
2
0
2
0( )! ( )!
( )
!k
k
k
k
k
k
ke
k
k
kk k
k
k
.2797954925972408684... 1
102 1 5 1 5
1
531
( ) ( )i ik kk
1 .2798830013730224939... ei
e ek
ke e
k
i icos sin(sin )sin
!1
1
12
GR 1.449.1
1 .27997614913081785... 2 2 1 8
22
1
1
erfi( ) ( )
!e
kk
k
k k
k
.2800000000000000000 =
7
25
1
1
6
2 3
p
pp prime
.28010112968305596106...
41
2
0
2
ee
x
edx
x( )
sin
7 .2801098892805182711... 53
.28015988490015307012... ( )k
k
kk 21
.2802481474936587141...
3
1
61
3
21
3
2
1
2 331
i ik kk
.2802730522345168582... 1
63 9 3 2
2
3
1
3 11
21
log ( )( )
( )
k kk
.28037230554677604783... 5
32 2
3
2
1
2 22
log hgk kk
=( ) ( ( ) )
1 1
2 12
k
kk
k
1 .28037230554677604783... 8
32 2
3
2
13
22
log( )k kk
.28043325348408594672... 1
36
1
3
1
6 41
21
( )
( )
kk
1 .28049886910873569104... 1
2111
1
2
1
2
1
1620 k
kk
HypPFQ { , , },{ , },
.280536386394339161571... ii eei
2log2log2
1sin)1(
=
2
1)(sin
k
kk
k
6 .28063130354983230561... k kk
( ( ) )2
2
1
.2806438353212647928...
2
12log2log2 2 l Berndt 8.17.8
.28086951303729453745...
1 )72(
1
7
2log2
735
352
k kk
.28092980362016137146...
1 2!
)1()21(
2
1log
kk
k
k
ke
=
1 2!
)12()1(
kk
kkk
kk
B [Ramanujan] Berndt Ch. 5
.2809462377984494453...
64
27
1
2
3
428 3
64
27
12 33
43
1
( ) ( )( )kk
.2810251833787232758... 2
3 2124
25192
14
k kk
.28103798890283904259... 2 31
22
1 22
2 1
1
1
log3
2
( )
( )
k k
k k
kk k
2 .281037988902839042593... 33 1
3 1log
.28128147005811129632... 1
22
1
2
1 1
1
log( cos )( ) cos
k
k
k
k GR 1.441.4
.28171817154095476464...
20 )1(!
1
)!2(3
kk kkkk
ke
=k
k k k k kk k!( ) !( )( )2 6
1
2 30 0
1 .28171817154095476464...
0
2
)2(!4
k kk
ke
.28173321065145428066...
1
21
2cos
2cos
!
sin)1(
2
)2cos(sin
2
1
k
k
k
k
e
e
e
e
.28184380823877678730...
8
3sinlog
8sinlog22log
8
7cot
8
3cot
4
=( )
14
1
21
k
k k k
.281864748315611781912...
4
3,
2
1
4
1,
2
122log2
8
1
=( ) log( )
1 2 1
2 1
1
1
k
k
k
k
.2819136638478887003... log( )
224
3 1
2 2 1
2
2 21
k
k kk
=( ) ( )
12
1
3
k
kk
k k
= 1
0
logarctanh dxxx
.28209479177387814347... 14
.28230880383984003308... ( )
1
3
1
1
kk
kk
H
k
(3)
.282352941176470588 =
0
4sin
85
24xe
x
1 .282427129100622636875...
)1(12
1exp , Glaisher’s constant
1 .28251500934298169528... 3 6 21
1 230
log( )
( )( )
k
k k k
.28253095179252723739...
2 3
22
2
3
8 2
32 2 3 4 2
1
43 3
log loglog log log ( )
Li
4
1
43Li
=log ( )
( )
2
120
1 1
x
x xdx
1 .2825498301618640955... 6
2 ( )
.2825795519962425136... 12
141 2
0
Ik
k kk
( )( !)
.28267861238222538196... 1
631 kk
.282698133451692176803...
3log913log32,
3
22
12
1 2 H
=
1 )13)(13(k
k
kk
H
.2827674723312838187... 1
332 kk
.282823346997933107054...
1 )1(2
1
k kkk
k
.2829799868805425028...
0
110 )!1(!
2)1(
2
)22()2;2;(F
k
kk
kk
J
.2831421008321609417...
( )( )
k
k kk
1
2
.28318530717958648... 2 6 2
0
1
arcsin arccosx xdx
6 .28318530717958648... 21
6
5
6
=1
14
340 ( )( )k kk
= log 1 6
0
x dx
=d e dx
e
x
x
2 10
2 6
cos
/
6 .2832291305689209135... 2 2 coth
2 .28333333333333333333 = 137
60 5 H
.2833796840775488247... ( )
( )!sinh
2
2 1
11
1 1
k
kk
kk k
.2834686894262107496... 113
1 31
1
ek
k
kk
/ ( )!
5 .28350800118212351862... 2 1 23 4 4
0
/ log
x dx
.28360467567550685407...
01 )3)(1(2
)1(
)1(32
33log32log3
kk
k
kk kkk
k
.2837571104739336568... log( )
log
2 2
11
2
1
22 1
k
k k kkk k
.28375717363054911029... 6
251
6
5 61
logk Hk
kk
.28382295573711532536... 4,2,1)4,2()4(1
36
49
6)1(
42
2
k k
.2838338208091531730... 1
41
1 1 2
2
1
120 0
1
0
1
e k
x
edx
xdx
k k
kx
( )
( )!
sinh
coth
1 .28402541668774148073... e4
.28403142497970661726... ( ) ( )
( )!
2 2 2
21
k k
kk
=1
1 11 1
2 22 k k kk
cosh
.28422698551241120133... ci ci( )sin ( )sin ( )cos ( )cos( )1 1 2 1 1 1 2 1 si si
=sin xdx
x10
1
6 .2842701641260997176... k Hk
kk
2
1 2
.28427535596883239397... 2
22 2
2122
3
6
2 3
4
1
22
log
log log loglogLi
1
2
1
2
21
4833Li ( )
=log ( )
( )
2
20
1 1
2
x
x xdx
2 .28438013687073247692... ( ) ( )3 4
.28439044848526304562... ( )
( )!sinh
k
k k k kk k2 1
1 1 1
2 1
2 .28479465715621326434... 811
3 .28492699492757736265... F
kk
k !!
1
.28504535266071318937... 2 2
24
log
1 .28515906306284057434... 1
21 k kk ! ( )
.285185277995789630197...
0
24/9
sin2
22 2
dxxe x
.28539816339744830962... 4
1
2
1
4 1
1
21
( )k
k k J366, J606
= sin cos sin cosk k
k
k k
kk k
1
2
1
= x xdxarctan0
1
.2855993321445266580... 11
1 .2856908396267850266...
0
4
)!2(32
1
32
15
k k
k
e
e
.285714285714285714 =27
1 . 2857644965889154964...
0
4422 )!(4
)!4(
8
7
8
5 kk k
k
.285816417082407503426...
13
2
)3(
1
162
11
543
)3(
k kk
1 .2858945918269595525... ( )!!
1 12
1
k
k
k
k
.28602878170071023341... 2
21
2
16
3 2
4 2
8 1
4 4 1
log
( )
G k
k kk
=k k
kk
( )42
.2860913089823752704... ( )3 G
.28623351671205660912...
1
23
12
2
)1(
8
1
24 k
k
kk
.2866116523516815594... 6 2
16
CFG D1
.2868382595494097246... 1
4HypPFQ { , , , },{ , , },
( !)
( )!( )1111
1
22 2
1
4 2 2 2
2
20
k
k kk
.286924988361124608425...
4 2 2 8 2
2
9
2
2 1
22
2
22
coth( )
cschk
kk
= ( )( )
12 1
21
1
kk
k
kk
1 .28701743034606835519... e erfk k
k
1 32
1
12
1
4 2
1
4/
!!
1 .287044548647559922... 11
22
( !)kk
1 .287159051356654205862... e k
k
( )
1
2
1
1 .2872818803541853045... 2 17
183
3
3 2
1
e k
k
k
k
( )!
.28735299049400502... c4 Patterson Ex. 4.4.2
1
12010 2 20 3 15 2 30 4 24 55 3 2 2( ( ) ( ) ( ) ( ) ( ))
2 .2873552871788423912... e
k
k
k
k
sinsin
!
/
12
42
1
1 .287534665778537497484...
2
1
2
1
4
2log2 22
iLi
iLiiG
2 .28767128758328065181... 2 11e e ee ei icos cos(sin )
3 .2876820724517809274...
1 4
11,1,
4
13log2log2
kk k
J117
=
2log
0 012
2
12 1313 xx e
dx
e
dx
xx
dx
xx
dx
=7
1arctanh2
)12(7
12
4
1Li
3
)1(
0121
1
1
kk
kk
k
kk
10 .287788753390262846... k
k
e
k !
0
1 .28802252469807745737... log1
4
2 .288037795340032418... ( )
.2881757683093445651...
5
1
5
6hg
= 53
2log
4
5
4
5log5
5
21
25
=
1
1
12 5
)1()1(
5
1
kk
k
k
k
kk
3 .28836238184562492552...
1
3
!
)(
k k
kt
.288385435416730381149...
2 22
1)1(
)()1(
)1(
)(
k k
k
k
k
kk
k
.28852888971289910742... i i i
kk6
2
3 3
2
3 3
1
3 2 120
( )
.2886078324507664303... 2
=
0
)(2
x
dxee xx GR 3.463
=
( )ex
dx
xx 2 1
120
GR 3.467
=
0
cos2
dxx
xe x
.2886294361119890619...
0 )62)(12(
1
20
3
5
2log
k kk
.2887880950866024213... 11
21
kk
=
12/)3(2/)3( 22
2
1
2
1)1(1
kkkkk
k Hall Thm. 4.1.3
.28881854182630927207...
2 )1(
)(log
k kk
k
1 .288863135...
primep pp 22 )1(
11
.28893183744773042948...
42
0
2
ex x x dx sin(tan ) sin tan
/
GR 3.716.8
.2889466641286552456... 1
4 2 11k
k k( )
.2890254822222362424...
201
sin x
edxx GR 3.463
.289025491920818... 1
one ninth constant
1 .28903527533251028409... 4
3
1
3 2 1
3
2 2 120
2
20
( )
( )
( !)
( )!( )
k
kk
k
kk
k
k k
=
3 3
310
275
13 1
2
20
log( )kk
Berndt 32.7
=
3 33
10
27
5
9
1
3
21log ( )
=
4
3,2,2,
2
1,1,1,1,1HypPFQ=
2
12,,
3
1
3
1
5 .28903989659218829555...
1
5
.2891098726196906... ( )
log
12
2
k
k k k
.289144648570671583112... ( )
1 1
1
k
kk F
.28922492052927723132... 2 3 3
481
3 3 10
( )( )( )
k
k k k
2 .2894597716988003483... 2 log
.289725036179450072359...
1001
3455
1
1
2 4
2 2
p p
pp prime
2 .2898200986307827102... li( )
.2898681336964528729... 2
2 2 2113
31
1 1 2
k k
k
k kkk ( ) ( )( ) K Ex. 111
= k kk
( ( ) )
3 11
1 .2898681336964528729... )1)1()1()((2)2(223 2
2
kkkkk
= 11
12 21
k kk ( )
2 .2898681336964528729...
1
2
2)2()1(13 k
kkk
= dxe
ex
x
0 1
1 GR 3.411.25
=
1
10
1 x
xxdxlog GR 4.231.4
3 .2898681336964528729... 2
233
2 21
2
( )( ) ( )k k
kk
=x dx
x
xdx
x
xdx
x
2
2
2
20
2
211 1sinh
log
( )
log
( )
GR 4.231.4
=log ( ) log( )2
20
1
0
11 1
x
xdx
x
xdx
=x dx
e ex x
2
0 2
=
0 13/1xe
dx
.28989794855663561964... 6 15
CFG C1
.2899379892228521445... 11
1 11
1
1
1
e
eH
e
kk
kk
( log( ))( )
( )
57 .289961630759424687... cot cot1180
.290000000000000000000 =29
10017 62
1
k kk
.29013991712236773249...
1
1
2
)1(
2
1arctan
3
2
kk
kOk H
1 .29045464908758548549... 2 21 1
0
/ ( ) / !e k
k
k
.29051419476144759919...
0 2
sinxe
dxx
=1
41 1
1
211
1
22 22 1 2 1F i i F i i, , , , , , ( ) cos(log( ))
csch
.29053073339766362911...
1
1)1(2
22log3
4
1
k
k
kk
2 .29069825230323823095... e erfkk
( )( )!
11
120
.2907302564411782374...
12 34
1
6
1
2
3coth
34 k k
.29081912799355107029... 4 81
2
1
4 2 30
arctan( )
( )
k
kk k
.29098835343466321219... 1
2 2
1 1
1 10
1
e
e x
e xdx
log( ( ) )
( )
.291120263232506253...
121
22
2
1
4
2log
24 kk k
GR 4.295.17
= 1
02
2
)1(
)1log(
xx
dxx GR 4.295.17
=
log( / )1 22
0
1 x
xdx
.29112157403119441224... 1
5 1
3
531 1k
k
kk
k
( )
.2912006131960343334... 2 2 2 2 2 12
2 1 11
1
1
cos sin ( )( )!( )
kk
k
k
k k
.2912758840711328645... 1 1 2
12
1
2
1
42 2
1
1 1/( )coth
kk k
dx
x
1 .291281950124925073115... 3
24
1 .29128599706266354041... 1
1 0
1
k
dx
xkk
x
GR 3.486
5 .2915026221291811810... 28 2 7
.29156090403081878014... G
.29162205063154922281... 1
4 16
2
4
7 3
8 1
2 2 3 2
20
1
log ( ) arccosx x
xdx
.29166666666666666666 = 7
24
1
2 1
1
214
0
1
( )
( ) ( )
k
kk k
dx
x
.29171761150604061565...
( )2
31
kk
k
.29183340144492820645... 4/14/14/1
3coth3cot4
3
1107
635
= 3 4 13
314
2
k
k k
kk
( ( ) )
.2918559274883350395... ( )k
kLi
k kk k3
23
1
1 1
.2919265817264288065... cos ( )( )!
2 12 1
1
1 1 12
2
kk
k k GR 1.412.2
= ( )
( )!( )
1 4
2 1
1 2
1
k k
k
k
k k
1 .29192819501249250731... 3 2
2124 1
arctan xdx
x
.291935813178557193... ( )( )
!/
11
11
2
1
2
k
k
k
k
k
ke
k
.292017202911390492473... 3
4 2 0
1
G
x x x dxarccot log
.29215558535053869628... ( )
!( ( ) )
112
k
k k k
.29215635618824943767... 1
2 6 2
12
3 2
2 21
coth
( )
/
k kk
.29226465556771149906... ( ) ( )2 1 2 1 11
k kk
.292453634344560827916...
2 1
1)(2log3
2
1
k k
k
=
kk kk
log 11
11
22
.2924930746750417626...
2 5 5
1
2
1
5 121
coth
kk
.29251457658160771495... 1
844
2 2
1
e ek k
kk
cos cos(sin )sin cos
!
.29265193313390600105... H
kk
kk
( )3
1 4
.2928932188134524756... 11
2
= ( ) ( )( )!
( !)
11
4
21
2
41
1
12
1
kk
k
kk
k
k
k
k
k
.2928968253968253 = 7381
25200 10
1
10
1
2510
21
26
H
k k kk k
2 .292998145057285615802...
1
0
2
12
log!
121,2,2,2,1,1,12 dxxe
kkPFQ x
k
3 .293143919512913722...
1
2/3
21,
2
3,
2
1
2
1
kk
k
3020 .2932277767920675142... 7
.29336724568719276586... log( )1
1 30
1
x
xdx
.29380029841095036422... sinh
cosh1
4
14 5
1
x
dx
x
1 .2939358818836499924... ( ) log ( )k kk
2
.293947018268435532717... 2log726)3(632log96)1(48192
1 23 G
=
14
3arctandx
x
x
.2941176470588235 =5
17
.2941592653589935936... 10
51
50
1
2521
coth
kk
J124
.29423354275931886558...
2
223)1()1(
)1(
2)2()2(
2 k kkii
i
=
1
1 1)12()1(6k
k kk
=
0 1
sinxx ee
xx
.29423686092294615057... ( ) ( )
log
1 1 11
1 12
2 2
k
k k
k
k k
k
k k k
.29430074446347607915...
0 12
)1(
kk
k
.2943594734442134266... Ik
k k kk
20
21 2
( )!( )!
.2943720976723057236... k k
kk k
k k
6 3
3 32
2
113 1
( )( ( ) )
.29452431127404311611...
1
22
3 )(
)3(
32
3
k k
kk J1061
= dxx
xxx
05
3)cos(sin Prud. 2.5.29.24
.2945800187144293609...
1
23
12
2
)1(2log2
124
k
k
kk
.29474387140453689637... 1
4
2 1
12
( )
( )
k
kk
.29478885989463223571...
22
2
)(
111
)(
)1)((
kk kk
k
= 112
( )
( )
k
k kk
2 .2948565916733137942... ( )kk
2
2 .294971003328297232258... e2
4 .29507862687843037097... k
kkk
3
1
.29522056775540668012... 1
0
00 arcsin2
)1()1(2
dxxee
LI x
1 .295370965952553590595... 2G
.2954089751509193379...
6
.29543145370663020628...
12 252
11
3
2log2
k kk
= dxx
x
14
)1log(
= dxx
x
1
0
2 11log
3 .295497493360578095... e
.2955013479145809011... 11
12
22
12
2
( )( )k k
H
k
.2955231941257974048...
( )( )
11
k kk
57 .2955779130823208768... 180
, the number of degrees in one radian
.29559205186903602932... ( )( )k k
kk
1
41
7 .29571465265948923842...
1
2
2
22
23csc3
k kk
kk
.295775076062376274178... ( )
1
4 4 5
1
21
k
k k k
.29583686600432907419... x
dx
x
ee
xx
0
31333log3 GR 3.437
3 .29583686600432907419... dxx
xxxx nnnn
1
0
133/13/21
1
33log3 GR 3.272.2
.29588784522204717291...
log log
( )( )2 2 2
21
2
2 12
2
123
1
2
k
kkk
.29593742160765668... 10 14 22 1 2
2
1
log( )( )
k
k kkk
5 .2959766377607603139... cosh
sinh( )!
12 2
24 4
24
0
e e
k
k
k
.29629600000000000000 = 37037
125000, mil/mile
.2965501589414455374... dxx
x
3
1
0
3 11log
12
39
.29667513474359103467...
2
25
2
53
2
5cos
1 11
=
12
11
k kk
.29697406928362356143... 7 312
4
12
41
( )
( )
k
kk
.29699707514508096216... 1
2
3
21
2
121
0
e
k
kk
k
k
( )( )!
.297015540604756039... 12
312
32
14 4 22
1
tan
k kk
.2970968449824711858...
1
1
12 !)!12(
!)!2()1(1,
2
3,2,22
k
k
k
kkF
1 .2974425414002562937... 2 211 20
ek k
k
( )!
3 .2974425414002562937... 22 1
2 20 0
ek
k
pf kk
kk
k
!
( )
.29756636056624138494...
0
1
246
1
612
)1(
k
k
k
.2975868359824348472... 2
22
206
31 5
2
1
2 2 1
log( ) ( !)
( )!( )
k
k
k
k k Berndt 32.3
.29765983718974973707...
0
3
3
3
4
3
1xe
dxx
1 .29777654593982225680...
1
1
2)1(
1k
k
k
k
.2978447548942876738... Hk
kk
3
1 6
.29790266899808726126... 1
0
2arctan)223log(224
1dxx
.2979053513880541819... H
kk
kk
(2)
41
.29817368116159703717...
02 1
)1(2
2
x
dxxeEi
e x
4 .298268799186952004814...
2
3
1)2(
)(
k k
k
.29827840441917252967... 1 2
22 k
k
kk
log
.2984308781230878565...
1
2
log1)2(
1
k k
k
3 .298462247044795629253... 2
3 29
2
2
20
1
log( )log( )
x
xdx
.29849353784930260079... ( ) ( )
1
1
1
2 12
k
kk
k
j kjkk k
3 .2985089027387068694... 32945
49
, volume of the unit sphere in R
.2986265782046758335... log6
6
.29863201236633127840...
0
2
)!3(
2
8
5
8 k
k
k
e
.2986798531646551388...
3 33
3 3 13 230
log( )
xdx
e x GR 3.456.2
=
x x dx
x
log
( )1 3 230
1
GR 4.244.3
.29868312621868806528... ( )( )
12 1
11
1
k
k
k
k
.2987954712201816344... 1516
24 33 3
81
3 41
log( )k kk
1 .29895306805743878110...
0 22
1
22
1arcsin
77
8
7
8
k k
k
k
.29933228862677768482...
1
2
4
)1()1(42
4 kk
k kkG
1 .2993615699382227606... ( )k
kk2
2 2
.29942803519306324920... ( ) cot2
1
2 2 2 2
1
2
2
4 21
k
k kk
= ( ) ( )2 2 2
21
k kk
k
.29944356942685078... ( )
153
2
k
k k
.2994647090064681986... 6
3 3e e J132
8 .29946505124451516171... 1
211 1
0
e e ek
ke e
k
/ cosh cosh(sinh )cosh
! GR 1.471.2
2 .2998054391128603133... 3 1 12
120
logx
dx
.299875433839200776952... 2 2
4148 3
1
3
2
4
G x
xdx
log arctan
5 .29991625085634987194... 3
12
3
1
2
3
3
2
3
1)
3
1,
3
1(
.300000000000000000000 = 3
10
1
2 31
0
2 1 3
cosh log( ) ( ) logk
k
ke J943
= sin cos3
03
0
x
edx
x
edxx x
1 .30031286185924454169...
1222 )1(
11
2
41cos
2
41cos
1
k kk
ii
.300428331753546243129...
1
0
12
)()(
kk
kk
.300462606288665774427... 12
13
.30051422578989857135...
( )
( ) log log3
42 3
2
32 2
1
22 2
1
23
2 3
Li Li
[Ramanujan] Berndt Ch. 9
= log log
( ) ( )
2
31
2
1 1 1
x
x xdx
x
x x xdx
= log2
1
2
1
x
xdx
[Ramanujan] Berndt Ch. 9
=
1
02
2
1
logdx
x
xx
= x dx
e x
2
20 1
= dxx
x
1
0
2 )1(log
= log ( ) log( ) log2
0
1 2
0
11 1
x
xdx
x x
xdx
= logsin tan/
x xdx2
0
2
2 .300674528725270487137... 2 3 ( )
.300827687346536407216... H
kk
kk 2 2 11 ( )
1 .3010141145324885742... ( ) ( )
( ) ( ) ( )3 4
390
41
41
13
1
k
k
k
kk k
HW Thm. 290
.301029995663981195214... log10 2
42 .30104750373350806686... 0
1
( )
!
k
kk
.3010931726... primep kkk
pp 1
1
.30116867893975678925... sin cos1 1
=
1 1
11
)!2(
)12()1(
)12()!12(
)1(
k k
kk
k
k
kk
= 3
1
1
0
1sinsin
x
dx
xdxxx
= dxx
xxe
1
logsinlog
1 .301225428913118242223... )4()2(360)4(60)2()4(1080090 2462
10
=
12
2log)(
k k
kk
1 .30129028456857300855... 2 11
11
1
1
4
20
20
log logx
xdx
xdx
=
022 )1)(2/1( xx
dx
2 .3012989023072948735...
)1(
2cos)}Re{sin(log
2sinh iii
= 11
2 1 20
( )kk
.301376553376076650855... 6
2
92
0
1
x xdxarcsin GR 4.523.1
.301544001363391640157... 12
21
2
2 11
2
1
sin( )
( )!( )k
k
k
k
k k
.301733853597972457948... 1
5 1 51
0
1k
kk
k
k
( )
.301737240203145494461... log log( )2
0
13
4
1 2
1 2
x
xdx
1 .301760336046015099876... 22tanharcsin2
arctan2 2 gde
1 .3018463986037126778... log ( )( )
loge e k
k kk
k k
2
12
111
12
1
= logsinh
log log
1 1i i
11 .30192195213633049636... Ik
k
k0 2
0
44
( )( !)
.301996410804989806545... 840
3 3
2 1
161
k
k
kk
k
.302291222970795777178... 1
2
1
2
2 1
2 11
logcotcos( )
k
kk
GR 1.442.2
.30229989403903630843...
0 )33)(13(
)1(
36 k
k
kk
= dx
x
x dx
x x
dx
x30
4 21
2 28 1 3
( )
=
012
7
012
3
11 x
dxx
x
dxx
.30240035386897389123...
2 2
1)(
kk
k kH
2 .302585092994045684018... log10
5 .30263321633763963143...
03
43
)2(3
)(
12
4
32)3(56
k k
=
334
)()2)(1(
kk
kkk
.30276089444130022385...
1
1)2( dxx
1 .30292004734231464290... 2
2
0
1
122
2
1
loglog( / )x
xdx
.303150275147523568676... log2
3
.303265329856316711802... 1
2
1
2
1
1e k
k
kk
( )
!
3 .30326599919412410519... 2 5 5
2
5
3
5csc sec
.303301072125612324873...
2
1coscosh
2
1cossinh
2
1sincos1
= 1
1
21
21 21
1
cos
sin
( )cos
!(cos ) /e
k
kk
kk
1 .303324170670084184712... 1)(2
)2(
kHk
k
.303394881754352755635... 4
4 11
1
2
20
2
2 2(log ) log( )
x
x
x dx
x GR 4.298.19
.30342106428050762292...
2 log)1()1(
1
k kkkk
1 .303497944016070234172... log!
11
1
kk
.30378945806558801568... 8
9 3
2 2
3
1
3 2
1
1
log ( )
( / )
k
k k k
.303826933784633913104...
2 2 2
5
61
2 1
21
1
coth ( )( )
kk
k
k
= 1
2 122 kk
.30389379330748584659... 21
3
1
2 2 33 3
2
0
1
Li Lix
x xdx
log
( )( )
.3039485149055946998... 2
2 2
0
1
12
14
27 x xdxarcsin
.30396355092701331433... 3
2
.30410500345454707706... 3 2 21 2
12
1
( )!
( )!
k
k kk
.304186489039561045624... 24
2
4
1
16 1221
log
k kk
.304349609021883684177... 1
2 2
1
22 2log
sinhlog ( ) ( )
i i
= ( )( )
12 1
21
1
k
k
k
k
2 .304632878711566539649...
12 12
112csc
2 k kk
2 .304988258242580902703... 3 1 2
2 318 2
1
1 4
( )
( / )
k
k
k
k
.305232894324563360615... log log log ( )2 2 2 22
3
x dx GR 6.441.1
.3053218647257396717... G
3
.305490036930133642027... 5040 185471
ek
k kk !( )
.305514069416909279097... )3(2log212log263
2log22log2
232
=
1
)2(
)2(2kk
k
k
H
.305548232301482856123... 1
23 kk !
.30562962705065479621... 3 2
2
1
31
2
2 1 3 2 11
arcsin( )!!
( )!! ( )
k
k kkk
.305808077190268473430... 1
3 22 k kk log
.305986696230598506245...
1
3
6451
138
kkkF
.306018059984358556256... 63
2 3 12
2
0
1
( ) log( ) logx x dx
.306119389040098524366... ( )
( )!
/k
k
e
kk
k
k
2 1
11
1
32
.30613305077076348915... 4 3
27
1
2 12 30
dx
x x( )
1 .30619332493997485204... 3 2
4
3 2
2
3
22
1 4
23 2
3
1
sin cos( )
( )
( )!/
J
k
k
k k
k
13 .30625662804500320717... 128
9
1 3 1 1
42
1
Gk
kk k
kk
( ) ( )( )( )
.306354175616294411282... 1
4
3
42
0
4
x e dxx
1 .306562964876376527857... 8
5sin2
8sin
8cos
2
22
= 11
4 20
( )k
k k
.306563211820816809902... 12
3
1
30
1
21
Jk
k
kk
( )
( !)
8 .3066238629180748526... 69
.306634521423027913862...
3
11579
1215
1 )3(4
= log3
3 21
x
x x xdx
.30683697542290869392...
122
22
)1(
1
2
1csch
4coth
4 k k
.306852819440054690583... 1 21
122 2
log( )
( )( )k
k
kk
kk
k
= k
kkk
1
22
J145
= 1
2 11k
k k k( )
J149
= 1
4 22
1
2 21
1
31k k k kk
k
k( )
( )
( )
=
1 )32)(12(k
k
kk
H
= dx
x x k k
dx
x xk3 2
12
12
1
1
= dx
e ex x
10
= e ee
x
dx
xx x
x
2
2
0
GR 3.438.3
= ( ) /x e edx
xx x
1 2
0
GR 3.435
= xx
x
dx
x
1
0
1
log log GR 4.283.1
= xe
edx
x
x
20 1
GR 3.452.4
= log x
x xdx
2 21 1
GR 4.241.8
= log(sin ) sin/
x xdx0
2
GR 4.384.5
1 .306852819440054690583... 2 20
1
log arccos logx x dx GR 4.591.2
.30747679967138350672... 3 1 2
4
1
4 2 1 2 3
2
0
arcsinh
( )
( )( )
k
kk k k
k
k
= x x dxarcsinh0
1
.307654580328819552466... 3 3
8 48
1
16
1
2
2 1
31
( ) ( )
( )
k
k k k
.307692307692307692307692= 4
13
.307799372444653646135... 111
1
1
ek e
ek
kk
/ ( )
!
.30781323519300713107...
log(cos )( )
sin1
21 1
2
1
k
k
k
k J523
.308169071115984935787... arccot
( )
( )
1
2 12 11
k
kk k
27 .308232836016486629202... cosh( )!
42
16
2
4 4
0
e e
k
k
k
AS 4.5.63
.308337097266942899991... ( )( )
13 1
1
1
kk
k
k
.308425137534042456839... 2
20
2232
1
4 2
1
2
( )
( ) ( )
k
k k
kk
k
= x x
xdx
x x
x
x
x
log log arctan4
0
1
41
20
1
1 1 1
.30850832255367103953...
0 03
1
20
2
!
)1(
)!(
)!2()1()2(
k k
k
k
k
kk
k
e
I
1 .3085180169126677982...
1
0
1
0
1
0
2
12log6
43 dzdydx
xyz
zyx
.30860900855623185640...
022
)1(
k
k
k
2 .308862659606131442426... 4
1 .308996938995747182693...
12 94848
1
12
5
k kk
=
0
3/
12
)1(
k
k
k Prud. 5.1.4.5
.309033126487808472317...
Li2
1
3
= 1
23 2 3 4 4 2 2
1
42 2
2log log log log
Li
= 1
63 3 3 4 6
3
42 2 2
2
log log Li
= ( )
1
3 4
1
21 1
k
kk
kk
kk
H
k
=
1
0 3
logdx
x
x
.30938841220604682364... 3
2 6
9 3
29
1
3
2
3 21
log
k kk
= ( )( )
12
31
1
kk
k
k
4 .309401076758503058037... 24
3 CFG A10
2 .30967037950216633419...
1
2 .30967883660676806428...
0
2
2
2
)2log(
22
2log3
x
x
16 .309690970754271412162... 6e
.309859339101112430184...
11
2 6
)2(
16
1
6cot
622
1
kk
k
k
k
.30989965660362137025... )()( ii
.309970597375750274693... log
( ) ( )2
2
1
8
1
2
1
2
i ii i
= sin2
0 1
x
edxx
.310016091825079303073... ( )( )
( )( ) log
12 1
11 1
111
1
22
2
k
k k
k
k kk
k
.31034129822300065748... 2
12sin
1
)1(
2
1cos
2
1sin)1cos22log2(
2
1
1
1
k
kk
k
1 .310485335489191657337... k
Hkkk 21
1 .310509699125215882302...
2 2
23
2
3 8
3
8 42 3 coth coth ( )csch csch
= ( ) ( ) ( )
1 22
2
k
k
k k k
3 .310914970542980894360... 9
1 116
1
17
1e
k
k
k
kk
k
k
k
( )!
( )!
.311001742407005668121...
159
1820 2 1414
1
8 221
cot
k kk
1 .3110287771460599052...
4
1
24
1 2
, the lemniscate constant
= dxxK
2/
0
223
sin12
1
2
2
4
3
22
1
=
1
04
1
14
3
4
5
x
dx
2 .311350336852182164229... 1
0 k kk ! !!
2 .31145469958184343582... 2
11
2 20
e
xe x
dx
log log
.311612620070115256697... 1arcsinh4
221log
22
1
.311696880108669610301... 1cosh8
sinhsinh
2csch
16
1 22
=
122
2
1
1
)14(
4
4
)2()1(
kkk
k
k
kk
.31174142278816155776... sinsin
coshcos
sinhcos
sinsin
(cos ) /
1
2
1
2
1
2
1 1
21 2
e
= ( ) sin
!
1
2
1
1
k
kk
k
k
.311770492301365488... )!1(
)1()1()1()1(1
1
1
k
HEieEie
ek
k
k
.311821131864326983238... 2
3 3
1 3
2
3
3 3
1 3
22 2
i
iLi
i i
iLi
i
= 5
36
1
6
1
3 1
21
20
1
( ) logx x
x xdx GR 4.233.4
= x
e e edxx x x( )
10
GR 3.418.2
.31182733772005388216... 7
7
2
7
8
1
5 821
tanh
k kk
8 .31187288206608164149... 7
.311993314369948002... 12
3 2
22
2
2
2
log
loglog
=( ) ( )
1 1
11
k
k
k k
k
1 .312371838687628161 = 1920
14631
1
71
1
111
1
19
= 2 11
31
1
71
1
111
1
192 2 2 2
[Ramanujan] Berndt Ch. 22
.312382639940836992172... 3 3
4 62 4 2 8
1
2 1
2 1
31
( )log
( )
( )
k
k k k
2 .3124329444966854611...
1
3!
22},,2,2,2,2{},1,1,1,1{2
k
k
kkHypPFQ
.312500000000000000000 = 5
16 5
1
2
1
113 1
23 2 2
2
k
k k k k kkk k k ( )
= k kk
( )2 1 11
2 .3126789504275163185... ( )2 1
21
21
2
k
kLi
kk k
.312769582219941460697... ( )
! ( )
1
1
1
1
k
k k k
.312821376456508281708... 2 3
13 2 2
e
x
xdx
cos
( )
.31284118498645851249...
1
1
2
1)2()1(
22
22log
kk
k
k
kii
.312941524372491884969... 12
1
6
2
6 41
log arctan x
xdx
.313035285499331303636... (coth )!
1 12
1
2 22 2
1 0
e e
B
kkk
kk
k
1 .313035285499331303636... coth cot( )!
11
1
4
2
2
22
0
e
ei i
B
k
kk
k
AS 4.5.67
2 .313035285499331303636... 2
11
22
21
0
e
e
B
kk k
k
k
( )!
.313165880450868375872... arccsch
.31321412527066138178... log arctan( )( )
34
10
3 2 9 12 21
k
k kk
[Ramanujan] Berndt Ch. 2
.313232103973115614670... 1
531 kk
.313248314656602053118... 1
11 1
2 21 1e e k e ke
k
ekk
k
( )
.313261687518222834049... log( )
11 1 1
1
e e k
k
kk
=
1 1xe
dx
1 .313261687518222834049... log( )1 e
.313298542511547496908...
( )
( )
3 1
2 1
.31330685966024952260... )2(2
1)2(
3
24,3,
2
1F
2
1120211 I
eI
e
=
k
k
kk
k 2
)!2(
1)1(
0
.3133285343288750628...
1
211
)!1(
)!()1(
24
1
32 k
k
k
k
=
2/
04
2tan2
cotcos
cos22
dxxx
xex x GR 3.964.2
.3134363430819095293... 234
21
41
1 31 1
ek k k kk k!( ) ( )! !
.313513747770728380036... 2 23
23
3
2
1
6 21
log log
k kk
= dxx
x
1
02
6 )1log(
.313535532949589876285...
1
21
31
3
1
3 31
i i
k kk
=
1
1
3
)12()1(
332
1
kk
k kii
.313666127079253925969... log( )
( )
2
3
.313707199711785966821... ( )
! ( )
1
2 2 1
1
1
k
kk k k
Titchmarsh 14.32.3
30 .3137990863619991972... 1
216 3
1
4 1 16
2
40
1
, ,log
/
x
xdx
.313799364234217850594... 3
3
2
2
3
1
3
hg hg
2 .31381006997532558873... 4
12 372
2 21
2 1
1 1
log( )H k
k k kk
k
.3139018813067524238...
( )
( )
( )
( )
( ) log2
2
2 2
22 3
2
21
k k
kk
1 .31413511119186663685... 3
4 16
2 4 3
30
x
xdx
sinh
.314159265358979323846... 10
.314329805996472663139... 7129
22680 9
1
99
21
H
k kk
1 .3145763107479915715... 1036
225 3
22
5 2
H ( )/
.31461326649400975195... H
k kk
kk
( )
( )
3
1 2 1
2 .31462919078223930970... e e Eik
kk
( )( )
( )!1 1
1
11
.31506522977259791641... ( ) ( )
1
2
1 3
1
kk
kk
H
.315121018556805747334... 1
3
1
3
1
3 2 1
1
1
sin( )
( )!
k
kk k
2 .31515737339411700043...
0
)2/1(
11
2
3
2 xe
dxxi
1 .31521355573534521931... 1
1 51 ( )
kk
.315236751687193398061... e2
2 .315239207248862830396... 3
22
2016 4
24
loglog x
xdx
.315275214363904697922... ( )( )
( )
13
2
2
2 11
3
3 211
kk
kk
kk k
k
.31546487678572871855...
1
9 332log
xx
dx
.315496761830453469039... 3 3
6 3
6 3 3
6
3 3
6 3
6 3 3
65 6
7 6 2 3
5 6
7 6 2 3
i i i i/
/ /
/
/ /
3
3 31
1
35 6 1 3/ /
=
1 1
13 3
)3()1(
13
1
k kk
k k
k
.31559503344046037327... 9
8
15
161
2 51
eerfi
k
k kk
!( )
.31571845205389007685...
primepk ppk
k
k 111log)(log
)(
2
=
2
)(
k
p
k
k
.315789473684210526 =6
19
.3158473598363041129... tan xdx
ex0
1
.315888571364487820746...
32 12
113sin2csc
6
1
k kk
.3160073586544089317... log( log( ))1 10
1
x dx
.316060279414278839202... 1
2
1
2
2
0
1
ex e dxx
=
13
152
1cosh
2
11cosh
x
dx
xx
dx
x
1 .316074012952492460819... 31 4/
.3162277660168379332... 3
1arctansin
10
10
.3162417889176087212... log (loglog
) log (log
)2 32
23
3
2
= ( )( )
121
kk
k
k
k
.316281418603112960885...
loglog
log( )3
2
1
30
1
1
x dx
x
k
kkk
GR 4.221.3
.316302575782329983254...
122 )1(
)2(2)3(3k
k
kk
H
.31642150902189314370...
1
2411
2 14
)24(
14
)2(
2
1
kk
kk
k
kkk
.3165164694380020839... 1
41
40
2
21
Ik
k kk
( )( !)
6 .31656383902767884872... Ei e Ei e dxex
( ) ( ) 10
1
3 .316624790355399849115... 11
.316694367640749877787... 2 2 2 2 22 1
81
( log log( )
k
k kkk
=( )!!
( )!
2 1
2 21
k
k kkk
.316737643877378685674... 1
3
1
3 3 41
e
dx
x
e
.316792763484165509320... cosh sinh
( )!
1 3 1
16 8
1
16 2 1
4
1
e
e
k
kk
1 .316957896924816708625... arccosh arcsinh2 21
22 3 log
= log3 1
3 1
.3171338092998413134... 1 2 1 2 21
12
1
( ) log log
( ) ( )
( )
k
k
k
k k
5 .317361552716548081895... 3
.317418282448647644165...
13
3
)3/2(
1
8
28
33
2)3(13
k k
.317430549669142228460...
1
2
1
14
)1(2
2/sinh21
k
k
k
1 .31745176555867314275... 13
48 24
11
122
1
2 2
50
1
log
log
( )
x
xdx
111 .317778489856226026841... e i i3 2 33
2
3
2 / cosh sinh
.317803023016524494541... 67
120 1
4 3
3 21
3
0
log
( )
x
x xdx
x
e edxx x
.31783724519578224472... 3 2
1 .3179021514544038949... eEik k
k H
kk
k
k
( )! ( )!
11
11 1
=1
11 1 0
1
( )! !loglog log
k kxdx e xdx
k
e
x
1 .318057480653625305231... 1
211
1
211
1
2
1
2 12 1 2 1 20
F i i F i ikk
k
, , , , , ,( )
1 .318234415786588472402...
2
3
3 3
2 2 3
log GR 8.366.7
.318309886183790671538...
044 396)!(
)110326390()!4(
9801
221
kkk
kk
Ramanujan, Borwein-Devlin p. 74
02/)33(3 640530)!()!3(
)13591409545140134()!6()1(12
kk
k
kk
kkBorwein-Devlin p. 74
1 )1)(1(
)(
k kk
kk
J1061
= x xdxsin0
1
.318335218955105656695... 1
83 22 2 3 2 coth coth csch csch
1
81 1 11 1 2 2i i i i i i i ( ) ( ) ( ) ( )( ) ( ) ( ) ( )
=
232
22
1
1
k k
kkk
= ( ) ( ) ( )
1 2 2 11 2
1
k
k
k k k
2 .318390655104304125550... cosh
( )
5
14
2 1 21
kk
5 .318400000000000000000 = 3324
625 4
2 2
1
F kkk
k
1 .31851309234965891718... ( )1 7
6
37 .31851309234965891718... ( )1 1
6
.318876248690727246329... 3
422
2
0
2
ex x dx cos( tan ) cos
/
GR 3.716.5
.318904117184602840105...
2
53
2
535log24arccsch
55
222 LiLi
= ( )
12
1
1
k k
k
Hk
k
.319060885420898132443... ( )( )
( )!cos
12 1
21
11
1 2
k
k k
k
k k
.319135254455177440486...
0 )52(2
1
3
141arcsinh24
kk k
1 .319507910772894259374... 41 5/
.319737807484861943514...
21
2 log)1(2
)1(2
)1(2log2log
k
k
kk k
k
k
k
.31987291043476806811... arccot cot csc ( )sin
1 2 1 12
1
1
kk
k
k
k
.32019863262852587630... logk
k kk3 2
2
.32030200927880094978... 2 2
3
1 3
1 3
2
0
1log log( )
x
xdx
.320341142512793836273... ( )( )
11 1
22
21
k
k k
k
kLi
k k
.32042269407388092804... 1
11 1
21 ( )!( )
log
!n
k
kk
n
kn
9 .320451712281870406689... k
Fkk
1
.320650948005153951322...
Lik
k
kk
3
1
31
1
3
1
3
( )
.320756476162566431408... ( )( )
( )!sin
12 1
2 1 2
1 1
21
2 11 2
kk
k k
k
k k k
1 .320796326794896619231... 2
1
4 2
22
1
ie
k
ki
k
logsin //
=i e
e
k
k
i
ik2
1
1
2
21
logsin/
/
1 .320807282642230228386...
2
21
4
2
10
cothsin
x
edxx
.32093945142341793226...
1
2)22()2(k
kk
.320987654320987654 = 26
81 1 40
2
dx
x( )
.321006354171935371018... e k
k kk
1 4
14 4
/
!
.321027287319422150883...
1
4
8623
200
kk
kF
.32104630796716535150... cot1
2
.32111172497345558138... 7
48 42
3 2
4
1
1
2 2 2
21
loglog log( )
( )
x
x xdx
1 .32130639967764964207... 4
5
2
5 5
2
5
3
5 1 5 20
csc /
dx
x
2 .321358334513407482394... 4
3
2
3 32 3
4
31
16
9 1222
2
log ( )
k
k k
kk k
.32138852111168611157... 2
1
3
1
3 2 1
1
2 10
erfk k
k
kk
( )
! ( )
.321492381818579142365... 16
63 37
37
2
1
7 321
tan
k kk
.32166162685421011197...
( )
log ( )k
kk
k
1
.3217505543966421934...
1
12
1
)12(3
)1(
42arctan
3
1arctan
kk
k
k
=
1
21
)1(
2arctan)1(
k
k
k [Ramanujan] Berndt Ch. 2, Eq. 7.5
= arctan( )
1
2 1 21 kk
[Ramanujan] Berndt Ch. 2, Eq. 7.6
=dx
x21
2
1
1 .32177644108013950981... HypPFQk
kk
k
{ , , }, , ,
( )
1111
22
1
4
12
10
1 .321779532040728089443...
1
021
1coscos
2
1sin
x
dx
xx
1 .321790572608050379284... 2 3 4 ( ) ( )
2 .32185317771930238873... 1
2111ijk
kji
.321887582486820074920... log5
5 4
2
1
F
kkk
k
2 .321928094887362347870... log2 5 3 .321928094887362347870... log2 10
.32195680437221399043... 3
3
2
31
21
1
5 721
tanh
k kk
1 .321997842709168891477...
1
24
242
12
221sin1sincsch
2
1
k kk
kkii
1 .32237166979136267848...
2csc55112533755252521475
525555
22/5
harc
=F
k
k
k
k
2
1 2
.322467033424113218236... 2
2212
1
2
1
2
kk
= 2
22
22
24
1
2 Li e Li ei i( ) ( )
= H
k k kk
k ( )( )
1 21
= ( )( ) ( )
11
21
2
k
k
k k
= ( )cos
1 12
2k k
k
.322634...
12
0 )()(
k k
kk
2 .32272613946042708519... 1
12
2
2
e
e Borwein-Devlin p. 90
1 .322875655532295295251... 2
7
14 .32305687810051332422... 2 202
0
2
I e dxx( ) cos
.32314082274133397351...
1
)2(
3 )1(28
)3(5
2
12
kk
k
kk
HLi
.323143494240176057189... ( ) ( ) ( )( )
2 2 3 41
2
41
k
kk
.323283066580692942094... ( ) ( )
!
k k
kk
2
2
3 .323350970447842551184... 15
8
7
2
2 .323379732200195668706...
1 )1(
log
k
k
kk
kH
1 .32338804773499197945... 9
251
1
3 21
sinh
( )
kk
.323409591142274671172... ( )
1
3 10
k
kk
.3234778813138516209... 1
23 2 2 1
11
2
log log ( ) ( )
k
k
k
kk
.323712439072071081029... sinh( )!
1 1
2 1 2 10
k k
k
AS 4.5.62
6 .323732825290032637102...
31csch31csc1sin1sin ii
=
124
24
22
22
k kk
kk
.32378629924752936815... log log( )2
0
15
8
1 4
1 4
x
xdx
3 .3239444327545729800...
1 21
kkkH
.323946106931980719981... arccsc
.324027136831942699788...
0
)12()1(1cosh2
1
k
kk e J943
6 .324033241647430377432...
1
23
24
9
4
216
49
360
17
3
)3(11
k
kk
k
HH
.324137740053329817241... 2
2 2 216 2
1
4
1
2
Li e Li ek
ki i
k
( ) ( )cos
GR 1.443.4
.32418662354114975609... ( )
( )!sinh
2 1
2 1 2
1 1
22 11 2
k
k k kkk k
.32420742957670372545...
15
3015
491
770
1
8 121
cot
k kk
4 .324426956609796143497... 23 2
0
1
8 21
loglog( ) log
/
x x
xdx
6 .324555320336758664... 40 2 10
1 .3246052049335865...
2 log)32)(1(
)1(
k
k
kkk
1 .324609089252005846663... cosh( )
4
11
4 2 1 20
kk
J1079
3 .324659569140513181898...
0
2
2
2
32csc3sinh
2
3
k k
kh
1 .324666791899989156496... ( )
( )
1
1 3
2
1
kk
k
.324686338305007385255... )1()2(12)2(144)2(126 4222
6ammaStieltjesG
=
1
log)(
k k
kk
1 .324717957244746025961...
3/1
3/2
3/1
2
69
2
9
3
1
2
693
2
27
3
1
= the plastic constant, unique real root of 013 xx
2 .32478477284047906124...
1 1 11 11
0
1
3
2
1
12
1
12
)(
2
)(
i j kijk
k jjk
kk
kk
kkt
.32525134178898070857... 3 6
3
1
2
1 12
1
( ) ( )!
!
k
kk
k
k
.325322571142143252138...
0
41
)1(12
4 x
dxxx
.325440084011503774969... arctane dx
e ex x
2
2 20
1
2 8
8 .32552896478319506789...
1
2
12
)(
kk
k
.325735007935279947724... 1
3
.325921357147665220333... ( )
( )!cosh
2 1
2
11
1 2
k
k kk k
.326149892512184321542...
4
73
4
7322arccsc
2
7log22
77
422
iLi
iLii
=
1 22k k
k
k
kH
.326190726563202248839... ( )
!
k
kk
1
1 .326324405266653433549... ( )( )
( )!sin
1 22
2
11 2 1
1
2
1
k k
k k
k
k k
.3264209744985675... H ( )
/2
1 6
1 .32646029847617125005... 224
3128
1
16
2 2
0
kk
k
k
.32654323173422703585...
1
2
22 cos
)2(2
1cos dxx
xsi
.32655812671825666123... 1)1(3
)1(62log2
34
1
kk
k
k
1 .32693107195602131840... ( )4 2
2 2 11
2
41
k k
kkk k
3 .326953110002499790192... e ( )3 43 .327073280914999519496... )(zofzeroofpartimaginary
9 .3273790530888150456... 87
.32752671473888716055... 17
42 2 6 3
1
2
3 21
( ) ( )log
( )
x
x xdx
.3275602576698990809... 13
2 3 121
( log log ) ( )( )
kk
k
k
3 .327629490322829874733... 4
3 ( )
3 .32772475341775212043... 8 41
1 2
1
1 2
1
2
1
21
Gk k
k k
k
( )
( / )
( )
( / )
7 .32772475341775212043... 82
13
41
Gk
kkk
, Adamchik (28)
129 .32773993753692033334... 2 56 31
42
13 21
43
0
( )( )
( )
kk
.327982214284782231433... 1
1 122 k
k
kk
log
.32813401447599016212... 11
21 1 ( ) ( )i i
= ( ) ( ) ( )
1 2 12
k
k
k k
.32819182748668491245...
k
k
kIIe
kk
2
4)!1(
11
2
1
2
111,2,
2
1F
11011
=( )!!
( )!!( )!
2 1
2 11
k
k kk
2 .328209453230011768962... 256
35
4
1 2
/
1 .32848684293666443478... dxx
x1
03
33 arcsin
162
2log3
= 2/
03
3
sin
cos
dxx
xx
.328621179406939023989... ( )k
kkk 21
13 .32864881447509874105... 23
.328962105860050023611... 2
22
032 2 2
1
2 2
log( )k
k k
=
0 2/1xx ee
dxx
.328986813369645287295... 2
30
.32923616284981706824... 2/
0
2 sinlog)3(72log216
1
dxxx
= 1
0
1
0221
)1log(dydx
yx
xy
1 .329340388179137020474... dxxe x )1(2
5
4
3 2
0
2
1 .329396468534378841726...
1 )32)(2(
log
k kk
k
.32940794999406386902... | ( )| k
kk 41
.3294846162243771650... 19
1 2 2 2 2 3 121
( log log ) ( )( )
kk
k
k k
3 .329626035009534959936...
3 2
2
1
0
1
loglog
log logx
x xdx GR 4.325.11
3 .329736267392905745890... 2
3
39
12
1
1
2 2
0
1
( ) logx x
xdx
.329765314956699107618... arctanh1 1
2 12 10
kk k( )
.329809295380395661449...
11 122
)12(2 kk
kkk
kk
.3302299...
primepk ppkkk
k
)1(
1
)23(
)(2
12
.33031019944919003837...
162
23
22
2 22 2 2coth coth
csch csch
= ( )( ) ( )
( )
12
2
2 2 1
2 12
1
2 2
2 31
k
kk
k
kk k k
k
.330357756100234864973...
2
1
2
1036
225
7
2
( )
3 .330764430653872718842... 2 1 11
e Eih k
kk
( )( )
!where h k Hi
i
k
( )
1
1 .33080124357305868479... 11
1 61
( )( )k kk
23 .330874490725823342456... 45 5
2 1
4
0
4
21
( ) log
x dx
e
x
x xdxx
.330893268204054533566... log( )e 2
20 .33104603466475394519... )1sinh1cosh1cosh()1sinh()1cosh(2
1 ee
)1sinh1cosh1sinh(2
1
= e e
e
k
k
e e
k
2 1
12 1
/ sinh
( )!
1 .331265897019480833836... 27 3
2
27
2
1
1 921
log
( / )
k kk
Prud. 5.1.25.20
1 .331335363800389712798... 1 4/
31 .331335637532090171911...
32
3
1 2 3
3 44
,( )( )( ) ( )
k k k kk
k
.331746833315620593286...
1
01
121
sinhsin)!4(
2)1(1sinh1cos1sin1cosh
2
1dxxx
k
k
k
kk
.33194832233889384697... 4 4 3 1 32 2
0
dx
x x( )( )
6 .33212750537491479243...
2 2
3 3
2
3
2
1
6log
log Berndt 8.6.1
1 .33224156980746598006... ( )3 1
2 2 113
1
k k
kkk k
.33233509704478425512... 3
16
1 3
1
1 12
1
( ) ( )!
( )!
k k
k
k
k
.33235580126764226213... ( )4 3
4 14 31
kk
k
1 .33271169523087469727... 4 2 .33302465198892947972... log3 2
.333177923807718674318... 2
.333333333333333333333 = 1
3
= 1
3 3
1
3 1 3 4
1
2 1 2 51 00k k k k k kk kk( ) ( )( ) ( )( )
= 1
5 6
1
4 4 321
21k k k kk k
= ( )
( )
1
21
22
1
1
1
1
k
kk
kk
k
kk
k
=
11 19
3)2(
13
)(
kk
k
kk
kk
= 2
2 5
120 1
k
k kk k k
kk
k
!( )( )
= k k
k kk
( )
( )( )
3
1 21
J1061
= 12
11
4
2 122
1
k k kk k( ) ( )
J1061, Marsden p. 358
= dx
x
x
xdx
( ) ( )1 140
2
40
=
0
5
0
2
33 xx e
dxx
e
dxx
= dxe
x
e xx
0
20
sinh
1
1
1 .333333333333333333333 =
0
2
)2(4
1
2
1,2
3
4
kk k
k
k
=
122
11
kk
.333550707609087859998...
1 2
1)2(
2
2log3
2csclog
kk k
k
.33357849190965680633... 1
4 11k
k
( )
.33373020883590495023... sinh sin1 1
.333850657748648636047...
6
12 2
( )
.33391221206618026333...
0
1
812
)1(
12
2log
8
1
312 k
k
k
2 .333938385220254754686... ( )k
kk
1
2 11
1 .334074248618567496498... 3 2
4 16
2 2
21
log arctan
Gx
xdx
.334159265358993593595...
10
51
50
1
2520
coth
kk
.33432423314852208404...
84 4 2 4 1 6 2 arcsinh log
= arctan arcsinx xdx0
1
44 .33433659358390136338... 622
2
1
ek
k
k
k
!
2 .33443894740312597477... log( )k
kk
22
1
1 .334568251529384309456... e /2
3 .334577261949681056069... 2 2
.33491542633111789724... 11
21
kk !
.335158712773266366954... 41
41
2
1
40
1
7 221
tan
k kk
.33516839399781105400... e
ee e e
k
e kkk
11
10
log( )( )
1 .335262768854589495875... 6
720 volume of unit sphere in R12
.335348484711510163377...
( )2
21
kk
k
1 .335382914379218353317... 1
2
1
4
1
2
3
4, ,
10 .335425560099940058492... 3
3
.335651189631042415916...
2
11
2
)()1(2log2log
kk
k
k
k
= 1
2 11
21 k kk
log
1 .335705707054747522239... 2
2e
.336000000000000000000 = 42
125 6
2
1
kk
k
.336074461109355209566... 46
75
2 2
5
1
2 51
log
( )k kk
.33613762329112393978... 4 41
1 2
1
21
Gk
k
k
( )
( / )
2 .33613762329112393978...
1
32
1
)4/1(
)1(46
k
k
k
kG
2 .33631317605893329200...
22
2
)()1(
log
kk
kkk
k
.336409322543576932... ( )k
k kLi
kkk k
1
2
1 1
221 1
2
.336472236621212930505... Li12
7
3 .33656672793314242409... dxx
xLiLi
1
0 21
2
3
32
3 )(
log
2
12
3
2log
3
2log)2(2
1 .3366190702415150752... e
e2 1
.33689914015761215217... 14
3 32 2
1
2
2
0
2
log( cos )
sin
/ x
xdx
.336945609670253572675... ( )
( )!/2 1
11
1
2 1 2
2
2k
kk e k
k
k
k
.336971553884269995675... Likk
k5 5
1
1
3
1
3
.3370475079987656871... 3
2 4 2 3 33 1
12
121
log log( )!
( )!( )
k
k kk
2 .337072437550439251351... ( )
( )!!
/2
2 21
1 2
21
2
k
k
e
kk
k
k
.3371172055926770144...
1 21)2(
k
k
kk
.337162848489433839615... 4
8 6 2 CFG D1
.337187715838920906649... k
k kk
1
2
.33719110708995486687... 6
25 5
1
2 1
1 12
12
1
( ) ( )!( )!
( )!
k
k
k k k
k
=( ) ( )!( )!
( )!
1
2 2 1
1 12
12
1
k
k
k k
k
.337264799344406915731...
2 2
3
3
21
3
2 22
sinkk
.337403922900968134662... )1(Re)2()!2(
)1()1(
1
Eikk
cik
k
AS 5.2.16
1 .337588512033449268883... 2
0
1
48 4
2
2
loglogarccot x xdx
.337610257315336386152... dxx
xxx
0
2/9
2)cos(sin
21
4
.3376952469966115444... 2 11
2 1 21
log/
e
dx
ex
.33787706640934548356...
3
22
2 1
11
log( )
( )
k
k kk
1 .33787706640934548356... ( )
( )( ) log
2
11 1 1
1
1
22
22
k
k kk
k
kk k
= 2log2
1
.33808124740817174589... 2
3 21
3 216
2 21
logk k
dx
x xk
1 .338108062743588005178... ppf k
kk
( )
22
.33868264216941619188... 68 1
51
ee
x
dx
xcosh
.338696887338465894560... 1
1 20( ) !e
B k
kk
k
.33876648328794339088... 18
241
12
0
1
x
xxdxlog log
=log( ) logx x
xdx
13
1
8 .338916006863867880217... 12
2 3 14 3 1
1
22
2
2
32
( ) ( )
( )k k
k k
k kk k
.3390469475765505037...
2 2
8
3
2 1
2
0
2
x dx
ex
.339369340124745379664...
22
1)1(
)2()(
k k
kk
.339395839527289266398... ( )
log2 1
4
11
1
412
1
k
k j jkk j
.339530545262710049640... 16
15
2
1 2
/
.339732142857142857143... 761
2240
1
1625
kk
.339785228557380654420...
1
2
!2
23
8 k k
kke
1 .339836732801696602532... 2
31 2
1 3
21 2
1 3
2
2 31 3
1 32 3
1 3
//
//
/( ) ( )
i i
2
32 2
2 32 3
//
= 4 3 14
4321
k
kk
kk
( ) )
.339836909454121937096... arccsc3
.339889822998532907346... ( )
!( )!
k
k k kI
k kk k
1
1
12
1 1
21
2
6 .340096668892171638830... 2
1
( )
!
k
kk
.340211994871313585963...
1
2
222
6
log)())2((12)2(
36
k k
kk
.340331186244460642029... 2
29
.340430601039857489999... 1
9
2
3
4
271
1
9
4
3
4 2
9 2
1
3 11
21 2
21
( ) ( ) ( )
( )
g
kk
=
13
2 1
log
3
)()1(dx
x
xkk
kk
.340505841530123945275...
08
24
1
)1(222
4dx
x
xx
.340530528544255268456... ( )
!( )
k
k kk
1
12
.340791130856250752478... Likk
k4 4
1
1
3
1
3
.34084505690810466423...
1
2
1 1
1
1 12
1
( ) ( )! ( )
( )!
k k
k
k
k
1 .340916071192457653485... 1
21 F kkk
1 .340999646796787694775... 1
3
5 3
27
12
11
k
kk
.34110890264882949616... 27
1 2
1
1 12
1
( ) ( )!
( )!
k k
k
k
k
.341173496088356300777... 9
16
1
2
4
0e
k
k
k
kk
( )
!
.34121287131515428207... ( ) logk kk
1 11
1 .341487257250917179757... 5
2
15 2
1
kk
/
.34153172745006491290... 1
2
1 3 3
26 31
1
1k k
k
k
k
kk
( ) ( )
4 .341607527349605956178... 6
1 .341640786499873817846... 3
5
.34176364520545358529...
3/13/4
3/23/23/1
2
1132
2
)2(2)33(2)1(
36
1 i
3/13/2
2
12)3(32
36
33 i
=1
2
1 4 2
26 21
1
1k k
k
k
k
kk
( ) ( )
.34201401950591179357...
1 1
12
2
)1(
)1(
)1(22log
12 k k
kk
kk
kk
H
kk
H
80 .34214769275067096371... 43 13
0
ek
k
k
k
( )
! GR 1.212
.342192383062178516199... log ( ) k
kk
2
1 .342464048323863395382...
0
)1(k
kJ
2 .34253190155708315506... e e
2 .342593808906333507124... k k k
kkk k
( )
( )
3 1
2
2
2 11
4
3 21
.3426927443728117484... ( )4 1
4
1
415
1
k
k kkk k
=
1
41
1
21
21
1
21
2
i i
52 .34277778455352018115... 945
32
11
2
.34294744981683147519... 1
02
323 arctan
64
)3(105
32
2log3
644
3dx
x
xG
4/
02
3
)(sin
dxx
x
.343120541319968189938... 8 2 3 41
4 5 31
log ( )
k kk
2 .343145750507619804793... 8 4 2
1 .34327803741968934237... 3152
2796 3
1
2
1 40
1
( )
log/
x
xdx
.34337796155642703283... sin
( )
x
xdx
1 20
.3433876819728560451...
0
1
12
2log
4
1
312412
)1(
k
k
k
.343476316712196629069... dxx
xx
kk
e
k
k
1
3
0
logcoslog2
)2()!2(
)1(1sin101cos612
.343603799420667640923... 1 1
10
e
e
e
k
e k k
k
( )( )!
1 .3436734331817690185... 1
2 311
kk
24 .3440214084656728122... 2 4
2
7
12012 2
9 3
2 log
( )
=
log log1
1
0
1
3
xxdx
16 .344223744208892043616... k k
kk
3
2
( )
!
.34460765191237177512... 2
3 2118
11
54
1
3
k kk
.34461519439543107372... H k
kk
( )3
1 4
2 .34468400875238710966... 1
4
1
2 10
e erfie e
erfi ek
k kk
cosh
!( )
.344684541646987370476... 265 72061
ek
k kk
!( )
.344740160430584717432... 2
12
2
20
2
cos
cos
x
xdx
5 .3447966605779755671...
csc5
1
5
4
5
= log 1 5
0
x dx
.34509711176078574369...
4 1 40
2
exe x dxx
/ sin
.3451605044614433513... 9 4
11900001
2564
5
sinh
kk
.34544332266900905601... sin cos
( )( )!
1 1
41
2
2
1
k
k
k
k
1 .345452103219624168597... 1
2 22 1 2
0
e ek
ke e
kk
/ / sinh
!
.345471517734513754249... ( ( ) ) log k kk
1 2
2
.345506031655163187271...
1
2 2 2 2
2 1
2
1
2 112
2
cot
( )k
kkk k
1 .345506031655163187271... 1
2 2 2 2
2
2
1
2 112
1
cot
( )k
kkk k
1 .3455732635381379259...
1
1)2()1( dxxxx
.345654901949164100392...
logcos
sin
arcsin/
2 21 10
2
0
1
G
x x
xdx
x
xdx
= K xdx
x( )
20
1
GR 6.142
.345729840840857670674... 1
32
1k
k
.3457458387231644802... 1
2
2
2
1
1
2 10
20
I
e k
k
k
k
k
( ) ( )
( )!
.345814317225052656510... 1
21
21
21
1eEi
H
kk k
kk
log ( )!
.346101661375621190852... 2
1
3
1
3 2 12 10
erfik kk
k
! ( )
1 .34610229327379490401... 2
0
1 x
xdx
1 .346217984368230168984... ( ) sin ( ) ( ) sin ( )/ / / /1 1 1 13 4 1 4 1 4 3 4
.346250624110635744578... 5
5
2
1
5
1
5 521
tan
k kk
.346494734701802213346...
222
log)(
)2(
)2(
k k
kk
.34654697521433430672...
0 2111 )!(
)1(1,
2
5,2
3
41
2
31
k
k
k
kFerfi
e
.34657359027997265471... logRe log( ) Re ( )
2
21 1 i Li i
=
0
12 )12(3
1
22
1arcsinh
3
1arctanh
kk k
AS 4.5.64, J941
=
111 2
1
2
1
2 kk
kk
kE
LiH
= ( ) cos /
1
2 2
2
0 1
k
k kk
k
k
=
( )( ) log
2 1
21
1
21
1
1 12
2
k
kk
kk k k
= dx
x x
dx
x x( )( )
1 313
1
= x
x x xdx
1
1 120
1
( )( ) log GR 4.267.4
=
002
112xx
e
dxx
e
dx
=
4/
0
4/
0 sincos
sincostan
dxxx
xxdxx
= ( )cos
10
ex
xdxx
= ( ) sin sinx x x dx30
1
GR 6.496.2
= x x
xdx
2
20
sinh
cosh
GR 3.527.14
=
0
22
sinsin43
dxxx
x Prud. 2.5.29.25
.346645900013298547510...
07
23
1
)1(
14
3cos
14cos
7
4dx
x
xx
.34700906595089537284... ( )
11
13
1
k
k
k
k
.34720038895629346817... 2 2 2 2 22 2
2
1
log log( )
H
kk
kk
1 .3472537527357506922... Likk
k
1 2
1
1
2 2/
.347296355333860697703... 218
2 2 2 2sin ...
[Ramanujan] Berndt Ch. 22
240 .34729839382610925755... 6 5
04 1 1
log
( )( )
x
x xdx GR 4.264.3
.347312547755034762174... 24 5 12 32
5 1
4 4
30
( ) ( )( )
x
ex
.347376444584916568018... 32 2131
6
1
2 50
log( )
kk k
.347403595868883758064... erf1
.347413148261512801365... ( )
logk
kk
k
1
2
2 .347560059129698730937...
1
1)1(2
k
k
kk
.34760282551739384823... 4
3 3 2
1
2 2
2
20
2
cos
( sin )
/ x
xdx
.347654672476249243683...
2
32 2 1 2
1
2
3
2
1 31 3 1
2 3 2 3
// / /3
/ /( ) i
2
31 2
1
2
3
2
1 32
2 3 2 3
// /3
/ /( ) i
= ( ) ( )
1 2 3 12
21
13
2
k k
k k
kk
.3477110463168825593...
1
1)1()(k
kk
1 .348098986099992181542... 3
23
.348300583743980317282... 2
3 902 3 16 2 16
1 4
2
2 4 1
1
( ) log( ) ( )k
kk
k
= 1
2 5 41 k kk
1 .348383106634907167508... iilog2
22
7 .3484692283495342946... 54 3 6
7 .348585884767866802762... 1
33
1
2
3
21
cot tank kk
Berndt ch. 31
.348827861154840084214... Likk
k3 3
1
1
3
1
3
.34906585039886591538... 9 1
7 2
90
x dx
x
/
.3497621315252674525... 42
3 21
4
1 1
421
1
1
log
( ) ( )
k k
k
k
k
kk
= hg1
4
5
4
= 1
0
4 )1log( dxx
.349785835986326773311... primepp 12
1
1 .3498073850089500695... coth21
21 1 2 1 2 i i
= 4 1
41 4 2 2 13
1
1
1
( )( ) ( ) ( )
k
k kk k
k
k
k
k
.349896163880747238828... 1
2
5
3
1
3
2
31
1
3
2
30 1
3
0
1
, , , e dxx
.34994687584786875475... k
kk
k2
2 11
( )
.350000000000000000000 = 20
7
.350183865439569608867... 11
220
k
k
.35018828771389671088...
4
1
2
1
2 1
2
20
4
arctancos
cos
/ x
xdx
.35024515950787342130... | ( )| 2
4 11
kk
k
.35024690611433312967...
6 3
2
3
3
6 131
2
log log xdx
x
.350402387287602913765... 2 1 2 21
2 11
sinh( )!
kk
2 .350402387287602913765... ee kk
12 1 2
1
2 11
sinh( )!
= e x dxxcos sin0
GR 3.915.1
1 .350643881047675502520... E1
2
141 .350655079870352238735... 52 42 15
1
2
1
ek
k
k
kk k
!
( )
! Berndt 2.9.7
.350836042055995861835...
2 1 31 3 2 3
2 3 2 3
6
2
61 2 1
4 2 2 3
4
/
/ // /
( )i
( ) /
/ / /
12
6
4 2 2 3
41 3
1 3 2 3 2 3
i
= ( )( )
13 2
2
1
21
5 211
kk
kk
k
k k
1 .3510284515797971427...
( )( )
( )
3
2
3
42 1 1
4 3 1
12
2
4 2
2 32
k kk k
k kk k
.351176889972281431035... 2 2
24
2
8
log Berndt 9.8
15 .351214888072621297967... ( ) ( ) ( ) ( ) ( )2 6 4 6 3 1 13
2
k kk
= k k
kk
2
42
4 1
1
( )
.351240736552036319658...
4 5 4 520
dx
x
.351278729299871853151... 21 20
ek
k kk ( )!
4 .351286269561060410962... K K kdk
k2
0
12
2
( ) GR 6.143
.35130998899467984127... 1
2
1 4 1
251
1
1k k
k
k
k
kk
( ) ( )
.35144720166935875333... dxx
x
1
0
22
1
arcsin
2
3216
.351760582118184854439... 4 2 3 1 3 121
e e eEi EikH
kk
k
( ) ( )( )!
.351867223826221510414... ( )
logk
k k kkkk
1 1
2
11
1
221
.351945726336114600425...
1 !k
k
k
E
1 .351958911849046348014... 39 3
4
81
2
1
1 92
2 21
( / )kk
2 .352009605856259578188... e2
.3520561937363814016... 1
4 1 422
11 ( )( )
k kkk k
k
1 .352081566997835463...
0 )32)(12(4
1
2
33log2
kk kk
.352162954741546234686... 1
3
1
3
1
3 2 11
sinh( )!
kk k
.35225012976588843278... 22
11 2e i isiin ei ei i
( ) / cos cos ( )
24 .352272758500609309110... 4
4
1 .352314016054543571075...
0
2
3
cosh8
)3(9dx
x
x
.352416958458576171524... 1
33
12
13SinhIntegral
x
dx
x( ) sinh
.35248587146747709353... 2
28
.352513421777618997471... arccot e ee k
k
kk
2
1
2 12 10
arctan( )
( )
= dx
e ex x
1
1 .352592294195119116937... 1
22 2 1
2
12
20
e Eik
k k
k
k
( ) log!( )
.35263828675117426574...
1
02)1(
)2()1(2
11
xe
dxEiEie
e x
2 .352645259278140528155... 54 193
3
1
ek
k kk !( )
2 .352694261688669518514...
2 1
2 312
6 321
1 4
( )
( / )
k
k k
.352809282429438776716... ( ) ( )
( )
1
2 12
k
kk
k
k
.35283402861563771915... Jk k
k
k
k
k
k
k2
0
3
20
21
21( )
( )
!( )!( )
( !)
LY 6.117
2 .35289835619154339918... ( )k
k kLi
kk k
1 1 12
12
1
1 .352904042138922739395...
22
3172
5 4
4
1
25 4 2
( )( ) ( )
H
kk
k
Berndt 9.9.5
.3529411764705882 = 6
17
.35323671854995984544... 1 1 11
2
p k
kp prime
probability that two large integer matrices have relatively prime determinants Vardi 174
.35326483790071991429... 2
1 100
1
( ( ) cos ) sin arcsinJ x xdx
.3533316443238137931... ( )
12 1
1 2
1
k
kk
k
.35349680070142205547... x dxx1
0
1/
1 .353540950076501665721... ( ) ( )3 2
.353546048172969039851... cos sin sin1 1
2 1e e
x x
edxx
.35355339059327376220... 8
sin8
3sin
8
8
4
2
.353658182777093571954...
4
1
2 6 2
12
4 21
4 21
cothk k
dx
x xk
.35382477486069457290...
1 5
)2(
5csc
5log
kk k
k
.353939237813914641441... log( ) ( )!( )
e Lie
Lie
B
k kk
k
1 1 21
21
2 322 3
0
= x dx
ex
2
0
1
1
1 .354117939426400416945... 2
3
.35417288226859797042... ( )4
4
1
4 114
1
k
kkk k
= 1
2 4 2 2 2
cot coth
.354208311324197481258... ( )( )
!
12
k
k
k
k k
.354248688935409409498... 3 7
.35429350648514965503... 3/23/23/23/13/1
212)2(1312)1(131212
1
ii
=
13 4
1
k k
.354880888192781965553...
1 )12(3
1
3
2log
3
1arctanh
3
32
kk kk
.35496472955084993189... 11 1
2
2 1
40
1
1
e
Ik
k k
k
kk
( )
!
.35497979441174721069... 2
2 12
1 2 0
1
log arctan arccosx xdx
.355065933151773563528... ii eLieLi 22
22)2(2
=
1
221
2 )12(2
14
)1(
1
kk kk
k
kk
=
212
23)1()()1(1)2(
1
k
k
kk
kkkkk
= ( ) ( )( ) ( )
1 21 1
21
1 2
k
k
k
kk
k kk k
= log log( )x x dx10
1
GR 4.221.1
.355137311061467219091... 7
1920
4 3
31
log x
x xdx
4 .355172180607204261001... 2 2 log
= xdx
ex
10
GR 3.452.1
=
02
2
1
)9log(dx
x
x
= x xdx
x
sin
cos10
GR 3.791.10
=
2
0 2sinlog dx
x
= log ( cos )2
0
1 x dx
GR 4.224.12
= log (cos )/
2 2
0
2
x dx
GR 4.226.1
1 .3551733511720076359... 5 3 1
6 2 1 12 50
dx
x /
.355270114150599825857... 3
2
2 1
11
11
1
22
2
log( )
log k
kk
kk k
.35527011687405802983... ( ) ( ) tanh3 2
3
3
2
15 4 3
1
k k kk
.3553358665526914273...
1
2
)!2(
)2()!(
k k
kk
.355379722375070811280... 1
3
2
3
3
2
1 4
32 2
1 30
2 3 1 3
e ek
k k
k
/ /
cos( )
( )!/
.3557217094508993824... )3(
)6()3(
.35577115871451113665... 1
1342 kk
1 .35590967386347938035... 11
41
kk
.3560959957811571219... coth kk
11
2 .356194490192344928847... 2arctan27
1arctan)1(arccot
4
3
=
12
2arctan
k k K Ex. 102
[Ramanujan] Berndt Ch. 2
=dx
x x20 2 2
.356207187108022176514... log7 2
.356344287479823877805... 4 2
1
21
2
4
1
4 11
211
coth ( )( )
kk
kk
k
k
.356395048612456272216...
112
228
1,
2
3,2,2
4
1
k k
k
kk
F
.356413814402934681864... ( )k
kk 41
.3565870639063299275...
161
32
21
3
2)1(1
)()(
)(
k
k
J1028
.356870185443643475892... 4
3
1
4 4
1
3
1
4 12
2
12
0
LiH
kk
kk
kk
( )
( )
.356892371084866755384... 2
1 )(x
dx
.3571428571428571428 = 5
14
.35738497664673044651...
0 12
1
3
123xe
dxx
1 .3574072890240502108... kk
k
2
1 3 1
. 35776388450028050232...
0
2
)32)(12(16
12
4
16
kk kkk
kG
6 .3578307026347737355... 3 3
224 2
27 3
2
log
log
=1
1 12
130 ( )( )( )k k kk
.357907384065669296994... 1 11
2
1
21
cot tank kk
Berndt ch. 31
= csc1
22
0k
k
k
.358145824525628472970... 36195
2 3
4
0
ek
kk
( )!
.358187786013244017743... 2
21
1
2 21
sin
kk
1 .358212161001078455012...
0
2 )tan2sin(1sinh
12 x
dxx
ee
=
0
2/
0
cot)tan2sin(cos)tan2sin(
dxxxx
dxxx
1439 .358491362377469674739... 61
128
7 6
0
x
xdx
cosh GR 3.523.9
4 .358830714150528961342... 3 32
( )k
k
.358832038292189299797... 2
1
2
8
( )kk
k
4 .358898943540673552237... 19
.358987373392412508032...
3/23/23/1
3/1212
2
11312)1(131
212
1 ii
=
13 14k k
k
5 .35905348274329944827...
1
21
24
)3(26360
17
k
kk
k
HH
1 .3590982771354826464...
00 xe
dxx
xe
dxxx
.359140914229522617680... e k
k kk21
2 41
!( )
1 .359140914229522617680... e k
k
k k
kk k2 2 1
1
21 1
( )!
( )
!
= dxex x
0
1 2
=
1
1 30 e x
dxx ( )
= log
( log )
x
xdx
e
1 21 GR 4.212.7
.359201008345368281651... 3 3 3 4 ( ) ( )
2 .359730492414696887578... 3 4/
3 .35988566624317755317...
11 )(
11
kkk
k kF
.359906901116773247398... ( )
!/k
ke
kk
kk
11
11
22
.359946660398295076602... arccot ( cos )cscsin
2 2 22
21
k
kkk
1 .360000000000000000000 =
1
31
2)1(
25
34
kk
kk kF
.360164879994378648165...
14
2
1 144
)24()1(
2tanh
8 kkk
k
k
kk
1 .360349523175663387946...
12 9/1)12(
1
4
3
k k
GR 1.421
1 .36037328719804788553...
1 )9(
11
2
77cos
4199
51840
k kk
.360553929977493959557... 1 2 3 4 ( ) ( ) ( )
.360816041724199458377...
0 )2(2!
)1(64
kk
k
kke
.361028100573727922821...
12 64
1
4
32coth
22 k kk
1 .361028100573727922821... 1
4 2 22
1
2 321
coth
k kk
.361111111111111111111 =
12 45
1
36
13
k kk
1 .361111111111111111111 = 3)2(
36
49H
.361328616888222584697...
e Eidx
e xx2
0
22
( )( )
.361367123906707805589... 22
1arcsin
.361449081708275819112...
1
32 84
1
1621
coth
kk
J124
.361594731322498428488...
1
2
)1(32
3log
2
3log
2
3
kk
k
k
kH
5 .3616211867937175442...
3 2 2 2 2 3
2 4
6 2
4
3 2
4
2
23
( ) log log log( )
=
log3
20
xdx
e x
441 .361656210365328128367... 162 11
7
1
ek
kk
( )!
1 .3618358603536914...
2 2 1
log
k nnk
k
6 .3618456410625559136...
0
3
)!1(
3
3
1
k
k
k
e
.3618811716413163894... 1
12 2 2 1 2
42 3 2 3 1 3 2 3
3
32
( ) ( )/ / / /
k
kk
.3619047619047619047 = )4)(1(4
12
105
38
0
kkk
kk
k
.3620074223642897908...
1 )12(
sin
k k
k
.36204337091135813481...
1
3
13
)3/1(
)1(
4
)3(39
36
527
k
k
k
.362131615407560310248... ( )( )
( )
12
2
1
4
112
12 2
1
k
k k
k
kLi
k
.362161639609789904943... Ik k
k0 2
1
2
31
1
3
( !)
.36219564772174798450...
1 )12(2
11
kk
.3623062223664980488...
11
1
)(
))((
)(
)1(
kk
k
kp
kp
kp
, p(k) = product of the first k primes
.362360655593222140672... 1
22k
k k( )
.362389718306640099275...
1
)2(12
)1(
)1(
6
2log
4
)3(5
k
kk
kk
H
.362532425370295364014... 1
31 ( )k kk
.3625370065384367141... 29
15 21
4
2 1 1
1612 1
( )
( ) ( )kk
kk
k
k k
.362648111766062933408... erfk k
k
kk
1
3
2 1
3 2 12 10
( )
! ( )
.3627598728468435701... 35
.3632479734471902766... dxx
x
1
0
22
2
)2log()2log3(log
2
1 GR 4.791.6
2 .363271801207354703064... 4
3
3
2
.363297732806006305...
242
91
24
)3(sinh)3(sin
k
.363340818117227199895...
1
01 log
1log
)1log(
x
dx
xe
e
ke
k
kk
GR 4.221.3
.363380227632418656924... 12 2 1
2
1
2 1
2
1
( )!!
( )!!
k
k kk
J385
= ( )( )!!
( )!!( )
12 1
24 11
3
1
k
k
k
kk
.36363636363636363636 = 4
11
.363715310613311323433... 1
12 k kk ! !
1 .363771665261829837317... 1
21 Fkk
.36380151974304965526... 2
23
1
7 3
41 1
2
4
2 1
( )( ) ( )
( )
( )k
kk k
k kk k
k
.363843197059608668147...
1
33/1
3!1(9
173
kkk
ke
.363975655751422539510...
234
234
1
2
32log)2()3(
xx
dx
kkk
.363985472508933418525...
01
2
1
1
sin
1
)1(
sinh22
1dx
e
x
k xk
k
sin
cosh
2
20
x
xdx
.364021410879050636089...
12 26
17cot
7214
1
k kk
1 .3641737363436514341...
0 2
22
)1(2
2
48
2log
16
)3(35
k
k
kk
kG
.364216944689173976226...
0 )1(2k
k
kk
k
H
7 .364308272367257256373...
1 )!1(
11
k k
2 .364453892805209284597...
0 )12(!
22
22
1
k
k
kkerfi
2 .364608154370872703765...
1 )!2(
42log)2(2
k
k
kkalCoshIntegr
.36481857726926091499... 12
2 21 21
1
log( ) ( )
k
k
k
.36491612259500035018... 3
3 6 7 CFG A22
2 .36506210827596578895... ee xdxx
4
2 2 1
0
2
log log
3 .36513890066171554308...
0
2 4/1
)1(
2csch2
k
k
k
.3651990418090313606... 31
3 1
1
3
1
332
1
1
( )( )
kk
=( ) ( )
1 1
3
1
1
k
kk
k k
1 .365272118625441551877...
1 1
k
k
2 .365272118625441551877... 1
1
.365381484700719249363... 2
)3(3
.365540903744050319216...
2
31
2
31
3
1
27 22
2 iLi
iLi
26 .36557382045216025321... 64
9
8
27
64
272
31 2
22
0
log ( )/x Li x dx
.365673715554748436672...
9
7
9
4]2&,1#1#1528323534848[
6561
1 )3()3(34 Root
= dxxx
x
133
3
1
log
.365746230813041821667...
22
11log
11
1
)1(
1)(
kk kkkkk
k
1 .365746230813041821667...
22 22 )1(
log)(
)1(
)(
kk jj
k kk
k
k
k
kk
k
=
2
11log
11
11
k kkk
.36586384076252029575... k
k
kk
2log
1
1
22
1 .365958772478076070095...
11
1 11
2)12()!1(
)2()1(
kk
k
kerf
kkk
k
2 .366025403784438646764... 13
3
= cos sin 6 6
= 1 12
6 31
( )k
k k J1029
.36602661434399298363...
( )3 1
2 13 11
kk
k
.366059513169529355217... 1
4
1 1
1 1
3
2 962 1
3 4
3 4 23 4
0
4
log( )
( )( ) sec
/
//
/
G i
iLi x xdx
.366204096222703230465... log 3
3 41
HOk
kk
.366210241779537772398...
036/13
3/1
63
2
x
dx
.3662132299770634876...
3
23log3log2log)2(
3
12
22 LiLi
=
1
1
12 2
)1(
3
1
kk
kk
kk k
H
k
.36633734902506315584... 3 22 4
7
9
8 1
4 1
2
22
log( )
Gk
k kk
= ( ) ( )k k
kk
1 1 1
41
.36651292058166432701... log log 2
.366519142918809211154...
dxx
xx
0
6
4)cossin
60
7
.366563474449934445032... 2 1 1 1 2 1371
97202 3
42 3
41 3
42 3
4
( ) ( ) ( ) ( )/ / / / Li Li Li
= 1
1296
1
3
5
6 13 3
3
31
( ) ( ) log
xdx
x
8 .3666002653407554798... 70
138 .36669480800628662109... 1160703963
8388608 9
9
1
kk
k
3 .36670337058187066843... ( ) ( )3 2 4
.36685027506808491368... 2
2
0
1
16
1
4 x xdxarcsin
2 .366904589024876561879...
00 cosh12
)1(2
4
3,
2
1
4
1,
2
1
xx
dx
k
k
4 .366976254815624405817... 4
G
.367011324983578937117...
134
1
12
2
1
2
)3()1()3(4log4
38
kkk
k
kk
k
.367042715537311626967... cos ( )( ) ( )
2
31
32
323
2
1
4
k
kLi e Li e
k
i i
.367156981956213712584...
1
1
12
)()1(
12
)2(2
kk
kk
kk
.367525688683979145707... k
k ks ds
k
1
132 2
3
log( )
1 .367630801985022350791... ( )k
kk 21
.367879441171442321596...
00
/2
)!2(
)1(
!
)1()1,1(
1
k
k
k
ki
kki
e
= ( )
!
( )
!
1 13
0
4
0
k
k
k
k
k
k
k
k
= 1
2 1 2 30 ( )!( )k kk
= ( )k
ekk
11
=
1 )1,(1
1
k kStirlingS
= k k e
k e kk
( )
( )( )
1 11
J1061
= dxex
x
e
dxx
02
1 )(
log
= 3
1
1
0
1sinhsinh
x
dx
xdxxx
1 .368056078023647174276... 2
22
20
1
34 2
1
log
log( )x
xdx
.368120414067783285973... ( ) log ( )
11
k
k
k
1 .3682988720085906790... sinh
( )! ( )!
2
2 2 2
2
2 1
2
2
2 2
0 1
e e
k
k
k
k
k
k
k
GR1.411.2
=
122
21
k k
.368325534380705489...
22
)1(
2
3sech
33
2log
k
k
kk
.368421052631578947 = 7
19
1 .368432777620205875737...
( )
( )
( )2
3 31
k
kk
Titchmarsh 1.2.12
=
primep pp )1(
11
.36855293158793517174... )1()1(2
1)(Re )2()2()2( iii
=
1
33 )(
1
)(
1
k ikik
.368608550929364974778... 2 2
2116
2
2
2
16
7 3
16 4 2
log log ( )
( )
H
kk
k
.36873782029464990409...
1 3!
1
k
k
kk
.3688266114776901201... 2
1 100
1
( ( )) sin arccos J x xdx
4 .369280326208524790686...
02 3/1
1
3coth
2
3
2
3
k k
.369669299246093688523... 12
2
21
1
21
1
log( )
( )k
k
k
k
= 1
21
11
1 kk
kk
log
.369878743412848974927...
3 3
4
33
2
31
4
9 622
2
log ( )k
k k
kk k
12 .369885416851899866561...
1
442
1)1(1
)3(4156 k
kk
k
2 .36998127831969604622...
0 2 ),2(
!
k kkS
k
.37024024484653052058...
6 2 1 1
2
120
8
120
x dx
x
x dx
x
.370408163265306122449... 363
980 7
1
77
21
H
k kk
.3704211756339267985...
0
31
331 3
1
)3(
1
13
)3(
kkkk kk
k
.3705093754425278059... dxx
x
kkk
1
03
6
12
)1log(
26
1
344
3log3
2 .370579559337007635161...
2/
0
224
cot)tan2(sin1
34
dxxx
e GR 3.716.11
3 .370580154832343626096... 2
1
k
kk k
1 .37066764204583085723... 11
1 51
( )( )k kk
3 .370690303601868129363...
22 1
11
2
5sec
k kk
.3708147119305699581... 5
21 1
1
4 120
log( )x
dx
2 .370879845350002036348...
1 )!2(
33sinh
2
3
k
k
k
k
.37122687271077216295...
1
02
sec2
14dxxx
G
.3713340155290132347...
1
22
2
1)2(2
2csc
82
2sin1
kk
kk
=
222
2
)12(
2
k k
k
2 .3713340155290132347... 12
2 8 2
2
2
22
1
sincsc
( ) k kk
k
= 2
2 1
2
2 21
k
kk ( )
4 .37151490659152822551...
1
22
)1(2
1)3(2
4 k
kk
kk
HH
.37156907160131848243... GH
k
kk
k
log ( ) /2
4
1
2 1
11 2
0
Adamchik (25)
=
12 )12(4
)2()14(
kk
k
k
k
= dxx
x
1
021
)2log( GR 4.295.13
= dxx
x
1
02
2
1
)1log( GR 4.295.13
=
0 cosh
)2/cosh(logdx
x
x GR 4.375.1
.3716423345722667425... 1 2 215 2
6
2 2
3
3
2
2 3
( ) log
log log ( )
= log ( )
( )
2
2 20
1 1
1
x
x x
.371905215523735972523... 1
44 2 1 4 1 2 22 (cos ) log( sin ) ( ) sin
=cos
( )
2
1 1
k
k kk
.37216357638560161556...
1
1
2
)1(
55
log4
5
12arccsch
55
4
5
1
k
k
k
k
.3721849587240681867...
12 )14(!
1)1(
421
k kkerfi
e
.372348858075476199725...
1
1
12
)()1(
kk
k k
.37236279102931656488... ( ) sin logcsc log sin 2 1 1 1 2 22
=
sin( )
( )
2 2
11
k
k kk
GR 1.444.1
.37252473265828081261...
3/1
3/2
3/1
3/1
2
)1(
2
1
2
1
3
1HHH
=1
2
1 3 1
241
1
1k k
k
k
k
kk
( ) ( )
1 .372583002030479219173...
0 )1(8
12221624
kk kk
k
.372720300666013047425... 1
41 12 2 1 1 coth ( ) ( )( ) ( ) csch i i i i
= ( ) ( ) ( )( )
( )
1 2 2 11
11
12 2
2
k
k k
k k kk k
k
16 .372976195781925169357...
0
22
3
12log4
3 xe
dxx
GR 3.452.2
.3731854421538599476... 1
4 1 41
11 ( )( )
k kkk k
k
1 .37328090766210649751...
.37350544572552464986... 2
1
2
)1(coth
2
)1(cot
4
1
2 4/34/34/3
iii
=1
2 1
1 4
241
1
1k
k
k
k
kk
( ) ( )
.37355072789142418039... 16
1
46
1
3
2log
33 1
kkk
J80
= ( )
1
3 20
k
k k K ex. 113
= dx
x
xdx
x
dx
e ex x31
30
1
201 1
.373663116966915687102...
6
56
3
23
3
16
6
13448
186624
1 )3()3()3()3(4
=
133
3logdx
xx
x
.37382974072903473044... 20
9
8 2
3
2 1
4 2 20
log
( )
k
k kkk
.37395581361920228805...
primep pp )1(
11 , Artin’s constant
.37404368238615126287...
( ) ( )log3
3
31
3k
kk
3 .37404736674013853601...
2/
0
22
cos12log
44
dx
x
xG GR 3.791.5
.37411695125501623006... 4 2 2
61
122
0
1
loglogx
xdx
=
14
2 )1log(dx
x
x
.374125298257498094399... ( )
( )
1
2 2 1
1
1
k
k kk
k
.374166299392567482778...
0 )4(2!
15896
kk kk
e
.37424376965420048658...
02
)1(
)(
1)(
k k
1 .37427001649629550601... 1
1
1 1 1
23 20
2 2 3 122k k k k k k kk kk
= ii i i i
1
41( ) ( ) ( )
= 1
2 4 2
1
4
coth ( ) ( )i i
.374308622850989741676... ( )k
kk
1 1
121
.374487024111121494658... 12 e
.37450901996253823880... cos log1 2
.374693449441410693607...
1 411
16log
kkk
k
L
.374751328450838658176...
12 1 )12(
)(
2
1
12
)(
)(2
1
kk kk kk
k
k
k
k
.37480222743935863178...
0
/1
!
11
kkk
e
1 .37480222743935863178... /1e
2 .374820823447451897569... dxxx
x
kkk
0
24
2
12 43
22
44
1
7
2
=
122 22 xx
dx
xx
dx
.37490340686486970379... ii ee 222
11
= 1)1()cos()1(1
1
kkk
k
1 .374925956243310054338... 1 2 3 13
01
21
Jk
kk
k
( ) ( )( !)
.375000000000000000000 = 3
82 1 3
0
e k
k
( ) log
=
1 )42(
1
k kk
=
13
3log
x
dxx
1 .3750764222837193865... 2 2 2
404 24
2 2
22
log
loglog xdx
e x
.3751136755745319894... dx
x
220
1
.37517287391604427859... ii eEieEi2
1
=
1 !
cos,0,0
2
1
k
ii
kk
kee
.375984929565308899765... 4
105
1
1
2 2
2 4
p
p pp prime
.376059253341609274676... 1
1
1
2 21k kkk
j kkj
.376148261725412056905...
1
1
42
51125log
2
55log5
515
1
kk
kk
k
FF
.37626599344847702381... Likk
k
1 2
1
1
4 4/
2 .3763277109303385627... 4
2log2
G
1 .376381920471173538207... 10
3tan
5cot
5
205
.3764528129191954316... 2 1 2 2 21 2 1
2
1
1
log log( ) ( )!!
( )!
k
k
k
k k
=
1
1 2
4
)1(
kk
k
k
k
k
.376674047468581174134...
12 54
1
5
6coth
2 k kk
.376700132275158021952...
0 )3()!12(
148
659
k kkee
.376774759859769486606...
1 )12(21arcsinh
2
21
2
1arctanh
2
11
kk k
k
=( ) ( )!!
( )!!
1 2
2 1
1
1
k
k
k
k
1 .37692133029328224931... 1
22 2
0
1
G x xdxlog ( ) sin GR 6.468
.377294934867740533582...
13
2
)2(
1
16
3
242
)3(
k kk
.377540668798145435361... 1
1
1
2
1
2
1
41 1 2
1eek k
k
tanh ( ) /
= 1
2
1
2 1
1
221
( )!k
B kk
J134
1 .37767936423331146049... 2
33 4 1 2
22
2
( ) ( ) ( ) ( )k
k
k k k
= 4 2 1
1
3 2
3 22
k k k
k kk
( )
.37788595863278193929...
22
log1
k k
k
.377964473009227227217... 7
7
.37802461354736377417... e
x xdxe(cos sin )log coslog
1 1 3
23
1
1 .37802461354736377417... e
xdxe(cos sin )coslog
1 1 1
2 1
.378139567567342472088...
01 )2(2
)1(
3
14
3
2log42
kk
k
kk kk
k
.378483910334091972067...
12 35
1
2
13tanh
131
k kk
.378530017124161309882... dxx
xsi
1
3
sin
4)1(1sin1cos2
.378550375764186642361... dxx
xsici
02)1(
cos1cos)1(1sin)1(
2
1cos1
1 .378602612417434327112...
1
000 !!)2(25151
5
1
k
kk
kk
FFJII
.378605489832130816595... ( ) ( ) ( )
1 11
2
k
k
k k
.378696261703666022771...
4
)3(
8
3
6
)2(36)2(7776)2()2(1295
=
1
2
3log)(
k k
kk
1 .378952026600018143708... ...2
1
2
1
2
1
1 212
12
kkkk kk
.379094331700329445471...
0
/1
!
cosh)1(
2
1
k
kee
k
kee
2 .37919532168081267241... 2 23
22
2 1
2 1 22 2
1
log log
H kk
k
.379347816325727458823...
0 1
363
3/5
)1(2log
32
3
k
k
xx
dx
k
.379349218647907817919... 3 3
22 6 2
42
23 243 2
2 2
( )log log log
= arcsin logx x dx3
0
1
.379600169272667639186... 2
26
.379646336926468163793... .log)12(
1
1
k kkk
.379651995080949652998...
10
22
11
12
)!(
)()1(
kk
k
kkJ
k
k
.379879650333370492326...
4/
0
6
sin1
cos
20
1
40
2
32
3 dx
x
x
.37988549304172247537... log( ) ( ) ( )
!
2
1
1 1 2 1
1
e
e
k
k
k k
j
=dx
ex
10
.37989752371128314827... 3
3
0
1
32
3
16 x xdxarcsin
.380770870193601660684... ( ) ( )
1 12
2
k
k
k
1 .380770870193601660684...
2 2
02
)1(
)(1)()1(
k k
k
kk
kk
.38079707797788244406... e
e
B
k
k kk
k
2
2
2 1 22
1
1
2 1
2 2 1
2
( )
( )
( )!
9 .3808315196468591091... 88 2 22
.38102928179240953333...
2
53
10
55
2
53
10
55
2
5tan
51
=
2
22 )1(
11)()1(
kkk
k
kkk
kkF
1 .3810359531144620680... 2/
0
2 coslog)3(72log216
1
dxxx
.381294821444817558571...
2 2
)()(
kk
kk
.38171255654242344132...
0 1
sindx
e
xxx
1 .381713265354352115152... 1
21 ( !!)kk
1 .381773290676036224053...
0
2/
1
2
!
)1(
)!2(
)2()1(1sin1cos
k
k
k
k
kk
k
.38189099837355872866...
log ( )( )
12
12
12
41
1
i i k
kk
kk
= log 11
4 21
kk
.381966011250105151795... 23 5
2
1 2
121
( ) ( )!
( !) ( )
k
k
k
k k
2 .381966011250105151795...
0 2
1134
2
57
k kF GKP p. 302
.382232359785690548677...
1 54
5log1
16
5
kk
kkH
.382249890174222104596...
22 22
1
22
)1(
kk
kk
k
2 .382380444637740041929... ( )
!/3 1
11
1
1
3k
kk e
k
k
k
.382425345822619425691...
12
1 15
1
5
)2(
5cot
522
1
kkk k
k
.38259785823210634567... dxx
x
14
arcsinh
6
)21log(2
.382626596031170342250... ( )3
.38268343236508977172... 8
3cos
8sin
4
22
.382843018043786287416...
1
21
1
2
12
4
1
14
)()1(
kkk
kk
k
.38317658318419503472... log
( ) ( )2
2
1
8
1 2
2
1 2
2
i ii i
= cos2
0 1
x
edxx
.383180102968505756325... log
1
1
1 k kk
.383261210898144452739... ( ) ( )
1 2 1 11 2
1
k
k
k
.383473719274746544956...
21 !
log
2!
1
k
n
nn k
k
n
.38349878283711983503... 8 2 4 2 161
0
1
Gx x
x
log
log( ) log
.38350690717842253363... ( ) ( )2 2 1 11
k kk
.383576096602374569919...
1 43
4log
3
4
kkkH
.38383838383838383838 = 7
18
.383917497444851630134...
12 16
122cot
24112
61
k kk
.383946831127345788643... 2
2
1
31
1
1
2
( ) ( )
( )
k
k kk
.38396378122983515859...
23 1k
k
k
H
1 .384161678... dxx
2
2 1)(
2 .38423102903137172415... 11
2
2
2 11
21
2 1
2 11 1
kk
k
kk
kk
Q k( )
.384255520734072782182...
12 2
11
k k
.384373946213432915603... dxxx 1
0
)(log)2(sin2
2log
GR 6.443.1
.3843772320348969041... 1
8 2 2
1
2
1
2
i i
= ( )4 1
4 4 114
1
k k
kkk k
.384418702702027416264...
0 )3)(1(3
1
3
2log12
4
21
kk kk
1 .384458393024340358836... dxxx
x
1
021
arctan)21log(
2
GR 4.531.12
.384488876758268690206... 1
3
2
3 6 12 62
2 2
40
4
G x
xdx
logcos
/
1 .384569170237742574585...
0
22
)1)(3(
log
8
3log
8dx
xx
x GR 4.232.3
.384586577443430977920... sin cos log( cos ) ( ) sin1
2
1
22 2 2 1 1
1
22
= sin( )
( )
k
k kk
1
11
GR 1.444.1
.384615384615384615 =5
13
.384654440535487702221... 3
2log
3
2
2
3log3log
2
2log3log2log)3( 2
322
Li
3
2
3
133 LiLi
=
1 1
)2(1
2 2
)1(
3k kk
kk
kk
k
H
k
H
7 .38483656489729528833...
3
2)2(
.38494647276779467738... log log2
4 2 220
x
x
dxx
xx
2/
02sin24
sin2/)(sinarcsin
Borwein-Devlin, p. 59
.3849484951257119020... ek k k kk k
73
13
11 21 1!( )! ( )! !
.384963873744095819376... ( ) ( )/ / / / / /
1
1
44 2 2 3 1
1
44 2 2 32 3 1 3 1 3 1 3 1 3 1 3 i i
1
3 21
1
22 3 2 3/ /
=
1 1 2
33 2
)1(1
14
1
4
)3(
k k k
k
k kk
k
.38499331087225178096... )1(!2
1)1(2
2
1
0
11
kkJ
kk
k
.385061651534227425527...
22 1
1
2
5arctan
3
2
3 xx
2 .385098206176909244267... 3
13
3 .385137501286537721688... 6
.385151911060831655048...
3
1
61 3 1 3
1
331
i ik kk
5 .38516480713450403125... 29
.38533394536693817834... log cos ( )cos
2 1 1 12
1
k
k
k
k
.38542564304175153356... 1)1()sin()1(222 1
1
kkeei
k
kii
.38594825471984105804... /1
.385968416452652362535... arccothee kk
k
1
2
1
2
1
2 12 10
log tanh( )
J945
= dx
e ex x
1
1 .386075195716611750758...
12
1
1 1
2)12()!1(
)24()1(
kk
k
kerf
kk
k
1 .38622692545275801365... 1
21
2
0
e x dxx ( )
.386294361119890618835... 2 2 11
2 1
1
2 2 401
log( ) ( )
k kkk k k
= 1
4 121 ( )k kk
J374, K Ex. 110d
= ( )
( )
1
1
1
1
k
k k k J235, J616
=
1 2kkk
k
L
= ( ) ( )
( )( )k k k
kkk
kk
kk
1 1
2
2 1
41
1 21
= 2
1
11log
x
dx
x
= E kdk
k( )
10
1
GR 6.150.2
1 .386294361119890618835... 2 2 43
4
3
411
log log
Lik
k
kk
=
101
2 2)1(2
1
2
1
kkk
kk
k
H
kkk
=
21
)1()1(2
)1(
kkk
kk
LiLik
=12 1
4 1
2
2 21
k
k kk
( ) J398
= 1 21
8 231
k kk
[Ramanujan] Berndt Ch. 2, 0.1
=
02 4/1)12()12(
1
k kk Prud. 5.1.26.10
=
11 4
12
)!2(
!)!12(
kk
k kk
k
kk
k
= x x x
xdx
n n n
1 1 2 2 1
0
1 2
1
/
GR 3.272.1
=log( )1
21
x
xdx
= log 11
0
1
x
dx
= log( )
11
20
x xdx
3 .386294361119890618835...
1 2
2log22k
kkkH
28 .386294361119890618835... 27 2 22 1
4
1
log( )
k
kkk
953 .386294361119890618835... 925 2 22 1
6
1
log( )
k
kkk
.3863044024175697139... k
kk 4 11
.38631860241332607652...
12
1
kke
.38651064687313937401...
4 4 2
1
2
22
32coth coth csch csch
= ( ) ( ) ( )
1 2 2 21 2
1
k
k
k k k
.386562656027658936145... 1
4 4 5
5
2
1
4 4 421
tan
k kk
.386852807234541586870... log6 2
.386995424210199750135... Lie kk
k3 3
1
1 1
3
.38699560053943557616... 1
0
24/
0
arctantanlog2)3(
8
7
2dx
x
xdxxx
G
.38716135743706271230...
1
2
310
4)!(8
)1(
4
)1(
kkk
kII
3 .3877355319520023164...
22log)1(
1
k kk
.38779290180460559878...
125
1
5
21
24
5log5
55
55log
4
5
k kk
=
112 5
)1(
5
1
kk
k
k
kk
11 .38796388003128288749... 2
34 3
22
22
0
( ) ( )x Li x dx
.38805555247257460868...
1 )!2(
!!1
k k
kk
.38840969994784799075... 2
1arcsinh21log
2
1 22
=22
1
!)!12(
!)!2()1(
0
kk
k
k
k J143
= arcsin x
xdx
1 20
1
=
1
021
arcsinhdx
x
x
.388730126323020031392... 34
15 CFG D1
7 .389056098930650227230... ek
k
k
k
k
k
k
2
0 0
12 1
3
2 1
!
( )
! LY 6.40
961 .389193575304437030219... 6
.3896467926915307996...
123
1
2
4
1
4
)2(
622log12
kkk kk
k
.39027434400620220856...
1 )13(
11
k kk
.39053390439339657395...
0
12 )12(3!
12
3
1
kk kk
erfi
2 .39064... 1
22 ( )kk
.3906934607512251312...
0 1
2
),2(
),2(
k kkS
kkS
2 .39074611467649675415...
0 2
)3(222log8
kk
k
1 .39088575503593145117... log
2
3
.3910062461378671068... 64
22
0
1
arcsin arccosx xdx
.391031136773827639675...
1
2
122
22 sin
)1)()(1(4
)1)(1(
kkii
i
e
k
eeee
eee
.391235129453969820659...
12 544
1tanh
8 k kk
GR 1.422.2, J949
2 .39137103861534867486... 2 3 2
0
2
3
2
3 1 2
log log log
( )( )
x
x x
.391490159033032044663...
1
11
keke kee
Li
.39173373121460048563... sinh
cosh1
3
13 4
1
x
dx
x
.39177... 1
122
k s
ks
.391801000227670935574...
22
1)2(
)()1(
)1(
)(
k
k
k k
k
kk
k
.391892330690243774136...
12
22
2
)1(22log2log4
62
kk k
k
.39190124777049688700...
2
00
22
cosh
cos
sinh
cos
2sech
4
dxx
xdx
x
xx
.39207289814597337134... 16 2 1
22
( )
( )
1 .392081999207926961321... 4
2/3
1 .392096030269083389575...
212 1)12(
2
51
2
51
2
1
kk kFHH
5 .392103950584448229216... 33
.39225871325093557127... e
e x xdxx4
1
0
2
log
.3925986596400406773... = 2 1 11
2 11
erfi Eik k kk
( ) ( )! ( )
218 .39260013257695631244...
1
4
!
44
k
k
k
ke
.392699081698724154808...
0 24
)1(12arctan
8 k
k
k
=
0 )34)(14(
1
k kk
= ( )
( )sin
1
2 1
2 1
2
1
21
k
k k
k J523
= ( ) ( )4 1 2
16
1
4 1
1
16 112
12
1
k
kk k k
k
k k
= 1
4 2 1 2 3
2
0k
k k k
k
k( )( )
=
8
4
8
1
8
6
8
7
2
1
12/)1(
kkkk
kkk
Plouffe
= arctan( )
1
1 2 21
kk
[Ramanujan] Berndt Ch. 2
=1
4 2 211 ( )s r
sr
J1122
= dx
x20 16
=
1 02413
1
04
04 )1(44 x
dxx
xx
dx
x
xdx
x
dx
=
022 xx ee
dx
= dxx
xx
0
2 2sinsin
=
05
222 cossindx
x
xxx
= 1
0
arcsin dxxx GR 4.523.2
=
0
coslog dxxxex x
4 .39283843336600648478...
1
4
8 )()4()4(4
2160 k k
krL
2 .39292255287307281102...
1
12
22 arcsin
28
x
xG
.393096190540936400082...
12
22
0 )!(
2)1()22(2
k
k
k
kJ
.393327464565010863238...
12
2
)12(4
2log
4
)3(7
k
k
k
H
1 .39340203780290253025...
1 )8(
11
)315()154(11
1532
k kk
.393469340287366576396... e
e k
k
kk
1 1
2
1
1
( )
!
2 .39365368240859606764... 2
6
.39365609958233181993...
3 !
1
k kk
1 .39365735212511301102... 2 2 1 2
22
2 22
k
k k
k
kLi
j
( )
1 .393825208413407109555...
1 2
)1(1
k
k
k
kk
.393829290521217143801... )3(34
18 .393972058572116079776...
0 !
8)1(50
k
kk
ke
.39416851186714529353... coth
8
4 .39444915467243876558... 3log4
.39463025213978264327...
1
)3(1
2
)1(
kk
kk
k
H
.39472988584940017414... 1)2(14
3log
1
kk
k
k
.394784176043574344753... 25
2
2 .394833099273404716523...
01 )!1(!
222
2
1
k
k
kkI
.394934066848226436472...
3
2)1(
2 1)3,2(3
4
5
6 k k
= 11
)(2
122
i
eLieLi
= ( ) ( )( ( ) )
1 1 12
k
k
k k
=
2 11
1)1(
j kk
k
j
k Berndt 5.8.4
=
134
logdx
xx
x
= dxx
xx
1
0
2
1
log
1 .39510551694656703221...
3
)(k
k
.395338567367445566052... 11
2
kk !
.39535201510645914871...
( )
( )cos/1
1 20
e
ee xdxx
1 .39544221511549034512... ( ) ( ) ( )( )( )1 1
1
5
8
7
8
3 1 1
82
k
kk
kk
.395599547802009644147...
1 )5(!44120
k kk
ke
1 .395612425086089528628...
0
3/1
3!
1
kkk
e
1 .395872562271002...
2 2 1
1
n knk
.395896037098993835211...
1
22
)!3(
2
8
19
8
3
k
k
k
ke
1 .3959158283010590...
2
12 )2/(11k
k
1 .396208963809216061296... 2/7)2(
2
311025
51664H
.39660156742592607639...
1
21
!
1)1()1(
k
k
k
k
1 .39689069308728222107... k
k
kk
2log
1
12
.39691199727321671968... 2 2 1 12
2 11
1
sin ( )( )!( )
kk
k
k
k k
.39710172090497736873... 1
2 3
2
31
1
3 121
csc sin
kk
J1064
.397116771379659432792... k
kk ( )2 21 1
Prud. 5.1.25.29
.397117694981577169693...
eerf
1
1 .397149809863847372287...
12
1
0 )!(
4)1()4(1
k
kk
kJ
.39728466206792444388... 3 3
4 2
3 2
20
sin x
xdx
.397532220692825881258... dxe
xx
e x
1
2 sin1cos2
.397715726853315103139...
0 )42)(12(
1
3
2log
6
1
k kk
3 .397852285573806544200...
1
2
)!22(4
5
k k
ke
.3980506524443333859... | ( )|( ) k k
kk
1
2
1
1
7116 .39826205293050534643... 4
3 8
.3983926103133159445... 1
0
)tanarctan(arc dxx
.39890683205968325214...
2 6
2
3
1
4
1
3 221
coth
kk
.39894228040143267794... 1
2
.398971548420202857300... 13
2 ( )
.3990209885941838469... 2
2 2 2 2 21
2
20
cos cos ( ) sin( ) ( )sin
( )
si cix
x
.39920527550843900193... 1
4 17
17
2
1
5 221
tan
k kk
.39931600618316563920... e
k k k
k
k
2
0
1
16
2
2 4
!( )( )
.39957205218788780748...
2
53
2
53)2arccsch(5log2
5
12arccsch2 22
2 LiLi
= ( )
12
1
1
k k
k
Hk
kk
.4000000000000000000 = 2
51
21
21
1
1
1
( ) ( )k kk
k
k kk
k
F kF
= 1
2 21 2 2 1
0cosh log( ) ( log )( )
k k
k
e J943
=
primep p
p2
2
1
1
= sin sin cos2
0
2
02
0
x
edx
x
edx
x
edxx x x
2 .400000000000000000000 = 12
5 42
1
F kkk
k
1 .40000426223653766564... 121
1)1(
kk
Hk
.40009541070153168409... log2
.400130076223970451846... e G
.40037967700464134050... e Eik
k k k kk k
1 11
1
221
20
( )!( ) !( )
= e x x dxx log0
1
.400450020151755157328... ( )( )
arctan
12 1
1 1 1
2 1
k
k k
k
k k k k
.400587476159228733968...
1
)2(log)(k
kk
.400685634386531428467... ( ) cos( / )3
3
33
1
k
kk
.400939666449397060221... 4 2
12 2
4 22 2
cos sincosh sinh
=sin2
40 1
xdx
x
.40130634325064638166... 1
62 1
1 22
( )
( )
( )( )
k
k kk
Adamchick-Srivastava 2.22
1 .40147179615651142485... ( ) ( )
!
//k k
k
e
ke
k
kk
k
11
2
11
1
.40148051389327864275...
3
2
1
12 4
)1(
48448
1
24
72log
k
k
k k
k
kk
.40162427311452899489... 912 1
3 21 30
e k k
k
kk
/
( )
! ( )
.40175565098878960367... 5 2
12
3
16 1
5
0
4
cos
sin
/ x
xdx
1 .40176256381563326007... log
1
3
1 .402182105325454261175...
6
1
3
1
6
1
6
5
3
4 1
=
06
1
03 11 x
dxx
x
dx Seaborn p. 168
1 .402203300817018964126... 3
46
23
0
23
0
1
x xdx xdxsin arccos/
7 .402203300817018964126... 3
4
2
.40231643935578326982...
3
4
2
42
1
222
csc
( )k
k k
.40268396295210902116... Li32 34 3
5
4
52
2
3
( )( ) log log
5 1
2
=5
)3(4
2
15log
15
2
2
15log
3
2 22
.402892520174484499083... 1
4 2 2 2 2
1
2
2
4
1
2
2
4
i i i i
= ( )
/
1
1 2
1
21
k
k
k
k
1 .4029920383519025537...
1 !2csc2
51
2
2
51
5
1
k
k
kk
FharcEiEi
3 .40301920828833358675... k
k kk
!
11
.403025476056584458847... ( )
!!
k
kk
1
2
.40306652538538174458...
03
12
1 273
2
27
2
27
12
15
1
39
2
x
dxx
kk
k
k kk
1 .40308551457518902803... e erfk k
k
1 18
0
12
1
3 2
1
3/
!!
6 .40312423743284868649... 41
1 .403128506816742370427... 0 1 0 20
11 2 1
Fe
Ie k ek
k
; ;( !)
1 .403176122350058218797... 2
1
2
2
( )
( )!
k
kk
2 .40324618157446171142... 1
8
1
3 13
2 230
1
log
( )
x
x xdx GR 4.244.1
.403292181069958847737... 98
2431
21
6
1
( )kk
k
k
.4034184004471487... ( )
1 1
3 31
k
k k k
.403520526654739372535... K x
xdx0
2
20
2 2
2 1
( ) log cos
.403652637676805925659...
0 02 )1()1(
)1(1xe
dxx
xe
dxEie
xx
1 .403652637676805925659... 2 11
3
0
eEix dx
e xx( )( )
.403936827882178320576... 1
2 121
k
k
.404063267280861808044... 1
3 1
1
3 11
1
1k
k
k
kk
( )
.4041138063191885708... 2 3 2 22 ( ) ( )( )
= log log
( )
2
3 21
2
0
1 2
01 1
x
x xdx
x x
xdx
x
e ex x
GR 4.26.12
2 .4041138063191885708...
1
2)2( )1()3(2
k
k
k
H
=
1 3
1
22
)1(
k
k
kk
k
=
1
)1( )(
k k
k
=log ( ) log2
0
1 2
0
2
21
1
1
x
xdx
x dx
e
x
x xdxx
=log2
0
1
1
x
x
= dydxxy
xy
1
0
1
0 1
)log(
3 .4041138063191885708... 2 3 19 2 1
1
4 2
2 32
( )( )
k k
k kk
= (( ) ( ) ( )2 1 2 2 12
1
k k kk
15 .40411960755222466832... 82
3 1 1
41
2
1
Gk
kk
kk
( )( )
( )
1 .404129680587576209662...
2
31
6
3
10
cothsin
x
edxx
.404305930665552284361...
5112csc57747190512csc57747190190
1805
2harcarc
=
1
1
2
)1(
k
kk
k
k
F
1 .404362229731386861519... 2 2 1
2 1
1 22 1
1 2
k
k k
k
k k k
( )
( )!sinh
.4045416129644767448...
2 )1(
1)()1(
k
k
k
k
.40455692812372438043...
csc ( )
2
1
2
1
1
1
21
k
k k
2 .404668471503119219718... k
k kk
2
1
.404681828751309385178... 11 1
1
Eie k e kk
k !
.404729481218780527329... 9 182
23
63
6 iLi e Li ei i( )
=
sin63
1
k
kk
GR 1.443.5
.40480806194457133626...
2
1
21 1 1coth i i
= )12()2()1(1
1
1
13
kkkk
k
k
k
k
.404942342299990092492... cosh sinh
( )!
2
2
2
2 2
2
2 11
k
k
k
k
.40496447288461983657... ( )2
.405182276330542237261... cos(sin ) cosh(cos ) sinh(cos )cos
!2 2 2
2
0
k
kk
.4051919148243804... ( )
11
12
51
k
k
k
k
.405226729401104788051... 3 1
4
1
2 1
1 2
1
cos ( )
( )!
k
k
k
k
2 .4052386904826758277... e /2
2
.40528473456935108578... 4
2
.405465108108164381978... log3
2
1
3
1
311
Likk
k
J102, J117
= ( ) ( )
( )
1
2
1
2 2 2
1
1 0
k
kk
k
kkk k
= 21
52
1
5 2 12 10
arctanh
kk k( )
K148
= dx
x x
dx
e
dx
ex x( )( )
log
1 2 1 2 10
3
01
.405577867597361189695...
2 15 3 520
dx
x
.40573159035977926877...
1
)3(
)1(2kk
k
k
H
2 .40579095178568412446... ( )
!
/k
k
e
kk
k
kk
2
2
1 2
210
1 .405869298287780911255... ( )( )
arctan
12
2 1
1 11
1 1
k
k k
k
k k k
.40587121264167682182... 4 3
31240
log x
x xdx
1 .406166544295979230078... ( ) ( )2 3
1 .4063601829858151518...
13
)3(
k k
k
.40638974856794906619... 2 27 3
2 1
2 2
20
1
G
x x
xdx
( ) arccos
2 .40654186013... sum of row sums in 1
k kb a
.406612178247656085127...
05
1
)5(1205/1xe
dx
.40671510196175469192... sinh
sinh2
4
1
22
0
1
xdx
1 .40671510196175469192... sinh
cosh2
4
1
22
0
1
xdx
.40690163428942536808... 5
1
2
1
2 5
3 2
10 1 5
2
2 5 1 5
5 5
20 1 5
5 5
5 5
arccsch arccsch
( ) ( ) (log
1
5 1 55 2 5 1
( )log log( )
=
5
1
10
7
10
1
= ( )
1
5 20
k
k k
2 .40744655479032851471... ( )
!/2
11
11
2k
ke k
kk
1 .40812520284268745491... 1
6 182
1
2 2
40
1
log
log
( )
x
xdx
.40824829046386301637... 6
6
.40825396283062856150... 11
21
kk k
.408333333333333333333 = 49
120 6
1
6
1
96
21
24
H
k k kk k
.40837751064739508288... 1
164
2 2 2
2 2
1
2
7 4 3 4 3 4
3 4 3 4 41
/ / /
/ /
sin( ) sinh( )
cosh( ) cos( )
kk
.408494655426498515867... ( )
1
1
1
41
k
k
k
k
.408590857704773823199... 14 5 e
.40864369354821208259... sinhcosh
1
2 81 1
12 4
1
erfi erfx
dx
x
.408754287348896269033...
1
2
2 )!1(1
6
1
k
k
kk
B
eLi
[Ramanujan] Berndt Ch. 9
=
12
1
kk ke
2 .40901454734936102856... e
e dxx2
1
0
2
97 .409091034002437236440... 4 3 1
290 4
( ) ( )
=
442
)()3)(2)(1(
kk
kkkk
193 .409091034002437236440...
2
196 )3(4
5 .4092560642181742843...
0 12
)3(93/1xe
dx
.40933067363147861703... e
x xdxe(sin cos )log sin log
1 1
23
1
18 .40933386237629256843... e2 Borwein-Devlin p. 35
1 .409376212740900917067... 3
22
.409447924890760405753... ( ) ( )
1 2 11 2
1
k
k
k
1 .409943485869908374119... 2
317
a k
kk
( ) Titchmarsh 1.2.13
.410040836946731018563...
2 1)(
1)(
k k
k
.41019663926527455488...
1
)4(
3kk
k
k
H
1 .410278797207865891794... 21
F
k
k
2 .410491857766297... ( )/
1 11
21
kk
k
e
k
.41059246053628130507... 11
2
1
2 1
2
20
1
arctan
tan cos
cos
x
xdx
.4106780663853243866...
1
1
2!!
)1(
kk
k
k
1 .41068613464244799769...
100 2!
2
)!12(
2!
!)!12(
1
2
1
2 k
k
k
k
k
k
kk
k
k
kerf
e
2 .410686134642447997691...
0 0 )!2(
!)!2(
)!2(
2!
2
1
21
k k
k
k
k
k
kerf
e
.410781290502908695476... sine
.410861347933971653047... 3 93
4
1
3 20
log( )
( )
k
kk k
.41096126403879067973... ( )( ) ( )
11 1
2
k
k
k k
k
1 .41100762619286173623... 1
1 k kk !!
.411233516712056609118...
1
222
2
4
1)()(
4
)2(
24 k kiLiiLi
= x dx
ex
3
0
2
1
= log logx
x xdx
x x
xdx3
12
0
1
1
= log log1 112
02
0
e dxx
dx
xx
= log(sin ) tan/
x xdx0
2
GR 4.384.12
.41131386544700707263... 1
21 1 1 1 1
0
1
cos cosh sin sinh cos sinh x x dx
2 .411354096688153078889...
0
2 3/1
)1(
3csch
2
3
2
3
k
k
k
1 .41135828844117328698...
0 21 )12()!(!
1)2(
k kkk
alSinhIntegr
2 .411474127809772838513... 2 318
1 8 8 8 8cos ...
[Ramanujan] Berndt Ch. 22
1 .41164138370300128346...
2 1
2
sinh (log ) sin ( log )i i
6 .41175915924053310538... 7G
.4117647058823529 = 7
17
.411829422582171159052...
13 1
11
2
23
2
23
2
3sech
k k
ii
=
12
2
1k kk
kk
.41197960825054113427... 3 3
8
24
1
log log( )
x
xdx
1 .41228292743739191461... csclog2
0
11
x dx
x
1 .412553231710159198241...
1
24
242
2
12sinh2csch
2
k kk
kk
.4126290491156953842... 1
2
1 1
12
( )
( )
k
kk
5 .4127967952491337838... kk
k
2
1 2 1
.412913931714479734856...
233 log
1
k kk
.413114513188945394253... ( )
(( )!)
2
2
1
2
2 212
10 0
k
kI
kJ
kk
148 .413159102576603421116... e5
.41318393563314491738...
( )4 1
4 11
kk
k
.41321800123301788416... 32log369
1
= 2
3
2 3
9
1
2 arcsinh
= 2 11
111
2
1
2
1 22
Fk
k
k k
k
, , ,( )
.413261833853064087253... 1
331 kk
.413292116101594336627... cos(log ) Re i
2 .4134342304287...
2 1
21
1n k
nk
.413540437991733040494... 2
3 3 5 52 3 30
/
dx
x
1 .41365142430970187175... 1
2 4
2
0
cos cosh( )!
k
k k
6 .41376731088081891269... 16
243 1
4 3
30
log x dx
x
2 .414043326710635964496...
0 )1)(2/1(!
11222
k kkkerfie
2 .414069263277926900573... e
e x xdxx
1
102
0
sin cos
.414151108298000051705... log12
6 42 1
1
H kk
k
.4142135623730950488... 8
5cot
8tan
2)!2(
!)!12(12
1
kkk
k
1 .414213562373095048802... 2
= 1
8
2
0k
k
k
k
= 11
2 11
( )k
k k GR 0.261
2 .4142135623730950488... 8
csc8
3sin21
.414301138050208113547... 1 2
3
3
6 3
1 3
4
3 3
4
log
( )
i
i
i i
2
6 3
3 3
4
1 3
4
i
i
i i
( )
= ( )
1
1
1
31
k
k k
.41432204432182039187... 1
31 k
kk
.414398322117159997798...
13)2/1(
1
2
3,38)3(7
k k
2 .4143983221171599978...
13
412 )(
6)3(7k k
k
8 .4143983221171599978... 7 31
12
31
( )( )
kk
.414502279318448205311... 11 1
21 1 1 1 1
eE i i i i( , ) ( ) ( ) ( , )
= ( )
!( )
1
1
1
21
k
k k k
.414610456448109194212... 7
6 2 6
3
2
1
2 322
cot
kk
.4146234176385755681... 939
10
3
4 2
= k
k
kk
k 21
1
( )
.414682509851111660248... k
primepp
k
2
)(
2
1
2
1
.41509273131087834417... 1)1(2
22log1
kk
k
k
=
2 )1(
)()1(2
k
k
kk
k
.415107497420594703340... 2
e
.41517259238542094625...
1
)3(
3kk
k
k
H
1 .415307969994215261467... ok kj
k k jk( ) ( ) ( ) )
1 1 1 12 11
.415494825058985327959... 11
6 2 22 2 4 4 1
1
coth cot ( )k
k
k
= 4
442 kk
27 .41556778080377394121... 25
9 1
2
6 50
log
/
x dx
x
.41562626476635340361...
1 2
43
3643
)1(
)14(1)3(
k k k
kkkkk
31 .415926535897932384626... 10
.415939950581392605845...
( )( )
( )2 1 1
3 36
122 4
22
22
11
f k
k jkk kj
1 .41596559417721901505...
1
22
31
)4/1(
)1(
2
1
k
k
k
kG
= 1
0
2 )( dxxE Adamchik (17)
.416146836547142386998...
0
1
)!2(
4)1(2cos
k
kk
k
1 .416146836547142386998... 2 1 1 2 14
22 1
1
sin cos ( )( )!
kk
k k GR 1.412.1
7 .4161984870956629487... 55
7 .416298709205487673735...
4
11
4
1,
4
1 2
.41634859079941814194... arctan 3
3 2 1020
dx
x x
.41652027545234683566... 3
16 2 2 12 30
dx
x( )
.416658739762145975784...
( )34
.416666666666666666666 = 5
12
1
222
k kk
.41705205719676099877...
( )2 1
4 11
kk
k
.41715698381931361266... | ( )| k
kk 4 11
.41716921467172054098... 7 3
2 2 84
32
27
1 1
4
2 3
12
( ) ( )( ( ) )
Gk k k
kk
.417418352315624897763... 3
81
4 2 30
erfi
e k
k kk
!( )
.417479344942600488698...
1
351 4
2
)2(
)4(Im
kkk kki
k
.41752278908800767414...
k
k
k
k
kk 2
)2()1(
)12(
112log4)2( 1
12
.417771379105166750403...
0
4
222 cossin
23
1dx
x
xxx GR 3.852.5
.41782442413236052667...
1
330 26
)1(
6
2log
36 xx
dx
kk
k
1 .417845935787357293148... root of ( )x 3
.418023293130673575615... 11
1
2
1
1
2 1 1 21
e
e
e ekk
k/
=
1 )1(
11
! k
k
k
kke
e
k
B GR 1.513.5
.418115838076169625908...
2
2024
23
42
2 1
2 1log ( )
log( )
( )
k
kk
Prud. 5.5.1.5
.418155449141321676689... )(Im)(Im ii .418168472177128190490... )(i
.41835932862021769327... 1
3293 5 7 2
2 2
0
1
( )arcsin arccos
x x
xdx
2 .418399152312290467459... 4
3 3
3
2 2 11
k
k
k
kk
!
( )!! J278
= area of unit equilateral triangle CFG G13
=
1
21
21
)!12(
)!()!(
k k
kk
= dx
x x
dx
x20
3 201 1
/
= dx
e ex x
1
.41868043356886331255... 35
2592
5 63
31
sin( / )k
kk
GR 1.443.5
.41893853320467274178...
1
1)1(2
212log
2
1
k
k
kk
k
1 .419290313672675237953...
3
3
2 2 3
1 3
2
1 3
2
2 2
tanh
i i i
i i i
2 3
1 3
2
1 2
21 1 ( ) ( )
= H
k kk
k2
1 1
97 .41935701625712157163... 4 4
.41942244179510759771... 11
2 11
kk
2 .419550664714475510996...
1
32
1)1(2
8)3(26
42log4k
kk
k
.41956978951241555130... e
e e
k
ekk
sin
cos
sin1
1 2 121
2 .4195961186788831399... sec(log )
2i i
.4197424917352996473... 1
4
1
2
2
4
1
2
1
2 1 2
1
1
cos sin( )
( )!
k
kk
k
k
148 .41989704957568888821... e e5 5
.42026373260709425411...
2
2
1)2(43
27
k
k
2 .42026373260709425411... 92
3
1
1 3
1
1 3
2 1
2
1
21
( )
( / )
( )
( / )
k k
k k k
1 .420308303489193353248...
1
3
)(2 2
)6(
)3(
k
k
k
HW Thm. 301
=
primep p
p3
3
1
1
3 .42054423192855827242... 2
0
2
2
loglog sin x xdx GR 4.322.1
.420558458320164071748...
0 )3)(1(2
12log3
2
5
kk kk
.420571993492055286748... ( )!
1 11
1
kk
k
k
k
.420659594076960499497... 1
532 kk
17 .420688722428817044006...
8 21
2
30
log
e x
edx
x
x GR 3.455.1
.42073549240394825333... sin ( )
( )!
1
2
1
2
1
1
k
k
k
k
.421024438240708333336...
02
2/
0
01
cossin)sin(tan)1( dx
x
xdxxxK
.421052631578947368 = 8
19
.421097686033423777296...
1 1
0
1 144
14
4
)(
14
12
k kkk
k
kk
k
k
6 .42147960099874507241... e
e
e
k
k
1
1 .4215341675430081940...
122
)1(1
kk
k
k
.4215955617044106892...
2 )2,(1
)2,(2
k kStirlingS
kStirlingS
.421643058395846980882... ( )( )
12 1
1 0
1
kk
k
k
.421704954651047288875... 2 21
2 220
Jk k
k
kk
( )( )
!( )!
.421751358464106262716... 3 2
2
9 3
16
6
30
log log sin
x
x GR 3.827.13
.42176154823190614057... e2 2 2 2 2 2 2 2 2 2cos sin( sin ) sin( sin ) cosh( cos ) sinh( cos )
=
ie e
k
ke e
k
k
i i
2
2 22 2
1
2 2 sin
!
1 .421780512116606756513... k H
kk
kk
2
1 2 2 1( )
.421886595819780655446... sin51
.42191... 1
21 1n
kn k
n
( )
.4219127175822412287... H
kk
k
2
31 1( )
1 .4220286795392755555... log log
( )3 2 2
3
2
3 6
2 2
34
1
23 3
Li
=x x
x xdx
log
( )( )
2
21 2 1
.422220132000029567772...
( )
( ) ( )
3
2 3
.422371622722581534146...
2
1122
1
2
log( )k
kkk
5 .422630642492632281056...
33 1
3 1
1
1 60
log( )
( / )
k
k k
.422645425094160918302... 1212log4
1
16 22
2
Berndt Ch. 9
.422649730810374235491... 11
3
1
2
2
2
1
1
( )k
kk
k
.42278433509846713939...
2
1)()2(1
k k
k K Ex. 124g
=
221
1
1log
11
kk kk
k
kkLi
=
0
logdx
e
xxx
= x
dx
xx
x
0 1
1sin GR 3.781.1
= cosxe x
dx
x
1
10
1 .42278433509846713939...
2
1)(22
k
k
k
k
1 .42281633036073335575...
122
1k
k
k
.422877820191443979792...
5
23Li
.422980828774864995699...
1 )2()!2(
4)1(2log)2(
k
kk
kklCosIntegra AS 5.2.16
.423035525761313159742...
1
21)2(k
k
.423057840790268613217...
012 )23(2
)1(
2
1,
3
5,
3
2,1
2
1
kk
k
kF
.4232652661789581641... ( )
( )3
360
4
31
k
kk
.423310825130748003102... e
1 .423495485003910360936... 2
)3()2(
.423536677851936327615...
2
12
12
1
2
1
4
12log
iiii
=
13
1)1(
k
k
kk
.423565396298333635... 2
1)3(2)1(
12
12)1log(2 232
eLi
eLi
eLie
=
0 0 !
)1(
6
1
)!2(k k
kk
k
k
B
k
B
.423580847909971087752...
1 2
1)1(
kk
k kH
2 .423641733185364535425...
0
32
)!3(
2
3
3cos2
3 k
k
ke
e J803
.423688222292458284009...
0 )4(2
1
3
322log16
kk k
.423691008343306587163... 1
0
2sinlog22log2
1dxxxlCosIntegra GR 4.381.1
.423793303250788679449...
3
2
1
7
)1(
21
57csc
72 k
k
k
1 .4240406467692369659... 11
1 41
( )( )k kk
.424114777686246519671...
245
2 4)3(6)4(624
k kk
= 1
0
3log)1log( dxxx
.424235324155305999319...
2
1)(k
kk
1 .424248316072850947362...
12
12
1 11)1()1(
kk
k
kLi
kk
k
.424413181578387562050...
2/1
1
3
4
.4244363835020222959...
2
1
2
121 4
1
1 0
2
eerfi
kk
k
e xdxk
k
x/
( )
!
sin
=
2
1Dawson
.424563592286456286428...
12 15
1
15
7
2
21tan
21 k kk
.424632181169048350655...
2
1)( 1)(k
kk
14 .424682837915131424797... 12 31
2
20
( ) /
x dx
ex
.42468496455270619583...
22 1
2
20
loglog sinx x
xdx
1 .424741778429980889761...
12
1
1 1arctan
12
)24()1(
kk
k
kk
k
=
2
11log
2
11log
2
11log
2
11log
2
iiiii
.424755371747951580157...
328 3 123 2csc cosh cosh sinh coshh
= ( ) ( )( )
( )
1 2 11
11 2
1
2 2
2 32
k
k k
k kk k
k
9 .424777960769379715388... 3
.425140595885442408977...
2
31
2
3122
3
2 ii
=
1 1
31
1
1)3()13()1(
k k
k
k
kkk
.425168331587636328439... dxx
x
e
22 1
2cos
.4252949531584225265...
2
1arctan2
2
3log
3
1
2)1(
1
21
kkkk H
3 .425377149919295511218... 2
coth
.425388333283487769500...
13
2
42log22log2
k
k
kk
H
.4254590641196607726... csch1
2 1
12 2 1
0
e
e e kk
GR1.232.3, J942
.4255110379935434188... B
k
k
kk
k !
2
0
2 .42562246107371717509... 2
3
7
42 3 2
3 3 1
1
2 2
32
( )( )
k k
k kk
= ( ) ( )( ( ) )
1 1 12
k
k
k k k
.425661389276834960423...
1
)2(
3kk
k
k
H
.425803019186545577065...
0 )33)(13(
1
4
3log
312 k kk
.42606171969698205985... ( )( )
12 1
2
1
213
1
kk
k k
k
k k
=
1
21
21
2
i i
.42612263885053369442...
0 2)!2(
)1(2
4
kk
k
ke
8 .4261497731763586306... 71
1 .426255120215078990369... 1)()1( xofroot
2 .42632075116724118774... 1
21 Fkk
.42639620163822527630...
1 )12(
log
k kk
k
.426408806162096182092...
2
53
2
15log
15 22
2
Li
Berndt Ch. 9
.426473583054083222216...
0
1
27
)1(
k k
.426790768515592015944...
1 )32(
1
3
2log2
9
8
k kk J265
1 .426819442923313294753...
G xx
dx
x
3 3
81 1
12
0
( )log( ) log
.426847730696115136772... 2
22 28
23
2
4 1
2 2 1
1
2
log( )
( ( ) )k
k k
k k
kk
k
2 .427159054034822045078... 12log22
.427199846700991416649...
1
21
)!2(
2)1(
4
2sin22cos2
k
kk
k
k
.42721687856314874221... dxx
x
4/
0
4
sin1
cos
812
12
.4275050031143272318...
3
2
6
52log2
3
=
12
1
3
)1(
k
k
kk
= dxxx
x
02
2
1
1log
= log( )
11
1 30
xdx
1 .427532966575886781763...
1
23
23
2
2 3cos)1(
2
1
124
9
k
kii
k
keLieLi
GR 1.443.4
.42772793269397822132...
0 22
)1(
2
1
2
22
k
k
k
.42808830136517602165... x xdxtan0
1
.428144741733412454868...
1
)2(3
)1(22
)3(
3
2log2log)2(
kk
k
k
H
2 .428189792098870328736...
133/23/1
11
)1()1(
1
2
3cosh
1
k k
Berndt 2.12
=
12
11
k kk
.42821977341382775376... eieRe
2cos
.42828486873452133902... 1
log!
1
2
k
k
kk
.428504222062849638527...
1 )1(
)1)1)(log(1(
kk
k
e
H
e
ee
.428571428571428571 = 7
3
.4287201581256108127...
11
1
1
1
1
1
1 2eEi
k k k k
k
k
k
k
( )( )
( )!
( )
!( )
=( )
( )! !
11 22
k
k k k
403 .428793492735122608387... 6e
.42881411674498516061... 2 2
20
1
12
1
2
2
2
2
2
1
1
log log log( )
( )
x
x x
1 .42893044794135898678...
2
3
21
1
1 2
0
x
e edx
x x
/
.42909396011806176818...
0
3
)!3(
3
54
172
k
k
k
e
.429113234829972113862... 23
2
1 .42918303074795505331... ( ) ( )1 2 1 2 i i
.42920367320510338077... 22
=1
4
12 1
43
201 k k
k
kk
( )
=( ) ( )!
( )!
1 1 12
121
k
k
k
k
= log ( ) ( )2 11 2
21
1
k
k
k
kk GR 8.373
=2
22 1
2
2 1 2 2 11 1
k
kk
kk
kk k
k
k k k
( )
( )!!
( )!! ( )
=
sin 4
1
k
kk
= arccos arcsinx x dx0
1
= 2/
0 cos1
sin
dxx
xx Prud. 2.5.16.15
=
02 cosh)1( xx
dx
GR 3.522
.42926856072611099995...
0
1
1
log
3
1
3
22log
3 xx ee
x GR 4.332.2
6 .429321984298354638879...
163
22
22
2 22 2 2csc cot cot csc
= kk k k
kkk
k
2
1
2 2
2 31
2
2
2 2 1
2 1
( ) ( )
( )
4 .429468097185688596497... 7
3
.429514620607979544321...
052 )1(256
35
x
dx
.429622300326056192750...
2
9log2log33log2log3
6
1)3(
2
3
2
1 2233 LiLi
2
3
64
729log
2
9log2log3log3
6
12
2 Lii
=
12
1
2
)1(
kk
kk
k
H
.429623584800571687149...
23
23
221)(2
kk
k
kkLi
k
k
1 .429706218737208313187...
1
00 !!)2()2(
k
k
kk
HKI
3 .42981513013245864263... G
4 .42995056995828085132...
22
1 3
31)2(33cot
2
32
kk
k
kk
1 .429956044565484290982... dxx
x
04
22
)1(
log
9
3
.43020360185202489933...
1 1222 4
1
4
)2(
k kk k
Lik
k
.43026258785476468625... 1
2 1
1 3
231
1
1k
k
k
k
kk
( ) ( )
.43040894096400403889... 2
5
1
2
2
5
1 5
2
12
1
1
arcsinh
log( )k
k k
kk
=
0
122!)!12(
!)!2()1(
kk
k
k
k J142
=
11 42 k
kkk
kkk
k
LF
k
F
=
0123 3xx ee
dx
xxx
dx
.43055555555555555555 =
1 )4)(3(
2
72
31
k kkk
k
.430676558073393050670... 2log5
11 .430937592879539094163... 48 64 211 2
2 20
log ( )/x Li x Lix
dx
1 .430969081105255501045... 5/16
.431166447015110875858...
1
1
)12(2
)1(
kkk
k
1 .43142306317923008044...
1
0
1
0
1
022212
2log3
4
33 dzdydx
zyx
zyxG
2 .43170840741610651465... 24 4
22
( )
.431739980620354737662... 1
2 6 21 2
2
2
coth ( ) ( ) ( )k
k
k k
= k
k k kk
3
2 22
1
1 1
( )( )
5 .43175383721778681406...
1k k
k
F
H
5 .43181977215196310741...
0 )(x
dxx
.43182380450040305811... 1
5
1
2
1
2
1
2 2 1
1
2 11
arctan( )
( )
k
kk
k
k
.4323323583816936541...
1
02
13
02
2
)!1(
2)1(
2
1x
e
k
kk
e
dx
x
dx
ke
e
.43233438520367637377...
4/34/3
4/14/1
2
1
2
1
2
1
2
1
24
iH
iHHH
i
=1
2
1 4 1
23 11
1
1k k
k
k
k
kk
( ) ( )
.432512793490147897378... ( ) log ( )
1 21
1
k
k
k
1 .432611111111111111111 = 4)2(
144
205H
.432627989716132543609... 2e
.432824264906606203658...
0
2
sin
log1
1x
x
.432884741619829312145...
1
0
1arctan
44arctan
xx ee
dx
ee
.432911748676149645455...
0
4
4
3
x
dxee xx
GR 3.469.2
33 .4331607081195456292...
2
3 1)()3(12)4(6)2(71k
kk
=
24
2
)1(
)14(
k k
kkk
1 .43330457796356855869... 81 3
2
189
4 1
2
1 30
1 ( ) log/
x
xdx
.43333333333333333333 = 13
30
1 3
31
arctan /x
xdx
4 .43346463275973539753...
2 2
2
2
1log
)1(
2311)(
1k k k
k
k
k
kk
k
k
.433629385640827046149... 413
.433763428189524336184...
02 19
11log12
3dx
x
.433780830483027187027... 1coshlog2
1log
2
e
e
=
ii
ii
21
21
11log
= 2 2 1
2
2 1 22
1
k kk
k
B
k k
( )
( )!
= 1
0
tanh dxx
.433908588754952738714...
1
2)2(
21
2sin2
k k
=
1 )2(
11
k kk
9 .4339811320566038113... 89
1 .43403667553380108311...
1 )7(
11
k kk
.434171821180757177936... 1)2(1)1(1
kkk
3 .43418965754820052243... log3
1 .434241037204324991831...
1
34/1
)!2(
!
2
1
128
79
64
45
k k
kkerf
e
.434294481903251827651... 10log
1log10 e
.43435113700621794951...
1
2
1
13
)1(
2
1
3csc
32 k
k
k
1 .43436420505761030482...
1 2
1
kk k
.434591199224467793424... ( )2 1
2 11
kk
k
1 .434901660887806674581...
01
2
0
2
)!2(
12)2(
6
5)2(
3
2
k kk
kI
eI
e
1 .4350443315577618438... 2 21 2 1 2 2 1
0
2 1sinh / / ( )!
k k
k
.4353670915862575299... 1
21 ( )!! !!k kk
.4353977749799916173... sin cos ( )
( )!
2
4
2
2
1 4
2 11
k k
k
k
k
.435943911668455273215...
1
5
2!
)1(
32
23
kk
k
k
k
e
.435953490803707281686...
6/
0 cossin32arctan2coth2
xx
dxharc
1 .435991124176917432356...
32
3
1 , Lebesgue constant
.43617267230422259940...
1
23
12 )1(2log2
121
k
k
kk
.436174786372556363834...
1
0 1
cosh
2
12log)1log( dx
e
x
ee
x
.43634057925226462320...
12
1)1()1(
)3/1(
)1(
6
1
3
2
4
19
k
k
k
.43646265045727820454... 2
22 2
125
2
5
1
5 2
1
5 3csc
( ) ( )
k kk
.436563656918090470721...
1
2
)!2(52
k k
ke
5 .436563656918090470721...
01
2
!
1
!2
kk k
k
k
ke
.43657478260848849014...
1
2
)2(1
2)1(
kk
kk
k
H
1 .436612586189245788936...
1
11
2
)2()1(
21
21log2
kk
k
k
kii
= 2 11
2 21
log
kk
1 .43674636688368094636...
3
1
2
3log3log2log
6)2( 2
22
2 LiLi
= log
( )log
2
20
1
02 11 2
x
xdx e dxx
=
1
0 2/1
logdx
x
x
.436827337720053882161...
22 2
1
4
3
2
7tanh
7 k kk
.43722380591824767683... 42 18 3
241
12
0
1
2
( )log logx
xxdx
2 .437389166862911755043...
2
)3()(3)2()4(k
kkk
=
24
23
)1(
12
k kk
kkk
.437528726844938330438...
4
2 14
)1(
1820
257
142
14csc
k
k
k
.43754239163900634264...
1
2
!
1)1(
k k
k
.437795379594438621520...
1
1
!
)1(1)(
k
kkk
e
k
HeeEie
.43787374538331792632...
4
5,4
4
3,4
4
5,21612816
128
3 2 G
= dxx
x
03
3
cosh
.438747307343219426776...
logx
edxx 10
1
.43882457311747565491...
0 1
2 168
1
)22)(12(
)1(
2
2log
4 k k
k
kkkk
=
1 12
)1(
2
)1(1
k
kk
kk J72
=
1
2/)1(
k
k
k Prud. 5.1.2.4
= 1
0
4/
02
arctancos
dxxx
dxx
.43892130408682951019...
coth
2
1
2
1
121
kk
.439028195096889694210...
1
21)1(
k k
k
.439113833857861850508...
1244
2
2coth2)2(24
k kk
=
1
2
)2(
)3(Im
3214Im
kki
kii
.439255388906447904336...
2
2
1
2
1)(2
12)1(
)12)(1(
2
3
72log4
kk
kk
k
kkk
2 .439341990042075362979...
)(,!
2
1
)(
kkk
k
number of prime factors of k
5 .43937804598647351638... 7
120
2
4
2
83
1
2 2
4 2 2 4
4
3
21
log log log
Lixdx
x x
.43943789599870974877...
1 1
11
2)1(
2
3
2
)1(2
k kk
k
k
k k
.43944231130091681369... e
e x
dx
x2
5
2
14
1
cosh
.43947885374310212397... 4 7 6 2 5 3 41
61
( ) ( ) ( ) ( ) ( ) ( )
( )
k
kk
1 .43961949584759068834... /1
.44001215719551248665...
2 log)1(
1
k kkk
.440018745070821693886...
0 )13)(2(
)1(
5
2log2
5
1
35 k
k
kk
.44005058574493351596...
0
1 4)!1(!
)1(
2
1)1(
kk
k
kkJ
.44014142091505317044... 2
12sin
1
1
2
1cos)1(
2
1sin)21cos22log(
2
1
1
k
kk
1 .44065951997751459266...
1
)22(Im
2tanh
2 kki
k
=
12 122
1
k kk
=
0 sinh
sindx
x
x GR 3.981.1
=
dxee
xxx
sin
2 .44068364104948817218...
02)1(!
1)(1
k
k
kk
eeEi
e
.440865855581020082512...
12
1
0 2)!(
)1()2(1
kk
k
kJ
.440993278190278937096...
2
3
24
3
Berndt 2.5.15
3 .441070518095074928477...
1
3 1)3(9)2(12)3(6)4(6k
kk
=
242
2
)1(
14
k kk
kk
3 .4412853869452228944... 8
3sin
4
1
4
32
4
3 14/34/5
.44133249687392728479...
22 log)1(
1
k kkk
5 .44139809270265355178... 31
13
230
( )( )k kk
= dxx
x
02
2 )3log(
3 .441523869125335258... e Ik
k
kkk
00
11
2
2( )
!
1 .441524242056506473167... )4(2)3(3
4 .44156786325894319020... )572161log(16
52log2arccsch
4
5
10
55
=
0 5/1
)1(
k
k
k
.44195697563153630635... ( ) ( ( ) ) log
1 1 11
11
1
1 2
kk
k k
H kk k
1 .44224957030740838232... 3/13
5 .44266700406635202647...
.442668574102213158178... ( )
! ( )
1
1
1
1
k
k k k k
.44269504088896340736... 1
21
1
2 1 2
2 2
3 1 2 21 21
1 3
1 3 2 31log /
/
/ /
kk
kk
k
k
k k
[Ramanujan] Berndt Ch. 31
1 .44269504088896340736... 1
2log
.44287716368863215107...
1 2
3)2()()1(
)1()3()2(
k k
k kkk
k
=
12)1(k
k
kk
H
.44288293815836624702... dxx
x
1
0 1
arcsin42
4 .44288293815836624702... 21
4
3
4
=
dx
e
ex
x
1
4/
= log log1 1 24
0
2
0
x dx x dx
= dxx
x
02
2 )2log(
=
x
dx2sin1
= dxx2/
0
tan
.44302272411692258363...
1
21221
)!2(
)12(2)1(1tanlog
k
kkkk
kk
B AS 4.3.73
.44311346272637900682...
00
2 42
4dxxedxex xx
LY 6.30
= dxxx
xxe x
2/
04
2tan
cotcos
cos12
GR 3.964.2
.443147180559945309417...
2
2 1
)1(
4
12log
k
k
k
k
.443189592299379917447...
12 24
12cot
224
3
k kk
= 1)()1(2
2
kL
kk
k
1 .443207778482785721465...
1
22
)12(
)1(2log2)3(2
k
k
kk
Hk
.443259803921568627450 = )15(8160
3617
.443409441985036954329...
0
2/)12(2/12/1
)1(2/1cosh2
11
k
kk eee
J943
1 .443416518250835438647...
2
1)()(k
kk
3 .443556031041619830543...
12!
124,2,2,2,2,
2
3,1,1,12
k kkk
kHypPFQ
1 .443635475178810342493... 2arcsinh
.4438420791177483629...
log ( )( )
!2
12
12
1
1
Eik k
k
kk
.444288293815836624702... 25
.444444444444444444444 =
1 49
4
kk
k
.44466786100976613366...
e
xe dx
x
xe
/1
0
/1 log11
1 .44466786100976613366... ee /1 , maximum value of xx /1
=
0 !
1
kkek
5 .4448744564853177341...
32 3
022 3
( )log xdx
ex
1 .44490825889549869114... ( ) ( )( )
( )
1 12
8
2 12
2
31
kk
k k
k kk k
k
1 .44494079843363423391... 2 03
1
3( )( )
k
kk
Titchmarsh 1.2.1
1 .44501602875629472763...
( )
( )
2
2 11
k
kk
3 .44514185336664668616... 9
3
.44518188488072653761... 32 3
3 3
2
1
3 21
log
k kk
=
4
3
1
31
1
31
1
hgkk
kk
( )( )
= 1
0
3)1log( dxx
.445260043082768754407...
1
2
65
6log
125
42
125
48
kk
kHk
.445901168918876713517...
1 3
)1(
2
3log1
4
3
kk
kHk
.446483130925452522454... )()(2
1)4()2(2
3
2
3
1 24
24
ii eLieLi
=
1
4
2cos
k k
k GR 1.443.6
.44648855096786546141...
0 )2()!12(
)1()21sin21(cos2
k
k
kk
.446577031296941135508... ( )
( !)
k
kI
k kk k2
20
1
21
1
.446817484393661569797...
2 )(
)1(1
k k
k
2 .4468859331875220877... k
dkk
1
2
.44714430796140880686... 2sin22cos2
=
1
1
33 )1()!2(
2)1()()(
k
kkii
kkeLieLi
.447213595499957939282... 5
5
.447309717383151082397... 128
27
32
3
4
3
24
27
144 2
27
2
G log
=log( ) log
/
11 4
0
1
x x
xdx
.44745157639460216262... 3 3
2 21 2
1 33 3
0
/
( )( )
dx
x x
.447467033424113218236...
2
23 2
112
3
8
1
21 1
1
2
( ) ( )k
k k
k kk k
2 .4475807362336582311...
3
2
3
1)2(
3
2 13/2
.44764479114467781038... 2
3tan
.44784366244327440446... ( )
( )log
1
3 11
1
31 1
k
kk
kkk
.447978320832715134501... 2
tanh122
3csch
6)1(csc
62coth
126/5
ii
= ( )
1 1
4 21
k
k k k
3 .448019421587617864572...
1
1
)2(2
)3(3)2(
k
kk
kk
HH
810 .44813687055627345203... 2sinh2coshsinh2sinh2coshcosh2
1
2
2
eeee
=
0 !
cosh
k
k
k
ke
.448302386738721517739... 4 2
12 1
2 1 20
cos( )
( )
k
kk
GR 1.444.6
.44841420692364620244... log
log loglog2 2
2 2
2
22 3
3
2
1
3
1
2
Li Li
=
3
263log32log3
6
12
222 Li
=
112
1
2
22
32
)1(
4
1
2
1
2
2log
12 kk
k
kk
k
k
H
kLi
= ( )
( )( )
1
2 1 2 20
k
kk k k
=
1
1
02
21
02
2
2
log
)12(
log
)2(
log
x
dxx
x
dxx
x
dxx
= log 120
e
dxx
.44851669227070827912...
1
22
2
1
2
)15(
5
5
)2(
5csc
5
2sin52
40 kkk k
kkk
1 .448526461630241240991...
122
12
1 1)2()1(
kk
k
kLi
k
k
1 .448545678146691018628...
1 2 ]2,[1
)1(
!
1
k k
k
k kSHk
.44857300728001739775...
1
333
cos
2
1
k
ii
k
keLieLi
.448623089086114197229...
1 2
11
k kk
k
.448781795164103253830...
0
3/1 3)!2(
)1(6
9
kk
k
ke
.44879895051282760549... 7 1
5 2
70
x dx
x
/
6 .44906440616610791322... 32
9 3 2 2
12
12
1
( )!( )!
( )!
k k
kk
.449282974471281664465... Li22
5
.449339408178831114278...
1322
1
2
1
x
dx
.449463586938675772614...
18 2
2
9 3
2
9 1 23 20
log
( )( )
dx
x x
2 .449489742783178098197... 6
110 .449554966820854649772... 41 11
6
1
ek
kk
( )!
.44959020335410418354...
12 2/14
1
22cot
221
k k
3 .4499650135236733653...
0
2
322
22 log
)3(42
3269
xe
xdxx
.45000000000000000000 = 20
9
3 .45001913833585334783... dxx
xxx
x
x2
0
222
)2(
2log)1(log
3
2log
GR 4.313.7
.45014414312062407805... log( )
( )( )2
3
3 2
1
2 1 3 20
k
k k k
2 .45038314198839886734...
1
1
!!
2)1(
k
k
k
.450571851280126820771... 1
2 22 2 2 2
2 2
cosh sin sinh cos
cos cosh
= ( )
1
1
1
41
k
k k
.45066109396293091334... 3 2
2
2 1
2 120
cos sin ( )
( )!( )
k
k k k
. 45071584712271411591...
0
2
)32(16
12
2
12
kk kk
kG
.45077133868484785702...
13
1
2
)1(
8
)3(3
k
k
k
=
022
3
2xx ee
dxx
= 2 3 3 22
0
1
Li i Li i Li xdx
x( ) ( ) ( )
4 .450875896181964986251... )2(216
36
.45091498545516623899...
( )2
21
k
kkk
.45100662426809780655...
3
3
0
1
83 6 arcsin xdx
.45137264647546680565...
0
3
3
2
3
1xe
dxx
3 .451392295223202661434... loglog( )
34
1
2
20
x
xdx
.45158270528945486473... log( )
log 2
2
41
1
412
1
k
k kkk k
Wilton
Messenger Math. 52 (1922-1923) 90-93
= iloglogRe
=
02
12log
2
1
kk k
k
k Prud. 5.5.1.16
= x
dx
x
x
log1
11
0
GR 4.267.1
= dxxx
2/
0
cot1
GR 3.788
1 .451859777032415201064...
0 !
sinhsin
4
1111
k
eeee
k
kkeeee
i iiii
.451885886874386023421...
0 212
)1(32log3
12
1
k
k
k
2 .451991067287107389384...
1
22
1)1(2
22log1
3 k
k
kk
k
22)2(
)1)3((2
2log
k k
kk
k
kk
.4520569031595942854... 4
3)3(
.45224742004106549851... pk
kk
p prime k
2
1
2
( )
log ( ) Titchmarsh 1.6.1
=
222 )1(
)()12(
k kk
kk Shamos
.45231049760682491399... 3
4 2
3
2
2
3
1
2 3
1
21
csch
( )
/
k
k k
.452404030972285668128... 1
42 4 2 4 1
1
coth ( ) ( ) ( ) ( )
i i k kk
=
2
12 1
1
k kkk
16 .452627765507230224736... 2
22
2 1
2 1
0
erfik k
k
k
!( )
.452628365343598915383... 1
6
1
2
1
3
1
2
5
6
1
3 20
, ,( )
k
k k
.452726765998749507286...
1
1)3(
)2(
k k
k
.45281681270310393337... 22
22log
28
5
2
2log5
28
5
16
51
=
12 )58(8
25
8
)(5)1(
kkk
kk
kk
k
.452832425263941397660... log2
3 20
1
e e
edx
x
x
.4529232530212652211... 2/2
1
2
1
ei
1 .452943571383509653781... 3 11
2
( )k
k
.453018350450290206718... 2
4
22
0
sin /x
ex
.45304697140984087256... e
6
9 .4530872048294188123... 16
3
42 2
40
sin ( )x
x GR 3.852.3
.453114239502760197533...
0
2 3
sin
3log1
1x
x
.453344123097900874903...
0
2cos)(
4dx
x
xxsi
.453449841058554462649...
4 3 3 4 4 3 320
21
40
dx
x
dx
x
xdx
x
.453586433219203359539... 4 2
3
1
3 6
3 3
4
3 3
4
logcot
( )cot
(
i i
=
1 124
1
8
2)1(
k k
k
kkkk
.453592370000000000000 = pounds/kilogram
.453602813292325043435...
cot log log7
88 2 2 2
2 2
2 2
= 1
4 2 221
3 42k k
k
kk
k
/
( )
.45383715497509933263... sin/
x dx0
4
.45383998041715262... ( ) log ( )
112
n
kn
nk
.4538686550208064702... cot(log )
i i
i ii i
.45395280311216306689... 3
4
5
21
1
3 2
0
x
e edxx x
/
.453985269150295583314...
2
1
2
1
2
1
8
2log
96
522
22 iLi
iLi
.454219904863173579921... Eik kk
k
1
22
1
21
log!
AS 5.1.10
4 .45427204221056257189... 2cschsinhcosh2
=k
k kk k
2
21
21
4
21
2
2
.454313133585223101836... ( ) ( )
1 2 11
2
k k
k
.454407859842733526516... ( )kk
12
2 .45443350620295586062...
( )
( )
3 1
4 1
.45454545454545454545 = 5
11 63 2
1
F kk
k
1 .4546320952042070463... 2 3 21
21
( ) ( )
( )
k
kk
.4546487134128408477... sin
( )( )!
2
21
2
2 11
42
02 2
1
kk
k kk k GR 1.431
=
/2
01cos1sin
.454822555520437524662... 6 8 22 2
1
1 21
log( ) ( )
k
k B kkk k
3 .454933798924981417101... ( )
( )!
/2
11
1
21
2
k
k
e
kk
k
k
.455119613313418696807... 1
9log
1 .45520607897219862104... 22
21
7 31
( )log
( )
x
x xdx
.45535620037573410418... i
i ii
i4
3
22
3
22
sinh( ) ( )
= e x
edx
x
x
sin
10
.455454848083584498405... 3
2
1)( 1 dxe x
.455698463397600878... ( )
logk
k k
k
kkk k
1
4
1 4
4 11 1
.45593812776599623677... i ei / /2 4
.4559418900357336741... 12
sinh
.455945326390519971498... 9
2
12 2
qq squarefree with an odd number of factors
Berndt 5.30
.456037132019405391328... 1
8
1
4
1
44
1
2i
ii
i itanh tan csc
1
84
1
2
1
4
1
4csc cot cothh
ii
i i
= e x
edx
x
x
sin
1 2
979 .456127973477901989070... sinh
2
2
2
21
1
kk
.45636209905108902378...
1
2
3
21
3
2 1 21
cos( )
kk
.45642677540284159689...
1
1
12 3
)2()1(
13
1
2
1
3coth
32 kk
k
k
k
k
1 .456426775402841596895... 1
2 2 3 3
1
3 120
coth
kk
.456666666666666666666 = 137
300 5
1
55
21
H
k kk
.45694256247763966111...
1
21 )3(
1
13
)2(
kkk k
k
.457106781186547524401... 1
2
1
4 1
3
0
4
cos
sin
/ x
xdx
2 .45714277885555220148... 2 21
2
1
log( )!
!
k
k kk
.45765755436028576375... cot 2
.45774694347424732856... log2 1
3
1
12
3 3
2
3 3
2
3
2
3
2
= ( )
1
332
k
k k k
.457970097201163304227... 12
cosh
.457982797088609507527... G
2
= log( sin ) log( cos )/ /
2 20
4
0
4
x dx x dx
Adamchik (5), (6)
= x dx
e ex x2 20
.458481560915179799711...
2 1
4 2112 2
1
2
1
sinh
( )k
k k k
.45860108943638148692... 2
1
3
4
4482
34
= ( )sin
1 14
41
k
k
k
k
.458665513681063562977... 7 3
42 1
1
21
3
( )( ) ( )
( ) ( )
kk
k
k k k
.458675145387081891021... 1 11
11 0
log( )ee k
dx
ekk
x J157
=
1 !k
k
kk
B [Ramanujan] Berndt Ch. 5
2 .458795032289282779606... 6 4 12 3 7 29
81 13
2
( ) ( ) ( ) ( ) ( )
k
k
k k
= 8 5 4 1
1
3 2
42
k k k
k kk
( )
2 .45879503228928277961... 7 2 12 3 6 49
81 13
2
( ) ( ) ( ) ( ) ( ( ) )
k
k
k k
= 8 5 4 1
1
3 2
42
k k k
k kk
( )
.45896737372945223436... cos( sin ) (coshcos sinhcos )1 1 1 1
= ee e
k
k
ii e e
k
i i
2 1
2
1
cos
( )!
22 .459157718361045473427... e Not known to be transcendental
2 .459168619823053134509... 1
84
2
2 2
2
22
2 0
1
K
K
E xdx
x
( ) GR 6.151
.459349015616034745... 7 6
45
3
2
12
51
( )( )
( ) ( )
k
kk
.45936268493278421889... 1
2
1
2
1
2 1 2
1
1
sin( )
( )!
k
kk k
.4595386986862860465...
122
11
kk k
.4596976941318602826... 21
21 1
1
22
1
1
sin cos( )
( )!
k
k k GR 1.412.1
= sinsin log
xdxx
xdx
e
0
1
1
= sin1
21 x
dx
x
3 .45990253977358391000...
1
1 .459974052202470135268... kk
kk
( )
( )
1
12
.460047975918868477...
312
6 21
3
2
3 21
log( )k
k k k
.46007559225530505748... 2224
22
=
022 )2)(1( xx
dx
= dxx
xdx
x
x
2/
02
22/
02
2
cos1
cos
sin1
sin
=
02 18
11log dx
x
.460265777326432934536... 9 e
.460336876902524181992... ( )k
k kk
2
.460344282619484866603...
012
8/1
2!)!12(
)1(
22
1
2 kk
k
kierfe
1 .46035450880958681289...
1
2
1 .4603621167531195477... Gx
xdx
log log( )2
4
1
120
= log x
xdx
2
21
1
1
GR 4.295.14
= x
e edxx x
2 20
=
cos sin
cos sin
/ x x
x xdx
0
2
GR 3.796.1
= arccot x
xdx
10
GR 4.531.2
= dxx
1
02x-1
arcsinh
= log tan/
10
2
x dx
GR 4.227.10
1 .46057828082424381571...
0
2 27
1arctan2
7
1
xx
dx
.460794988828048829116... tan
!
k
kk
1
.46096867284703433665...
1
1
)!2(
)!1()1(
4
1,2,
2
3,1,1
2
1
k
k
k
kHypPFQ
.461020092468987023150... 1
31 1
3 3
21 1
3 3
25 31 3
1 6 2 32 3
1 6 2 3
//
/ //
/ /
( ) ( )
i i
= 1
31
3 1
32 11
1
1k k
k
k
kk
k
( )( )
1 .46103684685569942365... tanh
!
k
kk
1
.46119836233472404263... )12sin(4)!(
)!2()1(
2
1sin
1cos2
1
02
kk
k
kk
k
Berndt 9.4.19
.461390683238481623342...
j
k
jk k
j
k
1
4
4
14
1
)!(lim
1
11
.461392167549233569697... 3
4 2
1 1
2 120
cos
( )
x
x x
dx
x GR 3.783.1
=
0
5 log2
dxxex x
.461455316241865234416...
eEi
2
1
2
1 .4615009763662596165... 2
12
( )
( )
k
kk
1 .46168127916026790076...
2 1
)(log
k k
kk
.46178881838605297087... 9 2 3 12 2
301
14
0
1
3
loglogx
xdx
.461939766255643378064... 8
3sin
3cos
4
22
.462098120373296872945... 2 2
3
1
3 1 3 20
log ( )
( )( )
k
k k k
= dxx
x
3
1
0
2 11log
= dxx
x
1
04
6 )1log(
.462117157260009758502... e
e
B
k
kk
k
1
1
1
22
2 1
2
22
1
tanh( )
( )! AS 4.5.64
36 .462159607207911770991...
1 .462163614976201276864... 4
27
8 2
9
2
2
G
( ) J309
=
1
22 )23(
1
)13(
1
k kk
=log x dx
x30
1
1
=
1
03
log1
12 dxx
x
x
1 .46243121900695980136... 11
2
p
p prime
1 .462432881399160603387... ( )
!!
k
kk
2
.46260848376583338324... ( )/
kk
k
11
2
1
2
1 .4626517459071816088... =
21
1
2 10
erfik kk
!( )
3 .462746619455063611506...
111 2
)(
)21(
1
12
11
kk
kk
kk
kp
.46287102628419511533... 205
44
1
4 1 20
log( )
( )( )
k
kk k k
.4630000966227637863... 1
21 2 2 csch
=
1
1 )22()2()1(k
k kkk
= )1(Re)1(
)1( )1(
222
ik
kk
k
= dxee
xxdx
e
xxxxx
00 1
cos
1
cos
320 .46306452510147901007... 3
6
.463087110750218534897...
4/1
4/34/14/3
3
1cot
3
1cot
3
1
4
i
=
1
223
1
k kk
=
k
k k
3
)24(1 1
.46312964115438878499... 21
22
1 5
2
12
21
21
arcsinh2
log( )k
k k
kk
3 .463293989409197163639... 6
.4634219926631045735... e Ei Eidx
e xx( ) ( )( )
2 1
10
1
55 .46345832232543673233...
1
8
85764801
319735800
kk
k
.463518463518463518 = 6
13
1 .463611111111111111111 = 5)3(
3600
5269H
.46364710900080611621... arctan arcsin1
2
1
5
= Im log( ) Re log2 12
i ii
=( )
( )
1
2 2 12 10
k
kk k
=( )!!
( )!!
2
2 1 51
k
k kkk
= arctan( )
2
2 3 20 kk
[Ramanujan] Berndt Ch. 2
= ( ) arctan( / )
12
11
21
k
k k [Ramanujan] Berndt Ch. 2, Eq. 7.5
.4638805552552772268... e
e
ek
k
k
( )1 1
1
3 .464101615137754587055... 3212
.46416351576125970131...
1
1
1 1
)1(
1
1
kk
k
kk ee
Berndt 6.14.1
.464184360794359436536...
23 6
)1(
k
k
k
1 .46451508680225823041...
2
2 )3()()1()3(4)4()2(9k
k kkk
.46453645613140711824...
1 )4(!249
k kk
ke
24 .46453645613140711824... e9
1 .46459188756152326302... 3/1
.4646018366025516904... 5
4
AMM 101,8 p. 732
.46481394722935684757...
22 1
1
3
4
2
3tanh
3 k kk
=
2
35
2
35
3
3 iii
=
1
)3()13(k
kk
.464954643258938815496... 21
2
3
2
1
2 2 1 2 20
arctan log( )
( )( )
k
kk k k
= log( )
11
2 1 20
xdx
1 .465052383336634877609... 2
2
0
21
1
4 21
sinh
/
i kk
.46520414169536317620...
1
3/1
3!3 kkk
ke
.4653001813290246588...
2
21)(k
k
1 .4653001813290246588...
2
2 )()(k
kk
2 .4653001813290246588...
2
0
2
2
)1(
)(1)(
kk kk
kk
.465370005065473114648... 2 23 3
4 121
12 1
4 31
log( ) ( )
k
k k k
.465461510476142649489...
)32(log
3
13 Discr. Comp. Geom 22:105(1999)
3 .465735902799726547086... 5 2log
.4657596075936404365...
0
002
2!
)1()1()1(
kk
k
k
k
ke
I
e
I
.466099528283328703461... )2()2()2()2(2
11 )1()1( iiiiii
=
1 2
22
21
)1(
13)1)12()(12()1(
k k
k
kk
kkk
.46618725267275587265...
( )( ) ( )
11
2
k
k
k k
k
.466942206924259859983...
1
1
1
1
kk
=
02
11
kk
Prud. 6.2.3.1
1 .466942206924259859983...
1
1
0k
k
.467058656500326469824...
3
1
3
222 LiLi
=
1
111log
11log
k kkk
1 .467078079433975472898... 2
12
)2(2442log3
2log622
, Porter’s constant
.46716002464644797643... 1 2 10
1
arcsinh arcsinh xdx
.467164421397788702939... ( ) ( )
!/2 2 1 1
11
1
2
2k k
k
k
ke
k
k
k
.467401100272339654709... 2
1
142 1
1
2
( )( )k
kk
kk
= ( )( ( ) )k k
kk
1 1
2 12
=( )!
( )!( )
k
k kk
1
2
121 2 1
x xdx xdx2
0
2
2
0
1
cos arcsin/
1 .46740110027233965471... 2
214
12
2 1 1
( )!!
( )!! ( )
k
k k kk
2 .46740110027233965471... 1arcsinlog4
222
i
=
122
2
1 12 )4/1()12(
2
2
)1(
kk kk k
k
k
kk
= ( )( )!
12
2
1
kk
k
k
k
=
0 sinh x
dxx
=
02cos1
sin
x
xdxx Borwein-Devlin, p. 50
= log
( )( )( )
x
x xdx K k dk
1 10 0
1
3 .46740110027233965471... 2
141
2
2 1 1
( )!!
( )!! ( )
k
k k kk
1 .46746220933942715546... E1
4
.467613876007544485395... 27 3
162 2
6
50
loglog
sin
x
xdx
1 .467813645677547003322... 2 11
1
2/k
k
.4678273201195659261... 11
22
1
kk
1 .467891949359644150296...
2 2 1
22
1 420
log
( ) sin( / )
xdx
x x GR 3.552.9
3 .467891949359644150296...
2 2 1
2
1
1 40
log ( )
/
k
k k
1 .467961779578029614731...
1 2
)3(
2
722log8
kk
k
.46804322686252540322...
6
31
18
1
921
coth
kk
J124
3 .468649618760533141431... 1
32 3
3
110
Ik k
k
k
!( )!
.468750000000000000000 = 15
32 5
2
1
kk
k
.46894666977688414590...
3
4
3
8
4
20
cossin x
xdx
.46921632103286066895... H kk k
k
kkk k
1
2 2
11
1 1( ( ) )
( )log
=
23
1
log2
)12()2(
kk kk
k
k
kk
=
1 1
)( 1)1(1n k
kn kH
= kkkkkk
log11
22323
2 .469506314521047562476... 1
1 kk !
32 .46969701133414574548... 4
3
.469981161956636124706... 2
21
1 .47043116414999108828... 2
51
1
2 21
sinh
( )
kk
.4705882352941176 = 8
17
.470699046351326775891... 3
2
9
4
1
2 3
1
1
( )
( / )
k
k k k
1 .470801229745552347742...
3
23
5
4
3
322
coth
kk
= ( ) ( )
1 3 2 11
1
k k
k
k
.4709786279302689616... 1
31k
k k ( )
.471246045438685502741...
15
342
)2/1()5(
8
31)3(
2
21
82 k k
k
.471392596691172030430... ( )( )
( )log arctan
12
2 11
12 2
11
12
1
k
k k
k
k k kk
k
1 .471517764685769286382...
0
14dxe
ex
403 .4715872121057150966... 1
4 2
2 22
0
e ee k
ke e
k
sinh
!
5 .47168118871888519799...
( )
( )
3 1
5 1
1 .471854360049...
12
1)(
)1()(
k kk
kk
8 .472010016955485001797...
1
2
2
12
222cscsinh
2
1
k kk
kk
4 .47213595499957939282... 20 2 5
.472597844658896874619...
Lik
k
kk
3
1
31
1
2
1
2
( )
3 .47273606009117550064...
0
2/3
22
3
4
3
2
3
2
5xx ee
x
=
0 13/2xe
dx
.47279971743743015582... 4 3
27
1
3 12 20
dx
x x( )
1 .47279971743743015582... 2
3
4
9 3
12 11
k
kk
1 .47282823195618529629... ( )( )
( )!sin
12
2 1
1 11
1 1
k
k k
k
k k k
.47290683729585510379... 6 4 1 4 11
2 1 1
1
1
cos sin( )
( )! ( )
k
k k k k
.47304153518359150747... si
x
dx
x
( )sin
1
2
12
1
.473050738552108261... ( )
1
1
1
51
k
k k
1 .47308190605019222027... 3
22
1
2 1 21
2
0
e k
k
k
kkk
kk
! ( )!
GR 1.212
.473095399515560361013...
1
1
1)1(1
2)1(1
2
6log
k
kk
kk
.473406103908474284834...
4 6
3
2
1
5
1
4 4 521
tan
k kk
.47351641474862295879... 7
1440
7 4
16
1
2
4 1
41
( ) k
k k
.473684210526315789 = 9
19
771 .474249826667225190536...
( ) ( )4 1
2744 5
1 .47443420371745989911... 7 3
42 2
42
4 2 1
2 2
21
( )log log
( )
H
kk
k
.47474085555749076227... 2
12
.47479960260022965741... 2 2 4 3 1 32
( ) ( ) ( ) ( ) ( ) ( )k
k
k k
= k
k kk
3
5 42
1
.47480157230323829219... 26
3 3
1
6
2
1
log
kk k
k
k
.4749259869231265718...
8
3
3
4
1
4 1
2
20
hg hgx
xdx
sin
( sin )
.474949379987920650333...
4
12 28/4/5 e , Weierstrass constant
1 .474990335830026928404... 1
1 k kk !
1 .475148949609715735331... 1
2 21k
k k
.475218419203912909153... ( )k
kk 41
.475222538807235387649... e
ek
kk
k
k
3
5 33 2
5 6 5 61
1
1 3
4 3
31 3
3
2
3
21
3
3
/
/
//
/ /
sin cos ( )( )!
1 .4752572671352997213... x dx
e xx
2
20
7 .475453086232465307416...
2
4
)12(
)(
k k
k
.475728091610539588799... Ai
1
2
1 .475773161594552069277... 71 5/
.47591234810986255573... dxxx
1
1)2(
.47597817593456487296...
12
)2(log)(
k k
kk
.4760284515797971427...
( )( )
( )
( )
3
2
1
82 1 1
1
12
1
2
2 32
k kk k
kk k
1 .476208712948717496147... 2 12 2
1
27
4
3
2
1
3 1 9
( )
( / )
k
kk
.476222388197391301...
k
k
ke
II
k
k 2
)!1(
)1(4,2,
2
1F1
)2()2(1
1
1
11210
1 .47625229601045971015... 3
2 2
( )kk
k
1 .476489365728562865499... 3
21
.47665018998609361711...
1
3 2 33
1
4 121
cot
k kk
1 .47665018998609361711... 2
3 2 33
1
322
cot
kk
.4769362762044693977... erf
1 1
2
.477121254719662437295... log10 3 5 .47722557505166113457... 30
.47746482927568600731... 3
2
.47764546494388936239...
sinh
log
=
1
1 )2()1(11log
kk
k
k
kii
28 .477658649975010867721... 2
2
2log
[Ramanujan] Berndt Ch. 22
.477945819212918170827... 8 6 2 7 22 1 2
22
1
log log( )( )
k H
k kk
kk
.478005999517759386434... 3 3
42
1
4
2
2
2
2
1
2
1
2
( )log
i i i i
= ( )
1 1
5 31
k
k k k
.478069959586391157131... G
G1
39 .47841760435743447534... 4 2
.478428963393348247270...
2
22
1)(2
)1(
54
23
4
)3(72log
8
3
kk
k
kk
.47854249283227585569... ( )( )
( )! /
11
2
1
22 1
2
k
kk
k
k
k k e
1 .47854534395739361903... 4
2 11
1
1
2 10
log ( )k
k k k
.47893467779382437553... 4 3 4 ( ) ( )
.47923495811102680807...
1
2
16/1
4/12
)1()1(Im2
22csc
2 k
k
k
.479425538604203000273... sin( )
( )!
1
2
1
2 1 22 10
k
kk k
AS 4.3.65
=
4
1
16120
( )
!( )
k
kk k k
.479528082151010702842... Jk k
kk
k2
1
0
2 2 12
2
( )!( )!
.47962348400112945189...
1
1
)!2(
)1()1(22
k
k
kkci
2 .47966052573232990761... ( )
( )!
/k
k
e
kk
k
k
1
1
2
1
1
.47987116980744146966...
3 3 2 41 3 30
/
dx
x
.480000000000000000000 = 12
25 4
2
1
Fkk
k
.48022960406139837837...
2 0 )(
1)(
k k
k
3 .480230906913262026939...
2 2
1
10
1
log log log
logx
dx
x
GR 4.229.3
.480453013918201424667... log( )
( )2
1 1
22 1
2 11
H
k
H
kk
kk
k k
k
= log( )( )
( )
x x
x
dx
x
1 4
2 20
GR 4.299.1
2 .480453013918201424667... 2 22 1
2
1
log( )
k H
kk
kk
1 .48049223762892856680... ( )2 1
2 12 11
kk
k
.48053380079607358092... ( )
! ( )
1
2 1
1
1
k
k k k Titchmarsh 14.32.3
2 .480548302158708391604... 22 e
.48061832131527820986...
1 )2)(1(1)2()3(3
2
1
k
kk
kkk
HH
1 .480645673830658464567... 2 2 1
11
2
2
2k
k
k
k
k
ke
( )
!/
1 .480716542675186062751... 2 3 1 12
213
2
k
k k
kk
k ( )
6 .48074069840786023097... 42
.48082276126383771416...
13
1
)!2(
!!)1(
5
)3(2
k
k
kk
kk
.480898346962987802453... 1
8log
.481008115373192720896... 2 2 2 2 5 21
22
5
4arctan log log arctan log
=
112
1
0 )12(2
)1(
)22)(12(4
)1(
kk
k
kk
k
kkkk
= log( )
11
2 20
xdx
2 .481061019790762697937... 0
1 11
1( )
! ( )!
k
k jkk jk
.481211825059603447498...
5
cos2log2arccsch=log2
51log
=
1
)5/cos(
k k
k
.48164052105807573135... 2 2 2 4 31
2
2 21
( ) ( )log
( )
x
x xdx
4 .481689070338064822602... ek
k
kk
3 2
0
3
2/
!
.48180838242836517607... 615 1 3
0
1
ee dxx /
.48200328164309555398... ( log )logsin tan
/1 2
22
0
2 x xdx
.482301750646382872131... 2 3 4 22 1
22
1
( ) log( )
H
k kk
k
.483128950540980889382... 24 1
2
2
0
loglog( )
cosh /
x
xdx GR 4.373.2
2 .483197662162246572967... 15
8
3
2
3
2 3
2
1
logk Hk
kk
.483204576426538793772... 1)(1
2 1
kakk a
7 .4833147735478827712... 56 2 14
6 .4834162225162414100... log
( )( )( )
3
2
3
21
k
k km
k m
1 .483522817309552286419... ( ) ( )
( )!sin
1 4 2
2 1
1
02
1
k
k k
k
k k
1 .48355384697159722986...
1
24
2
6
2 3
3
1
2
1
43
2 3
3Li Li( )log log
= 84 4
2
31
2
0
log x
x x
x dx
ex
.484150671381574899151...
2
1
41 2 1 2 ( ) (i i
.48430448328550912049...
1 21 2)!(
)!1(4
kkk
k
Dingle p. 70
1 .4844444656045474898...
0
3
222 log
3
3log
3
3log2
183 xe
xdx
.484473073129684690242... 3
64
.48451065928275476522... 2
1
3 1( )kk
.48482910699568764631... Ei
e
H
kk k
k
( )( )
!
11
1
=
log( )1
0
1 x
edxx
8 .4852813742385702928... 72 6 2
.485371256765386426359... 5
48 2
2
4
1
2
1
2
2 2
2 2
logLi
iLi
i
= log( )
( )
1
1
2
20
x
x x
1 .48539188203274355242...
1 )6(
1122sin
7
245
k kk
.485401942150387923665...
6 3
3
6
1
2 3 13 10
log ( )
( )
k
kk k
Berndt 8.14.1
= dx
x x2 12
.4857896527073049465... ( )
1
1
1
61
k
k k
.48600607487864246273... 2 3
3
1
1
3
20
logx
xdx
= log( )
11
3 120
xdx
=
0 cos2
cosdx
x
x
1 .486276286405273929718... 4 210
Gx x
xdx
logcos
sin GR 3.791.2
= log( sin )10
x dx
GR 4.224.10
4 .48643704003032676232... ( ( ) ( )4) 2 3 13
2 11
k
kk
k
9 .486832980505137996... 90
1 .48685779828884403099...
1
)1)()(k
kk
5 .48691648995407706207... log log24 44 4
1
k
k
kk
Prud. 5.5.1.15
.487212707001281468487... 10 161
2 30
ek kk
k
! ( )
.48726356479168678841...
2 23 33 4 40
/
dx
x
.48749549439936104836...
dxx
4/
0
tan22
223log
22
1 .487577370170670626342... 12 2
1
1 220
csch
( )
/
k
k k
3 .48786197086364633594... 2
22 1
( )
( )
k
kk
.48808338775163574003...
15
364 4 1414 143 4
4 4
/ csc csch
= ( )
1
1442
k
k k
.4882207515238431339...
( )k kk
k 21
.48837592811772608174... 3
42 2
3 3
22
0
1
loglog
log( )x x dx
.48854802693053907803... 2 2 25
8
1
4 1
4
20
4
arctancos
sin
/
x
xdx
1 .48858801763883853368... 2
1 3/
.48859459163446827723... H k
k
k
kk
k k
1
2
2
2
1 1
2 1
( )log
.48883153138652018302... 2 4 2 2 23
42 log log
( )
= ( ) log ( )1 12
0
1
x x
xdx
.488870533723461898816...
0
1
12
21
4
1
k k
k
8 .4889672125599264671... cosh ( )( )!
2 21
2
8
22 2 2 2
0
e ek
k
k
1 .48919224881281710239... 3
5
1 .489343461075086879741... 3
22
22
2
1
2
1
log ( )! ( )
e
Eik k k
k
k
.489461515482517598273... 32 2
2 32 32 2 3
0
( )logx x
edxx
2 .48979428564930390972...
1 !
)1(
k k
k
2 .48985263014562670196... eG
22 .49008835718767688354...
1
32
2)2()1(13 k
kkk
.490150432910047591725... ( )
12
1
41
k
k k
kk
.49023714757471304409... 13
16 12 12 1
2 2
1
log ( )
k
kk
k
Adamchick-Srivastava 2.24
.49035775610023486497...
2
1
2
40
9
5
2
( )
1 .490664735610360715703...
( )32
.49073850974274782075... 1
2 k kk ! ( )
.49087385212340519351... 5
32 12 40
dx
x( )
= sin5
30
x
xdx
GR 3.827.9
= dx
x( sin )5 20
2
= x xdx3
0
1
arcsin GR 4.523.3
.491284396157739513711...
( )
!log
k
k kEi
k k kk k
2 1
1 1 1
1 .491388888888888888888 = 6)2(
3600
5369H
535 .491655524764736503049... e i i2 4
.491691443213327803078... 1
3
12
3
2 3
1
3 20ee
k
k
k
cos( )
( )!
.491714676619541377352... dxe
xdx
x
xx
e x
0 022 1
)/sin(
1
cot
1
.49191653769852668982... 1
6 3 3 1 3
31 3 1 3 2 3 1 3
3
32 ( ) ( )/ / / /
k
kk
5 .491925036856047158345...
( ) ( )3 13
21 1
2
1
k
kk
k
.492504944583995834017...
41
1
2 1
2
0
4
cos
sin
/ x
xdx
43 .49250925534472376576... 16e
1 .492590982680769548804... 2
123
0
ex
xdx
sin( tan ) GR 3.881.2
.492860388528006732501...
4 2
2 21
16
1
821
coth
kk
.492900960560922053576... 2 2 1 2 2 2 1 2 21
2 2 30
arcsinh
log( )( )k
k k
2 .4929299919026930579... 23
2 622 2 2
0
x xdx
ex
log
5 .493061443340548456976... 3log5
.493107418043066689162... SinIntegralk k
k
kk
1
2
1
2 1 2 2 12 10
( )
( )! ( )
.4932828155063024... root of Ei x Ei x( ) ( ) 1
.493414625918785664426... 3
29 2
2
9
4
9 93
1 3
1 3 1 3 1 3
/
/ / /cos cos cos
[Ramanujan] Berndt Ch. 22
.493480220054467930942...
2
22 1)2(
)2(2
)4()2(
20 nT
,
where n has an odd number of prime factors [Ramanujan] Berndt Ch. 5
.493550125985245944499... 1
2
22
2 k
k
kk
log
.4939394022668291491...
43
3
3 21 0
1
156 2
1
( ) log logx
x xdx
x
xdx
= x
e edxx x
3
0 1
6 .4939394022668291491...
43
156 4 1 ( ) ( ) AS 6.4.2
= log3
0
1
1
x
xdx
GR 4.262.2
= x
edxx
3
0 1
Andrews p. 89
.49452203055156817569...
( ) loglog ( ) log ( )
( )2 2 2
2
3
3
4
1
12
3 2
20
1
x
x x
.494795105503626129386... 2 3 12
2321
k
kk
kk
( )
1 .495348781221220541912... 51 4/
.495466871359389861239... log
log
cos 1 2
0
x
dxx
.495488166396944002835... G8
.49559995357145358065...
1
02
2
4
log
2
1,3,
4
1
16
1
x
x
=i
Lii
Lii
2 2 23 3
.49566657469328260843... ( )4 2
4 4 11
2
41
k k
kkk k
=
8 2 2 2coth cot
.495691210504695050508... 1
2 11 ( )kk k
2 .495722117742122406049... 3
3 ( )
5 .495793565063314090328... 6G
.496100426884797122808... 2
125
4 53
31
sin /k
kk
GR 1.443.1
.496460978217119096806... 7
720
1
2
1
41 1 1
41 4 1 4 3 4 / / /csc ( ) csc ( )i
= ( )
1 1
8 41
k
k k k
.49646632594971788014... e
e
dx
x
x
1
4 1
22
20
sin ( )
3 .49651490659152822551...
1
22
2
8
11)3(2
4 k
k
k
H
.496658586741566801990...
16
2 2
2
2
21 1
cosIm
( )
( )
k
k
k
ikk
k
.496729413289805061722...
2 10 2 520
dx
x
.496999822715368986233... 31 6
32
7 4
8
2
2
1
2 4 2 2
( ) ( ) ( )coth tanh
= ( )
1 1
8 61
k
k k k
.49712077818831410991...
1
2
1
2
5
4
1
21 4
/
.497203709819647653589... sin
!
k
kk
kk 20
.497700302470745347474... 2 6 2 21
log log log ( )( )
log
k
k kk
Titchmarsh 1.1.9
=
primep p 21
1log)2(log HW Sec. 17.7
6 .497848411497744790930... 7 3
4
3
82
2
16 6 1
2 2 1
2 2
2
2
31
( )log
( )
( )
k k k k
k kkk k
.498015668118356042714... Im ( ) i
5 .498075457733667127579... ( ) ( )
1 14
2
k
k
k
2 .498215280831227494136...
0 2
)2(
k
k
k
k
.498557794286274894758... 12 2 1
22
1
1eEi
H
kk
kk
k
( ) log ( )!
.498611386672832761564... sin sinh( )
( )!
1
2
1
2
1
4 21
k
k k GR 1.413.1
.49865491180167551221...
3/13/2
3/1
6/113/4 2
3)1()33(
2
3132
32
1Hi
3/1
6/113/4 2
3
32
33H
i
=k
kk 2 330
3 .498861059639376104351... e ek
k
( )
2
1 .499005313803354632702... 1
2 22
kk k
( )
2 .499187287886491926402... eG
.49932101244307520621...
4
224
2
)1)(log
1dss
kk
k
k
.50000000000000000000 = 1
2
= 1
11 ( )! !k kk
=
1
)12()2(k
kk
= k k k
kkk k
( )
( )
2 1
4
4
4 112 2
1
= 1
3 30
2
1k
kk
k
k
= 1 1 1
2 122
312k k
k
k k k kk kk
( / )
= 1
4 121 kk
J397, K Ex. 109d
= 2 14 3
72
k k
k kk
= k k
kk
2
1
1
2
( )!
GR 0.142
= 1
1 3
1
2)0 1k k k k kk k!( )( ) !(
= 2 1
1 1 12 21
k
k kk
( )(( ) )
K 134
= 1 1
33 11
3 12k k k k k kk k
= ( )!!
( )!! ( )( )
2 1
2
4 1
2 1 2 2
2
1
k
k
k
k kk
J390
= 2
2 30
k
k k k k!( )( )
= ( ) ( ) ( ) ( )
1 1 2 2 12 1
k
k k
k k k
= ( ) ( )2
4
2 1
41 1
k k kk
kk
k
= ( )k
kk 2 11
= ( )2 2
4 11
k k
kk
= ( )sin
( )sin
( )sin
1 1 11
1
12
21
3
311
k
k
k k
kk
k
k
k
k
k
k
8 .525161361065414300166... log7!
.525200397399770342589... ( ) ( ) ( )4 3 2 11
5 41
k kk
= 1
21 4 11
1
( ) ( )k
k
k
.525223891622454569896... e
ek
k
k
k
3 2
5 3
5 6 5 63 2
1
1 3
4 3
33
3
2
3
2
3
3
/
//
/
/ //sin cos
( )!
1 .52569812813416969091...
3 2
3 2 1
12
3 1
2 11 2 1
//coth
( )( ) ( )
k k
k
k
.52573111211913360602...
2
2arctansin
5
3csc
2
1
55
2
.525738554967177620202... 1
3
3 .525956365299825031078... ( )
( )!
k
kk
11
2 .52605639716194462560... )5(9316
)3(3
64
2log 24
= 2/
0
3 coslog
dxxx
.526207036980384918178...
( )
( ) ( )
3
3 4
.526315789473684210 = 10
19
.52641224653331... ( )
log
1
2
k
k k k
.52693136530773532050... H
kk
kk 2 4
1
.526949261447891738811... ( )
1
150
k
k k
.526999672990696468620... 5
16
27
5
5
20
logsin
x
xdx
3 .527000471852952829762...
1
1
!
)(
k k
k
.52706165278494586199...
arctan1
1 e
1 .527525231651946668863... 7
3
2 1
70
k
k kk
.527600797260494257440... 1
21 1 1 2 2 1 1
11
( )( cos ) log( cos ) sinsin
( )
k
k kk
.527694767240394589488...
2
42
7
12
1
21 2 2 12
2
1 1
1
coth ( ) ( )
k
kk
k k
k
=
12 32
1
k kk
.527887014709683857297...
4 2 21
1 2
1
1
log( )
( / )
k
k k k
=( )!
( )!( )
k
k kk
1
2
12
121 2
1 .52806857261482370575...
22
1
21log
1)2(22csc2log
kk
k
kk
k
.528320833573718727151... log8 3
.528407192107340935857... 1
3
6 3 3
6
6 3 3
61
1
3
7 6 2 3 7 6 2 3
1 3
/ / / /
/
i i
= ( )
( )
3 1
3
1
3 113
1
k
k kkk k
.528482235314230713618... 24
0
1
ee dxx
.52862543768786425729... 1
21
12
1
SinhIntegralx
dx
x( ) sinh
.52870968946993108979... G
1 .528999560696888418383... 1
2 1 20k
k
/
.529154857165146540082...
4 2 2 4
4140
2
3
2 2
34 3 2
2 1
4
log log( ) log
k
k kkk
.5294117647058823 =9
17
.529416855888971505585...
1 3
1
k kF
2 .52947747207915264818... k
kk
!
!
12
. 52962384375569377748...
0
2
)22)(12(16
1224
kk kkk
kG
5 .529955849000091385901... 121
kk
k
.53001937835233379231...
1
1
1)1()1(
k
k
kk
.53019091763558444752... ( ) ( ) ( )31
2
1
22 2 i i
= 1 1
21 2 3 15 3
1
1
1k kk
k
k
k
( ) ( )
.53026303220872523716... 1 1
22 k
k
kk
.530277182813685398637... 1
21 ( )!k kk
1 .530688394225490738618...
22
2
2
322
2 3 1 1 14 3 1
1
( ) ( ) ( )( )
k
k k
k kk k
k k
1 .53091503904237871421... 1
2 11 k kk ! ( )
Titchmarsh 14.32.3
9488 .531016070574007128576... 8
.531249534072353755050... dxx
0
2 16
11log
6
1
3
1
.531250000000000000000 =
1
7
)1(32
17
kkke
k Prud. 5.3.1.1
.531275083024229991483... ( )( )
log ( )
1 21
1
k
k
k
kk
.531463605386615672817... ek
k ee
kk
1 1
1
/
!
2 .531895752688993200213... 3
2 2 1 3
3 4
40
/
/
dx
x
.531972647421808261859... 82
36
2 1
12 1 20
k
k k kkk ( )( )
.53202116874184233687...
2
3
2
20
1
12
1
2
1
3
3
83
1 3
Lix
x xdx( )
log
( ) ( )
.532108050640355849704... 41
22 1 2
1
1 4
1
1
log( )
/
k
k k
2 .532131755504016671197...
1
1
sin0 )1(2 dxeI x
.532598899727660345291... 34
3
2
2
21
k kk ( )
.53283997535355202357... 2 1 2 20
4
log cos log(sin )/
x x dx
46 .53291948743789139550... 7 1
3
3
1
3 2
1
e k
k
k
k
( )!
.533290128624160141531...
3
16
1
61 3
3 3
21 3
3 3
2
2
( ) ( )i
ii
i
= 1
21 3 2 1
11
15 2
1
( ) ( )k
k k
kk k
1 .533463799145879760767... ( )k
k
1
2
1
6 .533473364460804592338... 62 1623
4
1
ek
k kk
!( )
1 .53348137319375099599... 2
)3(322 3
Li
=log
( )( )
2
1 1 2
x
x xdx
1 .533582691060658478950... 1
0
00 arccos1)1()1(2
dxxeLI x
1 .53362629276374222515... e
.5338450130794810532... 2 1
2
( )
( )
k
kk
.53387676440805699500...
1 2
12sin
2
1
2
3sin2
2
1sin
51cos4
1
kk
k
1 .53388564147438066683...
1
)(
1 2
2
12
|)(|
kk
k
kk
k
7 .533941598797611904699... 1
8
.534195237508799281491...
1 2
)1(1
k
k
kk
k
3 .534291735288517393270... 9
8
.534304386409498279676... 16
3
4
31
41
1
log
Hkk
k
.53464318757261872264... dxx
x
kkkk
12
12 12
log
)144(2
1
2
1,2,
2
1
8
1
1 .534680913814755890897... 1
2 44 1 4
0
e ek
ke e
kk
/ / cosh
!
.534799996739570370524... logcos
sin
/
11
2 10
4
x
xdx
.535034887349803758539...
1
4
1
2
1
2
1
2
1
2
i i i i
= 1
21 4 1 1
11
15
1
( ) ( )k
k k
kk k
.535112163829819510520... sinsin
coshcos
sinhcos
sinsin(cos ) /1
2
1
2
1
2
1
21 2
e
= sin
!
k
k kk 21
GR 1.449.1
.5351307235543852976... 3
31
1
3 21
sin
kk
.535216927240908236283... 1
12 k kk !( ! )
1 .53537050883625298503...
1
421
4
2
1 2 1
1
1
15
15
1
kkk
kk
k
k kF
1 .535390236492927618733... ( )31
3
.5355608832923521188... Ai( )1
.535898384862245412945... 213)32(2 CFG D1
.535962843190222987031... 1
21 5 1
1
11
15
1
( ) ( )k
k k
kk
.5360774649700956698... ( )2 1
2
1
2 12
0
dx
ex
.536119444744722773227...
e
1 .53642191914104435977...
12 24
112csc
3
2
k kk
.536441086350966020557... 1
21 6 1 1
11
16
1
( ) ( )k
k k
kk
k
.5365207790738756208... 1
3 1 322
11 ( )( )
k kkk k
k
.536526945921177100962...
2
)(log dxx
.536683562016736660896... 1
21 7 2 1
11
1
2
71
( ) ( )k
k k
kk
k
.536784534109508703918... 1
31 1
1 3
21 1
1 3
22 3 1 3( ( ) ) ( ( ) )/ /
i i
=
1 2
21
)1(
1)13()13()1(
k k
k
kkk
kkk
.53685577365418220547... ( )jkkj
2 111
.536921592069942010965... k
kk 4 11
.536999903377236213702... 1
2 2
2
21
sinh
Re ( ) i
.537213193608040200941... 7 3
8
2
6
2
12
1
2
1
2
3 2
3 31
( ) log log
Likk
k
13
)2( 1
k
k
k
H
=log( / ) log1 2
0
1
x x
xdx
1 .53739220806790406178...
0
3/1
)!3(2
3cos
3
2
3
1 2/)3/1(3/1
k
k
kee
.538011176205005048612...
1 )48(5
1
2
51log
2
5
kk k
=
1
12
1
2!)!12(
!)!22()1(
2
1
kk
k
k
k J141
.5381869343911341187... 34/1
)3( )3(2764 H
1 .538395665719128167672... sin!
1
1 kk
.538461538461538461 =7
13
1 .538477802727944253157... 2 2 2 2coslog i i
.539114847327011158805... 1
8
1
4
1
44
1
2i
i i itanh tan csc
1
84
1
2
1
4
1
4i
i ii
icsch cot coth
= e x
edx
x
x
cos
1 2
.53935260118837935667...
( / )
( / ) ( / )
( )1 2
5 4 3 41
1
21
k
k k J1028
3 .539356459551305228672... 1
31
4 2 3 6
21
4 2 3 6
22 31 3
1 3 5 6 1 32 3
1 3 5 6 1 3
//
/ / //
/ / /
( ) ( )
i i
2
32 6
1 3
2 31 3
/
//
= 6 3 16
613
2
k
k k
kk
k ( )
9 .5393920141694564915... 91 8 .539734222673567065464... e Not known to be transcendental
.540235927524947682724... 3 3 1log
.540302305868139717401... cos cosh Re( )
( )!1
1
20
i ek
ik
k
AS 4.3.66, LY 6.110
.540345545918092079459... 4 4
1
2cosh
.540553369576224517937... 2
21 28
21
2
4 1
2 1
1
2
log( )
( ) ( )k
k k
k k
k
k
kk
.540722946704618254025... sin(cos sin ) coshcos
sinhcos
1 12
2
2
2
= ( ) sin
!
1 2
2
1
1
k
kk
k
k
6 .540737725975564600708... 37
4 2
2
8
3
0
k
k
kk
k
5 .540829024068169345604... ( ( ))3
.540946264299396859964... G7
8 .54097291002346216562... 3 2 3 31
3
30
1
( ) ( )log
( )
x
x
45 .54097747674216511051... 13 2 10 20 1
6
21
I Ik
kk
( ) ( )( !)
.541161616855569095758... 4 2
0
1
180
1
log( ) logx x
x
.5411961001461969844...
1 24
)1(1
8sin
8cos
2
22
k
k
k
J1029
.54123573432867053015... dx
x( )0
1
.54132485461291810898...
1 !
)1()1log(
k
kk
kk
Be [Ramanjuan] Berndt Ch. 5
.541608842204663463822... B
kk
k ( !)20
1 .541691468254016048742... 1
20k
k !
1 .541810518781157499421... log ( )( )3
2 6 3
4
81 3
1
3
2
3
26 3
27
31
= k k k k
k kkk k
2
2
2
313
36 9 1
3 3 1
( )
(( )
2 .541879647671606498398... 2 1
20
22
83
4
1
3 4
1
4
Gk
k k
kk
k
( )
( / )
( ) ( )
.542029902798836695773... 2
cosh
178 .542129333754749704054... 112 96 21 2 3
21
log( ( )( )H k k kk
kk
.542720820636303500935... 1
2
1
2
1
2 1 21
sinh( )!
k k
k
1 .54305583321647144517... p
pp prime !
.543080530983284218602...
2 4 2 8 2 2
2 2
coth
i i i
i i i
81
21
21 1 ( ) ( )
= H
kk
k 4 121
.543080634815243778478... 2
11
1 1sinh
)!2(
1
2
211cosh
x
dx
xk
ee
k
1 .54308063481524377848...
0
1
)!2(
1
2cos1cosh
k k
eei GR 0.245.5
= 14
2 12 20
( )kk
J1079
.543195529534402361791... 1
21 1 1 12log( ) log( )(sin cos ) e e e ii i i
= sin k
kk
11
.54323539510879497886...
1
)2(log)(
k k
kk
.543258965342976706953...
6
1
6
5
3
1 , Landau constant
.543383238748395175193...
1
22
)32)(1(2log22log4
64
k
k
kk
H
8 .5440037453175311679... 73
.544058109964266325950...
2tanh
2coth22
sinh
2
ii
.54425500107911929426...
1
)2(log
k k
k
.54439652257590053263... log
log sin( )/2
44
0
4
x dx
=
02
sinlogsin1
sindxx
x
x
=
4/
0 2coscottan
2
x
dxxxx Prud. 2.5.29.28
.54445873964813266061... ( ) ( )k kk
2 11
1 .545177444479562475338... 8 2 41
1 421
log( / )
k kk
.545454545454545454545...
1 411
6
kkkL
.5456413607650470421... dxe
eerf
ex
x
1
4/1
2
2
11
2
.54588182553060244232... 7
609 3
1 1
4 4
31
4
30
1
( )log
( )
log
( )
x
xdx
x x
xdx
1 .5459306102231431605...
2
1
22 2 2 32
0
arctandx
x x
.546250624110635744578...
21 1)()1(
2
5tan
5 kk
k kF
=
2 52512
)51(2
k kkk
1 .546250624110635744578...
21 1)(
2
5tan
51
kk kF
=
12
2 52512
)51(2
1
1
kk kkkkk
.54716757473605860234... 1 3 220
1
log log( )
/e
e
e edx
x
x
.54770133168797308275...
2
21
4
1
2 2
1
4
1
2 2
log
i i
=
02 1)12()12(
1
k kk Prud. 5.1.26.8
1 .54798240215774230466...
1
0
1
0
2
2
2 arctan4
1
arccos
2
)3(72 dx
x
x
x
dxxG
= 2/
0
2
sin
dxx
x
.54831135561607547882... 2
118
1 1
2
( )!( )!
( )!
k k
kk
= 2
1arcsin2 2
= dxx
x
0
3)1log(
11 .548739357257748378... i isin( log )2
.54886543055424064622...
1
21
12 )!12(
)!()1(
4
1,
2
3,2,2
k
k
k
kF
.54891491425246963638...
1 )12(2
1
kkk k
.54926923395857905326... log( )
( )
( )
( )2 1
2
0
01
1
2
1
1
Titchmarsh 2.12.8
.549306144334054845697... log
( )
3
2
1
2
1
2 2 12 10
arctanh kk k
AS 4.5.64, J941
= dx
x x
dx
x
dx
ex( )( )
1 3 1 202
2 0
=
12)2(
log
x
x
.549428148719873179229... 1
41 1 2 21 4 3 4csc ( ) csc ( ) cos( cosh(/ / (
( ) sin ( ) ( ) sin ( )/ / / / )1 1 1 13 4 1 4 1 4 3 4
= ( )
1
140
k
k k
7 .5498344352707496972... 57
5060 .549875237639470468574... 8
157! 8 7
1
81
7
0
1
( ) ( )log( ) x
xdx GR 4.266.2
2 .550166236549955... ( )/
1 11
31
kk
k
e
k
1 .550313834014991008774... 3
20
6 .550464382223436608725... 6
G
.550510257216821901803... 3 6
.550521946799569348190...
2
1)()1(k
kk
1 .550546096730430440287... 6
6
1
2 12
( )
.55066059449645688395... 1
2
1
2
12
22 2
0
coth( )kk
.550828461280021051067...
1
2
2
)(
kk
k
2 .551419182892557505235... )2()2()1(23636
4222
=
0 1
logdx
e
xxx
s
.551441129543566415517... 4 2
2
1
1 12 2e e sinh sinh cosh
J132, J713
.55154451810789954008...
2
5tan51252
10
12/)53(
H
=
21 1)(
kkk kFF
.5516151617923785687...
1
2
)!1(6
13
k k
ke
.5516932... primep p 1
12
.5523689696799021586... 8
1 11
4 31
erf erfix
dx
x
cosh
.55242828301478027042...
133
)1(dx
xx
x
1 .55260145083523513225...
1 )5(
11
2
21cos
8
k kk
.5527568077303040748... sin
11
21 kk
.55278640450004206072... 11
5
1 21
21
( ) ( )!
( !)
k
k
k
k
.55285569203859119314... 2 2 2 11 2
2 10
sin cos( )
( )!( )
k k
k k k
.553303299720515717371...
1 )12(2
12log1arcsinh2
kk kk
2 .553310333104003190087... 0
2 1
( )
!
k
kk
.5535743588970452515... arctan 2
2 2 520
dx
x x
.554204472098099291561... i i i i i
8
2
4
2
4
4
4
4
41 1 1 1 ( ) ( ) ( ) ( )
= ( )
( / )
1
1 4
1
2 21
k
k
k
k
4 .5544893692544693575... 2 1
21
k
k kkk
!!
1 .55484419730133345731...
1
2)12(
21
2cosh
3
1
k k
1 .555290108843124661542...
12
k k
k
kF
H
.55536036726979578088... dxxxx
dx
2/
0
3
02
sincos824
.5553968826533496289... 1
2
1 1
2 0
1
sin cos cos
e
xdx
ex
=coslog x
xdx
e
21
.55582269181411744686... 1
0
sin00 )1()1( dxeStruveI x
.5560460564528891354... 2
6
2
3
2
6
2 2
2 21
124
0
1
loglog logx
xdx
.55663264611852264179... 1
2 4 4Li e Li ei i( ) ( )
= ( )cos
1 14
1
k
k
k
k GR 1.443.6
.55683841210516333558...
2
5tan252
5
12/)53(
H
=
211 1)(
kkk kFF
.55730495911103659264... 21
2
1
2 1 1 21 21
log / /k
kk
1 .557407724654902230507...
0
2
)2(
)41(4)1(1tan
k
kkkk
k
B
6 .55743852430200065234... 43
.5579921445238905154... H k
kk
2
1 4
4 2 3
3
log log
.558110626551247253717... 1
6 6loglog e J153
.5582373008332086382...
132k
kk
k
H
1 .558414187408610494023... 1 2 2 22
2 10
cosh sinh( )!( )
k
k k k
.558542510602814021154... ( ) logk k
kk
1
21
1 .558670436425389042810... log
( )! ( )!
log
!
k
k n
k
kk n
n
k
1
1
12 1 2
.559134144418979917488... Jk
k
k0 221
2( )
( )
( !)
162 .559234176474304999069... 2222
3
1
ek
k
k
k
!
.559334724804927427919...
22
2 )1(
)(1
)1(
)(
kk kk
k
k
k
.55940740534257614454...
1 )2(!!
1
2
1
22
5
k kkerf
ee
.559615787935422686271...
1
1
11 4
3)1(
4
3
7
3
4
7log
kk
kk
kLiLi
3 .55989038838791143471... 2739 3
2
1
1 3
1
1 3
1
3
1
31
( ) ( )
( / )
( )
( / )
k k
k k k
.560000000000000000000 = 14
251
21
2
1
2
0
( )sink k
kk
x
k F x x
edx
.560045520492075839181... 1
2 2 11 k kkk ! ( )
Titchmarsh 14.32.3
.56012607792794894497...
1 3
11
kk
=
12/)3(2/)3( 22
3
1
3
1)1(1
kkkkk
k Hall Thm. 4.1.3
.560329854485890998637...
1 2
)1(1
k
k
k
k
.560511825774805757653... 9 121
3 1 21 3
0
ek k kk
k
/
! ( ( )
.560744611093552095664... 10
34 2 1
1
2
2
2 122 2
log ( )( )
( )k
kk k
k
k k
1 .560910575045920426397... 0
21
( )
( !)
k
kk
.561061199700803776228...
13
)2(3
)3/1(
127
3
1
2
127)3(13
33
2
k k
27 .56106119970080377623...
1
2
1
3
121
33
0
( )
( )kk
1 .561257911504962567051... 4 3 3 4 ( ) ( )
.56145948356688516982...
0 !
)1(
k
kk
ke
Euler-Mascheroni constant
= k
k
ek
/1
1
11
KGP ex. 6.69
=n
nn loglog)(lim
HW Thm. 328
=
nprimepn pn
11loglim Mertens
23 .5614740840256044961...
4 2 22 4
0
3
208 3
( )log xdx
ex
.56173205239479373631... 1
22 12
2
1
e ek
kk
cos cos(sin )cos
!
.561781717266978021561...
32 1
11
2
5sec
6 k kk
.56227926848469324079... dxx
xdx
x
x
2/
0
31
0
33
tan
arcsin
16
)3(9
8
2log
.562500000000000000000 = 9
16
= k kk
k kk k
( )( )
2 1 12 1
12
2
2 22
.56256294011622259263... 1
22 2 1
1 1
1
log cos( ) cos
k
k
k
k
.562610833137088244956... ( ) ( )2 4
.56264005857240015056...
12 14
11
2sin
2
1
k k
J1064
1 .562796019827922754022... ( ) ( )
1 2 11 2
1
k
k
k
4 .5628229009806368496... 2 23
2 23 2 2
22
1
log log( )kH k
kk
5 .5632758932916066715... pd k
kk
( )
21
.5634363430819095293...
0 0 )3(!)4(!
126
k k kk
k
kke
= 1
1 30 ( )! !k kk
8 .5636594207477353768... 1
4
1
6
2
3
11 11
32
0
( ) ( ) ( )
( )
k
k k
.563812982603190392613... 1
2
1
2cosh
1 .56386096544105634313... 6 2 6 43
3
3
8
6
1
2
42
( ) ( )
( )
( )
k
kk
= ( ) ( ) ( )( ( ) )
1 1 1 12
k
k
k k k k
.56418958354775628695... 1
2120
k
k kk ( )!
10081 .56420457498488257411... 17
16
8 7
0
x
xdx
sinh GR 3.523.10
2 .564376988688260377856...
3
1
22
1
4 Berndt 8.6.1
.564599706384424320593...
2 2
11log
1
1
1)()1(
k k
k
kkk
k
.56469423393426890215... sin k
k
kk 21
18 .564802414575552598704... erfik k
k
k
22 2
2 1
2 1
0
!( )
1 .56493401856701153794... 11
31
31 1
kk
kk
Q k( )
1 .564940517815879282638...
2
2
0
tanhsin
sinh
x
xdx
1 .565084580073287316585... 61 4/
.565098600639974693306... ee
ee
ee
11
11
11cosh cosh int sinh int
1
2
12
1
2
12
1
2
12e
ee
ee
ecoshint sinhint log
= H
kk
k ( )!21
.565159103992485027208... I1 1( )
.565446085479077837537...
1222
2 4/cos
2
1
2
1
2
1
192
11
k k
kiLi
iLi
.5655658753012601556... iii eLii
eLieLi 3
233 1
122
1
=
1
31 cos
)1(k
k
k
k
.565623554314631616419... 2 24 1
2
20
4
arctancos
sin
/
x
xdx
1 .565625835315743374058...
cos cosh( )
( )!2 2
1 64
4
1
0
k k
k k
2 .565625835315743374058... 1 2 2 12
41
6
1
cos cosh ( )( )!
kk
k k GR 1.413.2
2 .565761892331577699921... e e eEi Eik H
kk
k
1 1 1 11
2
1
( ) ( ) ( )( )!
1 .56586036972271770520...
2
33
3
9
3
9
2
3
1
33
2 3
3Li Li( )log log
= 83 3
2
31
2
0
log x
x x
x dx
ex
. 56588424210451674940...
0
2
)2)(1(16
12
9
16
kk kkk
k
.565959973761085005603... 2 2 2 12
11
21
Jk
kk
k
k
( )( !)
.565985876838710482162... 8
2
4
1
4 1 4 20
log
( )( )k kk
1 .566082929756350537292... Ik k
k0 2
0
21
2( )
( !)
.566213645893642941755... ( )k
kk 31
5 .566316001780235204250... 1
6
12 .566370614359172953851... 4
.5665644010044528364... e Ei Eix
edxx
21
0
1
2 1 21
( ( ) ( )) loglog( )
9 .5667536626468872669... sinh
2
21
22
1
kk
2 .566766565764270426298... k
kk
!
( !!)30
.566797099699373515726... log ( ) ( )( )15
8
3
21 1
1 3
22
k
k
k
k
k Srivastava
J. Math. Anal. & Applics. 134 (1988) 129-140
.566911504941009405083... arccot2
.567209351351013708132... 3 62
3
1
3 1 20
log( )( )k
k k k
.567384114877028322541... 1
21 ( )k kk
1 .567468255774053074863... ( )e
.567514220947867313537... log( )
log
k
k k kkk k2
12 1
1
212 1
.567667641618306345947... 1
2
1
21
2
121
1
e k
kk
k
( )( )!
5 .56776436283002192211... 31
.567783854352055695476... 2 32
3
4 2
3
2 1
4
2 3
31
( )log log
k
k kkk
.568074442278171...
2 )32)(1(
log)1(
k
k
kk
k
.568260019379645262338...
2
4 216 2
1
2
1
cothk kk
= 1
21 2 2 1
11
12
1
( ) ) Im( )
k
k k
kk k i
5 .568327996831707845285... 3 2/
.568389930078412320021... 1
2 2
1
22 12
( )kk
kk
jkj
.568417037461477055706... 3
41
3
2
2 3
0
1erf
ee dxx
/
.568656627048287950986...
0
12
020 12
)1(4
!)!12(
)1(2)1(
k
k
k
k
k
J
kH
.568685634605209962010... ( )( )
12
1
1
kk
k
k
1 .56903485300374228508... e
.569356862306854203563... 1
31
6 3 3
61
6 3 3
64 31 3
7 6 2 32 3
7 6 2 3
//
/ //
/ /
( ) ( )
i i
1
31
1
34 3 1 3/ /
= ( )3
3
1
3 113
1
k
kkk k
.569451751594934318891... 4 2 5 3 ( ) ( )
1 .56956274263568157387... 1)2()(sinh1
kkk
.5699609930945328064...
( )
( ) ( )
log log2
2
1
2 122
2
k
k
p
pk p prime
Berndt Ch. 5
=
12
)(
k k
k
.57015142052158602873... Eik kk
k
1
22
1
21
log!
.57025949722570976065...
1 33
1arctanh
2
3
kk
koH
.570490216051450095228... coshcos sinhcos cos sinsin sin (coshcos sinhcos cossin )1 1 1 1 1 1 1 1
= sin
( )!
k
kk
11
2 .57056955072903255045...
1
2
2
)(
kk
k
.57079632679489661923... 2
12
1
sin k
kk
= 2 1
4 2 1
2 1
2 2 11 1
k
k k
k
k kkk
k
( )
( )!!
( )! ( ) J388
= ( )
/
1
2 1 2
1
21
k
k k
= arccos
( )
x
xdx
1 20
1
= tanh x
edxx
0
= xdx
x x( ) sinh /1 220
GR 3.522.7
= x x dx
x
log( )1
1 20
GR 4.292.2
1 .57079632679489661923... 2
i ilog
= ( )
/
1
1 20
k
k k
=
122
21
02
4/1
)1(
4/1)12(
1
k
k
k k
k
k
= k
kk
!
( )!!2 10
K144
= 2
22
2 1
2 1
4 2 11
2
0 0
k
k
k
k kkk
kk
k
k
k
k k
( !)
( )! ( )
= ( )!!
( )!!
2
2 1 21
k
k k kk
=
1
)()(k
kk iLiiLi
=
0
2)12(
2arctan
k k [Ramanujan] Berndt Ch. 2
=
12)512(
1arctan
k k [Ramanujan] Berndt Ch. 2
= 4
2 1 2 1
2
1
k
k kk ( )( )
Wallis’s product
= dx
x
dx
x
dx
e ex x20
2 201 1 2 2
( )
= dx
x x20 1 2
/
= sin sinx
xdx
x
xdx
0
2
20
= x
xxdx
2
2 20
1
1
( )log GR 4.234.4
= cos/
2
0
2
x dx
GR 3.631.20
=
0
cot1
x
dxx
x Prud. 2.5.29.10
= arccosh x
x21
= log( tan )/
e x dx0
2
GR 4.227.3
= 1
2 20 e ex x
= xx
dxlog 11
40
= ci x xdx( ) log0
GR 6.264
2 .570796326794896619231... 2
1221
k
k k
k
3 .570796326794896619231... 2
2220
k
k k
k
.571428571428571428 =
primep pp
p42
4
1
1
7
4
1 .57163412158236360955...
2 4
2/3212/
1
k k
kkk
.57171288872391284966...
0
1
13
13log
2
1
12212
)1(
k
k
k
.571791629712009706036...
1
2
1
3
1
3
2 1
31
( )kk
k
= 1
3 121 k kk ( )
3 .571815532109087142149...
2
3 1)12(128
111
4
)3(9
k
kk
.572303143311902634417... 4
91
4
3 41
logkHk
kk
.572364942924700087071...
2
1log
2
log
=
e
e
dx
x
x
x
2
0 2
1
1 GR 3.427.5
.572467033424113218236...
2
112
1
42 2 1
k k kk
( ) ( )
= 1
2 2 2Li e Li ei i( ) ( )
=
223 1k kkk
k
= ( )( ) ( )
11
21
2
k
k
kk k
= ( )cos
1 12
1
k
k
k
k
1 .57246703342411321824...
12
2
14
3
12 k
k
k
H
.57289839112388295085... 4 2
8
4 2
20
sin x
xdx
.57393989404675551338... 3 3
23 2
( )( ) ( )
2 .57408909729511984405... 11
21
( !)kk
.574137740053329817241... 2 2
216
1
2
cos k
kk
.574859404114557591023... 1 2
3
1
31 1
11 3 2 34
1
( ) ( )/ /
k kk
= 1
21 3 1 11
1
( ) ( )k
k
k
5 .57494152476088062397... e ee k
k
1 1
1
/ !
.575222039230620284612... 10 3
.57525421205443264457... x x
xdx
sin
cos
/
2 20
2
.57527071983288752368... 2
23
30
1
43 2
3 3
8
1
log
( ) log ( )x
xdx
1 .575317162084868575203... 16 12 31
2
0
1
( )
log x
xdx
.575563616497977704066... 12
1
2
1
21 40
e
erfik
k
k
k/
( ) !
( )!
1 .57570459714985838481... log
3
4
.575951309944557165803...
21 1
2
1
21 1
41 1 ( ) ( ) ( ) ( )( ) ( )i i
ii i
= ( ) ( ) ( )
1 2 12
k
k
k k k
.576247064560645225676... 3
30
4
321
2 1
2 1
( )cos( )
( )k
k
k
k Davis 3.37
.576674047468581174134...
21 1 2 11
1
coth ( ) ( )
k
k
k
= 1
14 2 42
2 1kk k
k k
( ) ( ) J124
= sinx
e ex x
10
.576724807756873387202... Jk k
k
k
k
k
k
k1
0
1
20
21
1
1( )
( )
!( )!
( )
( !)
.57694642787663931446...
1 )4(
11
5
5sin6
k kk
.57721566490153286061... (Euler's constant, not known to be irrational)
= kHkk
loglim
Euler
=
n
kn
kkk1
))1(log(2
11lim Cesaró
=
n
kn
nk1
4log12
1lim2
=
3
1log
2
1lim 2 nnHnn
=
45
1
2
1log
4
1lim
22 nnHn
n
=
nn
n
1lim Demys
=
2
11log
11
k kk Euler
=
2
log)1(
2log
1
2
2log
k
k
k
k
=
1 )12(4
1)12(
2
3log1
kk k
k
=
2 2
11
k jjjk
= ( ) ( )
log
1 11
1
2 1
k
k k
k
k k k
=
2
1)(1
k k
k Euler
= k
kk
k kk k
11 1
1 1
12 2
( ) log Euler (1769)
=
1 12
1)12(
2
2log1
k k
k Euler
=
1 )12(4
1)12(
2
3log1
kk k
k Euler-Stieltjes
=
1 )12)(1(
)12(1
k kk
k Glaisher
=
1 )12)(1(
1)12(2log22
k kk
k Glaisher
=
2
1)()1(2log1
k
k
k
k
=
2
1)()1()1(2log
2
3
k
k
k
kk
Flajolet-Vardi
=
3
1)()2()1(
2
12log
4
5
k
k
k
kk
=
2 1
1)()1(238log
k
k
k
k
=
2 2
)()1(2
4log
kk
k
k
k
=
2 2
1)()1(2
9
16log1
kk
k
k
k
=
2 2
11)(
12log
2
3
kk
kk
=
2 3
1
2
11)(
13log
6
11
kkk
kk
=
2
),(log
ks k
sksH
= pprimeaofpoweraiskifpkk
k
k
log)(,1)(
2
1
2
= 1
11
21 n
k
k k
n
kn !
log
( )
Shamos
=
1
12
2
1
)1(
k j
jk
k jk Vacca, Franklin
=
k
k
k
k 2
1
log)1(
Vacca
=
0
logdx
e
xx
= dxx
xx
1
21
= log log1
0
1
xdx Andrews, p. 87
= log sinx x dx0
2
GR 4.381.3
= 1 2
0
2
sec cos(tan( )tan
/
x xdx
x
GR 3.716.12
=
0
log)1( dxxe x
= 1
0
)( dxxH
=
xe dxx e x
Prud. 2.3.18.6
= 1
0
/11dx
x
ee xx
Barnes
=
0
log4
2log22
dxxe x
= 1
22
1 12 20
xdx
x e x( )( ) Andrews, p. 87
= ( )/1 1
0
1
e edx
xx x Andrews, p. 88
=
1
0 )1log(
11dx
xx
= 1
1
1
0
e xe dxx
x GR 3.427.2
= 1
10
xe
dx
xx GR 3.435.3
= 1
10
xx
dx
xcos GR 3.783.2
= 1
1 20
xx
dx
xcos GR 3.783.2
= 0,logcoscos1
0
xxdxx
xdx
x
x
x
x
= 0,log1
0
xxdxx
edx
x
e
x
xx x
= babadxx
ee
ba
abba xx
,0,0,0
= dxxx k
k
1
0 1
2
1
11 Catalan
=
022 )1)(1(
22
1
xe
dxxx Hermite
=
02222 ))(1(
2lognxe
dxxnH
xn Hermite
1 .57721566490153286061...
10
si x x dx( ) log GR 6.264.1
= Ei x x dx( ) log
0
GR 6.234
.577350269189625764509... 3
3 6 tan
= 4
3
1
2 1 1 921 ( ) /kk
GR 1.421
= 2 1
6 1 20
k
k k kkk
( )( )
.577779867999970432229...
( )
( ) ( )
2
2 3
.577863674895460858955...
1
)1( 1
21
2 k
kk
ke
= dxx
x
021
cos AS 4.3.146
=
022 )1(
cosdx
x
x
= dxx
xx
021
sin
= cos(tan )cos/
x x dx2
0
2
GR 3.716.5
= cos(tan )sin
xx
xdx
0
GR 3.881.4
=
1
021
1coscos
x
dx
xx Prud. 2.5.29.11
= x x
xdx
3
2 21
sin
( )
Marsden p. 259
.5781221858069403989... 27 3
6 6
3
2
1
3
2
3 21
log
k kk
1 .57815429172342929876...
02
22
)(
log
3
3
ex
x
e
.578169737606879079622... ( )
( )
2
2
1
4
12
12 2
1
k
kLi
kk k
.578477579667136838318...
(sin sinh )
(cosh cos )
2 2
2 2 2 2
1
2
1
141
kk
= 1
21 4 1
11
12 2
1
( ) ( ) Im( )
k
k k
kk k i
2 .578733971888556748934... 3
2
( )
!
k
kk
4 .57891954339760069657... 4 log
.578947368421052631 = 11
19
.57911600484428752980...
2 4 2
12
412 24
7 3
( )( )
k
kk
1 .57913670417429737901... 4
25
2
=1
5 1
1
5 2
1
5 3
1
5 42 2 2 21 ( ) ( ) ( ) ( )k k k kk
6 .57925121201010099506... log !6
3 .579441541679835928252... 3
21 2 2
3
21
9
4 6
2
22
2
log ( ) kk kk k
.579480494370491801315... 1
12 k kk ( ! )
.5795933814359071842... 1
3 1 31
11 ( )( )
k kkk k
k
4 .57973626739290574589... 2
32
2
1
2
20
2
e e
ex dx
x x
x( ) GR 3.424.5
6 .57973626739290574589... 2
3 1
22
20
log( )
xdx
x GR 4.261.5
.57975438341923839722...
3
22Li
4 .579827970886095075273... 5G
2 .58013558949369127019... ( ( ( ( ))))2
.580374238609371307856... k
kk
k
k
k5
1 11
1
21 5 1 1
( ) ( )!
5 .580400372508844443633... e
k
k
kk
1
.58043769548440223673... arctan(tanh ) ( ) arctan4
11
2 11
0
k
k k
[Ramanujan] Berndt Ch. 2
.580551791621945988898... ( )( )
13 1
1
1
kk
k
k
6 .580885991017920970852... 2 21
0
e k
k
/ !
.58089172936454176272... 3
43
1
3 1 33
2
0
1
( )log
( )( )
Lix
x xdx
.58105827123019180794...
2 )1(2
11
2
3cos
2
k kk
.58106146679532725822...
2
1)(1
22log3
2
1
k
k
kk
.581322523042366933938...
036/13 33
2
x
dx
.581343706060247836785...
coth2
322cot1cot
66/1 i
ii
= k
kk
k
k
k
2
61
1
11
1
21 6 2 1
( ) ( )
.581824016936632859971... 29
64 4
1 4 3
1
e k
k kk
/
!
.581832832827806506555... k
kk
k
k
k
3
71
1
11
1
21 7 3 1
( ) ( )
.5819767068693264244...
12/1
1 /112
11
1
1
kk
kk k
eee
[Ramanujan] Berndt Ch. 31
=
1 !
)1(
k
kk
k
B
=
02
11
kk
e Prud. 6.2.3.1
= dxex
x
12)1(
log
2 .581931792882360146140... dxx
02 2
31log25
.581954706951074968255...
dxx
02 15
11log
5
12
1 .581976706869326424385... e
e e
B
kkk
kk
k
1
1 1
0 0
( )
!
3 .582168726084073272274...
1 !2
5cosh22
k
k
k
Le
.5822405264650125059...
122
22
2
1
2
1
2
2log
12 kk k
Li
J362
= ( )
1 1
1
k k
k
H
k J116
=
12
)2( 1
k
k
k
H MIS
= log( )
( )
log1
10
1
21
2
x
x xdx
x
x xdx GR 4.291.12
= logx
x
dx
x
12
=
log( )/ 1
0
1 2 x
xdx GR 4.291.3
=
log 1
20
1 x dx
x GR 4.291.4
=
log
1
2 10
1 x dx
x GR 4.291.5
4 .58257569495584000659... 21
1 .58283662444715458907...
12 14
113csc2sin
2
3
k kk
.583121808061637560277... 2G
.58326324064259403627... sin log1 2
.583333333333333333333 = 7
12
.58349565395417428579... ( )( )
12 1
1 1
1
kk
k
k
10 .583584148395975340846... 3 1
2
2
1
2 2
0
e k
k
k
k
( )!
.5838531634528576130... 2 1 2 12
22 1
2
1
cos ( )( )!
kk
k k GR 1.412.2
1 .584893192461113485202... 101 5/ 1 .58496250072115618145... log2 3
.5852181544310789058... 16 8 212
0
1
loglog( ) logx x
xdx
1 .58556188208527645146...
2)(2
1
kk
.585698861949791886453... ( )
1
130
k
k k
.585786437626904951198...
4/
0 1sin21
222
x
dx
.58617565462430550835... 7
3
4
3
4
31
3 4
0
1
,
/
e dxx
.586186075393236288392... 0
1 2
( )
( )!
k
kk
8 .586241294510575299961...
8
9
4 .586419093909772141909... 21
.586781998766982115844... 1
3
2
3 32 1
22
1
1
arccsch ( )kk
k k
k
.586863910482303846601...
0 )!2()!1(!
11},3,2{{},
k kkkHypPFQ
1 .58686891204429363056... G
2 .586899392477790854402... 512
63
5
1 2
/
.58698262938250601998... 8
tanh2
7 .587073064508709895971...
0
32/
)!3(2
3cos
3
2
3
1
k
k
kee
.587413281200730307975... log( ) log( )
log
k
k
x
xdxk
k
1
2
2
1 0
1
GR 422.3
.587436851334876027264...
12
2
)1(
)13(3
16
3
)(
363
2
9
1
2
3log
kkk kk
kkk
7 .58751808926722988286...
7
8
.58760059682190072844...
132
1
1cosh
)!2(2
1sinh
x
dx
xk
k
k
.587719754150346292672...
3 2 25 6 60
/
dx
x
.587785252292473129169... 10
3cos
5sin
.58778666490211900819...
1 35
9log
kkk
k
L
.5882352941176470 = 17
10
12 .588457268119895641747... 9 3 32 1
6
2
0
k
k
kk
k
1 .588467085518637731166... iiiiii 2,22,2)1()1( )1()1(
=
1
22 )(
1
)(
1
k ikiki
.58863680070958604544... ( ) ( )
1 2 11
1
kk
k
F k
.588645903311846921111... 3
2
3
2 3
1
1 3
1
21
csch
( )
/
k
k k
.588718946888426853122...
02 2)!2(!
122
kkkk
I
2 .58896621435810891117... 3 2
208 1
e
x
exdx
log
.589048622548086232212... 3
16
23
21
21
k k
k kk
( )
( )( ) J1061
=
032
02
6
0
5
)1(
sinsin
x
dxdx
x
xdx
x
x GR 3.827.12
=
0
2
23
0
4
0
3 sincossincossincosdx
x
xxdx
x
xxdx
x
xx
=
0
2
24
02
24 sincossincosdx
x
xxdx
x
xx
.5891600374288587219...
1
23 2/
)1(2log4)2(28
k
k
kk
6 .58921553992655506549...
6
7
5 .589246332394602395131...
0 !!
)2(1
21
k
k
kerfe
.589255650988789603667... 4/
0
3cos26
5
dxx
.589296304394759574549... log2
2 2
kk
k
1 .58957531255118599032... log
1
4
2 .589805315924375704933... e
xe
xdx
21 2
0
log log
.589892743258550871445...
)62(2
2
6324)31(
6
1 3/13/16/53/1
3/5 i
i
2
6324
6
)31( 3/16/53/1
3/5
ii
=
23 6
1
k k
.59000116354468792989...
12
2
)!2(
),2(
k kk
kkS
.590159304371654323... ( ) ( )
arctan
1 1
2 3
1 1
23
2
k
k k
k
k k k
.59023206296500030164...
00 )!2(
)1(
6
1
)!2(
)1(
k
kk
k
kk
k
B
k
B
.590320061795601048860... 243
518
2 2 1
60
k
k
k
kk
( )
3 .59048052361289410146... 2
9
. 59048927088638507516... 2arctan2
23
2
2
4
dxxx
x
1
022
2
1)1(
1arctan Borwein-Devlin, p. 53
.590574872831670752346... 6G
1 .590636854637329063382...
112
01 )!2(
2
)!()!1(!
1)2(
kkk k
k
k
k
k
k
kkI LY 6.113
4 .59084371199880305321...
5
1
4 .59117429878527614807... 2 2 1 4 2
0
2
arcsinh x dx
.591330695745078587171... log log( )1 2
21
12
0
1
2
xdx
x GR 4.295.38
= ( sin )/
1 2
0
2
x dx
GR 4.226.2
13 .59140914229522617680...
1
3
!5
k k
ke J161
.591418137582957447613... 1
5 22 2
12
0
1
log sin( log )li
xx dx GR 6.213.2
14 .591429305318978476834...
2 12
2
2 22
12
31
13csc32
k k kk
kk
k
1 .591549430918953357689... 5
5 .591582441177750776537...
5
6
9 .5916630466254390832... 92 6 .59167373200865814837... 3log6
.59172614725621914047... dx
e ex x2 2
0
erndt Ch. 28, Eq. 21.4
1 .59178884564103357816... 21
2
1
2120
e erfk k
k
( )!
10 .5919532755215206278...
1
22
)!2(1cosh
2sinh2
k
k
k
11 .5919532755215206278...
0
2
)!2(2coscosh
k
k
k
eei
AS 4.5.63
=
1
2
2
12 1
4
)12(
41
kk k
k
k J1079
1 .59214263031556513132...
6
743234)3(182 )2(3
433 .59214263031556513132...
6
134)3(182 )2(3
.59229653646932657566...
01
4/1 sinh)!2(
!
2
1
2
2
dxxek
kerfe x
k
.592394075923426897354... 12
2
( )
( )
k
kk
.59283762069794257656...
1 2
sin
)4/11cos1(2
1sin
1cos45
1sin2
kk
k GR 1.447.1
1 .5930240705604336815... 2
)223log(
2
2log
22
22log
2
2
2
2log
=
1
12
2kkkH
.593100317882891074703...
12 3/1
1
322
3
k k
.59313427658358143675... dxxx
x
12
2
)1)(1(
log
8
)3(7)2(
.593350269487182069140...
2 )1(
11
2
5cos
2
k kk
.593362978994233334116...
12
)3(
2kk
k
k
H
3 .593427941774942960255...
1
332
23
2log
3
2log)3(
kkkH
.593994150290161924318... 231 e
.594197552889060608131... ii ee
i
1sin2
1
.59449608840624638119... csch
2 1
4 2112
1
2
1( )k
k k k
.594534891891835618022...
1 2
)1(13log2log1
kk
k
k
.594696079786514810321...
2
35
6
33
2
35
6
331
iiii
=
1
)13()13(k
kk
.594885082800512587395...
0 2)!2(
164
kkk
e
2 .594885082800512587395...
1 2
)1(44
kk
kpfe
1 .594963111240958733397...
1
0
1!
)(
k k
k
4 .59511182584294338069...
4
5
.59530715857726908925...
1
)2(log)(k
kk
1 .59576912160573071176... 2
4
.5958232365909555745... )!12(
33)1(
4
11sin
12
1
13
k
k
k
k GR 1.412.3
.595886193680811429201...
22 )1(
)3(23k
k
kk
H
.596111293068951500191...
H 1(
1
3 .59627499972915819809... log
.59634736232319407434...
e Eik
kk
( )( )
!1
1
0
, the Gompertz constant
=dx
e x
x dx
e xx x( ) ( )
1 10
2
0
=dx
x xe x dxx
21 01
1( log )
log( )
4 .59634736232319407434... 4 11
4
0
e Eix dx
e xx( )( )
.5963475204797203166... ( )
12 1
1
1
k
kk
k
.596573590279972654709...
13)1(
log
4
1
2
2logdx
x
xx
1 .59688720677545620285... 11
22
1
kk
.596965555578483224579...
11
1)1(
kk
k
k
.59713559316352851048...
043 )1(3243
80
x
dx
.597264024732662556808...
1
2
)!2(
2
4
5
k
k
k
e
1 .597264024732662556808...
0
2
)3(!
2
4
1
k
k
kk
e
.597446822713573534080...
1 2
/12
2
11
!
)22()2( 2
k k
kek
k
k
kk
1 .597948797240904373890...
22/1
2
111)(
kk kkLikk
.598064144464784678174...
1
44 4cos2log
2
1)2sin2(log
k
ii
k
kee
.59814400666130410147...
0 )12(!
2)1(2
22
1
k
kk
kkerf
54 .598150033144239078110...
0
4
!
4
k
k
ke
1 .598313166927325339078...
2
422
246
2
324
)1(
14581)2(
128
127
48120 kk kk
kkkkk
8 .59866457725355536964...
1
2
4
3,1
224
kk
kk
G
.599002264993457570659... 2
1 1 1 1 2 1 11
12 2
2
1
cos sin sin log( sin ) sinsin ( )
( )
k
k kk
1 .59907902990129290008...
1 !
1
k kFk
1 .599205922440719577791... 1)3(log
.599314365613468571533... 3
)3(1
.59939538416978169923... log ( ) 21
kk
.5996203229953586595... 2 111 1 11
2
1
e Eik k
k
k kk k
( )( )! ( )!( )
=1
1 22 ( )! !k kk
= H
k k k kk
k k!( ) ( )!
2
1
11 1
1 .599771198411726589334... 2
11
k
kk k( )
4 .599873743272337314...
1
24
4
)4(17
360
17
k
k
k
H
Borwein & Borwein, Proc. AMS 123, 4 (1995) 1191-1198
.599971479517856975792... arccsch2
.600000000000000000000 = 3
5 42 1
1
F kk
k
1 .600000000000000000000 =
1 45
8
kk
kk LL
.600053813264123766513... 3
2
5
24
2
22
1
22
1
2
2 2
2 2
logLi
iLi
i
= log( )
( )
1
1
2
2 20
x
x xdx
.600281176062325408286... 3 6 2 91
1 3
1
1
log( )
( / )
k
k k k
.6004824307923120424...
1 2
22 1log
1)22()2(
k k k
k
k
k
k
kk
2 .600894716163993447115... e k
k
1
1
1/ !
.601028451579797142699...
1
1
)2)(1(2
)3(
k
kk
kkk
HH
=
0
1
2
0
1
0
22
))1sinh(log(
)1(log
)sinh(log
log
sinh x
dxx
x
dxx
xe
dxxx
.60178128264873772913...
22
12
1 111)1()1(
kk
k
kLi
kk
k
.601782458374805167143... 1
1 12 k
k
kk
log
.601907230197234574738... dxeK x
0
11
2
)1(
.602056903159594285399... ( )33
5
.602059991327962390428... log10 4
8 .6023252670426267717... 74
2 .60243495809669391311... 2sinhcsch2
1
sinh2
2sinh
=k
k kk k
2
21
21
2
11
1
1
.602482584806786886836...
2 5
51
10
1
521
coth
kk
175 .602516306812280539986... 24 5 50 32
3
5
21 1
4 24
2
) ( ) ( )
k kk
= 16 11 5 1
1
4 3 2
52
k k k k
k kk
( )
4 .602597804614589746926... 22
sinh
Berndt 7.2.5
1 .60343114675605016067... 2 2 21
23 arctan log
= log( )
12
1 20
xdx
2 .603451532529716432282...
1 )4(
212sin
26
k kk
2 .603599580529289999450... 7 3
4
1
2 1 4
3
2 31
( )
( / )
k
kk
2 .603611904599514233302... 11
1
k k
k
.603782862791487988416... log
!
k
kk
2
1 .603912052520052227157...
1
24
242
2
1
2
3cosh11
1
k kk
kkii
.60393978817538095427... 1
2
1
2 20
2
ee xdxx
/
/
cos
1 .604303978912866704254... dxx
03
3/1
1
21log13
3
2
3 .60444734197194674489...
1
1
12
)(
kk
k
.604599788078072616865...
1 13
1
23
1
33 k kk
=
3 3
2
2 1 41
( )!!
( )!!
k
k kkk
= 1
3 1 3 20 k kk
)( )
= 1
21 k
kk
k
CFG F17
=dx
x x
dx
x x
xdx
x x2 2 2 4 2 1 120
20
4 20
= dx
x x
dx
x x20
202 4 3 3
=
03
06
3
06
123 811 x
dxx
x
dxx
x
dxx
xxx
dx
= dx
x x x x x5 4 3 20 1
= dx
e ex x
10
10 .604602902745250228417... log !8
.604898643421630370247...
1
1)1(
2
1
2
1,
2
1
2
121
k
k
k
5 .604991216397928699311... 10
.605008895087302501620... 7
48
3
83
9 3
16
1
4
2
3
2 21
log
log ( )
.605065933151773563528... 9
4 2
1
1
2 2
0
1
x x
xx dxlog
.605133652503344581744...
0 )1()1()1(
2
12
xe
dxEierfi
e x
.605139614580041017937... 9
8
3 2
4 2 1
2
22
log
( )
k
kkk
1 .605412976802694848577... sik k
k k
k
( )( )
( )!( )2
1 2
2 1 2 1
2 1
0
AS 5.2.14
.605509265700126543353... 3 2 4 4 ( ) ( )
.6055217888826004477... dxxkkk
22
21)(
log
1
3 .605551275463989293119... 13
.605591341412105862802... 5
256
3 43
31
sin( / )k
kk
GR 1.443.5
.6056998670788134288...
12)12(
21
2cos1
2
2sin
k k
2 .60584009468462928581... dxx
11log2log2
6
1
0
222
.606125795769702052589... 4
11 2 2
0
ee x dxx cos
3 .606170709478782856199...
1 )1()3(3
k
kk
kk
HH
= dydxxy
yx
1
0
1
0
22
1
)log(
.606501028604649267054... ( )( )
!/
12 1
11 1
2
2k k
k
k
ke
.606530659712633423604... 1 1
2
1
20 0ei
k ki
k
kk
k
k
/ ( )
!
( )
( )!!
15 .606641563575290687132... 5
.60669515241529176378... 1
2 1
1
2 2 1
2
211 1 2
kk
k kk
k
kk
( )
( )
1 .60669515241529176378... 1
2 1 2
2 1
2 2 11
0
1 12k
kk
k
k
k kk
k
( )
( )
=1
21
2
211 2jk
kj
k
kk
( )
Shown irrational by Erdös in J. Indian Math, Soc. (N.S.) 12 (1948) 63-66
.606789763508705511269... log 21
2 10
xe
edx
x
x GR 3.452.3
1 .606984244848816274257... 1
1 k kk ! ( )
.60715770584139372912... erf
e k
k
k
( ) ( )
( )!
1 11
20
3 .6071833342396650534... 3 35
10241
51
( )
( )( )
a kk
where is the nearest integer toa k k( ) 3 . AMM 101, 6, p. 579
5 .607330577557532492158... 1536 5634
4
0
ek
k kk !( )
1 .607695154586736238835... 12 6 32 2 1
6 10
k
k kkk ( )
.607927101854026628663... 6 1
21
12 2
12
( )
( )k
k pk p prime
.607964336255266831093... 4
6081075
1
1
12
2 4 6 8 10 12
p p p p p pp prime
.60798640550036075618...
11212 )12(24
1,
2
3,2,
2
1
2
1
3
1
4
3log
kk k
kF
=
222 2xx
dx
=
04
3)cos(sindx
x
xxx
.608006311704856510141... ( ) ( )2 5
.608197662162246572967... 3
2
3
2 31
log
Hkk
k
.608199697591539684584... 1
1 2 21 ( )! ( )k kkk
Titchmarsh 14.32.1
.608381717863324722684...
primep primep
ppppp
p
pp
plog
1
1
)1)(1(
2
1
log2222
= )2(
)2(
)1)(1(
log)2(22
primep ppp
pp
.608531731029057175381... 2
31185
1
1
8
2 4 6 8
p p p pp prime
.608699218043767368353... log logsinh
( )( )e e k
kk
k
4 2
12 11
1
K Ex. 124f
= log( ) ( )
log1
2 21
12
2
i i kk
1 .6089918989086720365... !
1)2(2
1 k
k
k
k
k
.609023896230194382244...
2
1)(
k kH
k
.609245969064096382192... Hk
kk
3
1 4
4 .609265757423133079297...
2
221
12csc2
k k
1 .609437912434100374600... log54
51
Li
.609475708248730032535... 2 2
3
1
3 40
4
sin
cos
/ x
xdx
.6100364084437080...
2 log)1(
)1(
k
k
kkk
7 .610125138662288363419... cosh( )!
e e ee
ke e
k
k
1
2 2
2
0
.610229474375159889916...
2 4 3
302 15
9 31
( )( )
x
edxx
.610350763456124924754... log!
11
2
kk
.6106437294514793434... 23G
.61094390106934977277... 8 2 e
.611111111111111111111 = 11
18 3
1
3
1
33
21
24
H
k k k kk k
.611643938224066294941...
1
2 3
1
61 3
1 3
21 3
1 3
2
( ) ( )i
ii
i
= k
kk
k
k
k3
2
1
111 3 1 1
( ) ( )
.612018841644809204554...
1
3
)1(455
55log5
1048576
1953125log
5
2
kk
kk
k
FF
5 .6120463970207165486...
14
243 1
4 3
30
log xdx
x
1 .612273774471087972438... sinh
0
21
12
1i kk
2 .61237534868548834335...
12/3
1
2
14
2
3
k k
=
primep primep
pppp
p 4/932/33
2/3
1)6(1
1
.61251880056266508858... 142
3 453 3 5
1
1
1 12 4
2 52
( ) ( )
( )k k kk
1 .612910874202944353144... 11
32
( )kk
42 .612923374243529926954...
12
41sinhcosh
1
2
2sinh
k k
.613095585441758535407... 1
2 4
1
21
2 1
2 11
1
log tan ( )sin( )
k
k
k
k GR 1.442.3
.61314719276545841313... log6 3
3 .613387333659139458760... ( )
!
/k
k
e
kk
k
k
2
0
1
21
2 .613514090166737576533...
( )3
.613705638880109381166... 2 2 23
2
1
2
log hg
= 1
2 21 k kk
GR 0.234.8
= k
kkk 2 10 ( )
= ( )( )
( )
11
2
2 1
21 11
1 1
kk
k
k
kk
kk
= logx
xdx
1 20
1
= dxx )1log(1
0
2
= 4 1
1
1
2 1 22 20
1
0
1 log( )
( )
log( )
( / / )
x
xdx
x
xdx GR 4.291.14
= log(sin ) sin/
20
2
x x dx
GR 4.384.10
4 .61370563888010938117... 6 2 22 1
3
1
log( )
k
kkk
128 .61370563888010938117... 130 2 22 1
5
1
log( )
k
kkk
.613835953026141526250... 11
11
2
1
3
1
321
tanh
k kk
.613955913179956659743... 2
5 5 5 2
7 10
50
/
dx
x
.61397358864975783997... 1
22 2log
2 .61406381540519800021... 4 42 1
1
1log ( ) /
k k
k
.614787613714727541536... 11
3 11
k kk ( )
.61495209469651098084.... erfik kk
k
1
2
2 1
2 2 12 10
! ( )
.615384615384615384 =8
13
.61547970867038734107...
4/
02sin1
cos
5
3arcsin
2
1arctan
x
dxx
.615626470386014262147...
log(cos )( ) ( )
( )!1
1 2 2 1
2
1 2 1 22
1
k k kk
k
B
k k AS 4.3.72
7 .6157731058639082857... 58
.61594497904805006741... 2
1cot
2
1
2
1sin2cosh
2
1cos2cos
2
1sin2sinh
2
1cos
2
1cos2sin
2
1sin
2
=
2
1cot2cotcot
42/2/2/ ieeee
i iiii
kkk
sin1)2(1
Adamchik-Srivastava 2.28
1 .61611037054142350347... arctan
!
k
kk
1
9 .6164552252767542832... 8 3 ( )
54 .616465672032973258404... 2 4 4 4cosh e e
1 .6168066722416746633... 2 2 22 1
1
1 2
1
1log ( ) / /
k kk
k
k
k
.616850275068084913677...
1
2
2
10
2
)14(
4
4
)2(
1
)1(
16 kkk
k
kk
k
kkk
k
H
= x x
xdx
arctan
1 40
GR 4.531.8
5 .61690130799559924468... 2 3 6 2 31
12
130
log( )
( )( )
k
k k k
1 .61705071484731409581...
6
3
2
1
3
1
3 220
coth/
kk
.617370845099252888895... sincos
22
2
1
2
1 .617535655787233699013... 510
5 5
2 1 1
5 10
log( )
k
k kkk
.617617831513553390411...
1222 251
1
4
51
5
1
kk
k
k
FLiLi
4 .617725319212262884852... 8
.61779030404890075888...
2
1)1(1k
kk
k
1 .61779030404890075888...
2
22
2
1log
1
)1(
231)1(
1 kk k
k
kkk
kk
k
k
3 .6178575491454110364...
0 2
22
)1(2
4
4
)3(7
2
2log
2 k
k
kk
k
.617966219979193677004...
3
7
2
3log3
324
15
2 .61799387799149436539...
0 05/12 316
5xx ee
dx
x
dx
.618033988749894848204...
11 5 1
2
1 .618033988749894848204... 5
cos22
15
, the Golden ratio
2 .618033988749894848204... 1 2
.618107596152659484271... 1
21k
kk kH
1 .618157392436715516456... 4 2 22
20
log( )
k
kk
.61847041926350753801... 2
315
1
1
4
2 4
p pp prime
.61851608883629551823...
2 4
)(3)1(
4
2log9
8
31
kk
kk k
1 .618793194732580219157...
1 )2)(1(
11
2
3cosh
3
2
k kk
.61897071820759046667... 2/)51(2/)51(
55551010
1
HH
=
22 1)(
kk kF
.619288125423016503994... 16
3
10 2
3
2 1
8 20
k
k kkk ( )
4 .619704025521212812366... 2 1
1
k
k k kk
.619718678202224860368... 16 6
1
3 1 221
cot
/kk
.620088807189567689156...
1
22
12
)2/1(
)1(2
2csch2
6 k
k
kk
.62011450695827752463... log( ) ( )
!
e k
k
k
k
1
2
1 2 1
1
=
1 !
)12()1(
k
kkk
kk
B [Ramanujan] Berndt Ch. 5
= dx
e x10
1
.620272188927150901613... 4
9 3
1 3 2
1
e k
k kk
/
!
4 .620324229254256923750... 0
1 1
( )
( )!
k
kk
.62047113489033260164... ( )( )
arctan
12 1
2 1
1 11
21
k
kk
k
k k k
.62073133866424427034...
1 )5(
11
2
29cos
7
24
k kk
8 .62074214840238569958... 1
31
4 2 3 6
22 2 62 3
1 31 3 5 6 1 3
2 3 1 3/
// / /
/ /( )
i
( ) /
/
/ / /1
3
4 2 3 6
2
2 3
1 3
1 3 5 6 1 3
i
= 6 3 1 16
613
2
k
k k
kk
k ( )
1 .62087390360396657265...
0,2
1,
2
1
9 .62095476193070331095... 2 2e/
.621138631605517464637... 2 2
2 224 22
2
2
1
2
1
2
log
logLi
iLi
i
=4
2log32log
162
22
= log( )
( )
1
1
2
20
1
x
x x
.621334934559611810707... 1
5 5loglog e
2 .621408383075861505699... 1
45 2 13 4 5 50 12 5 2 65 20 5
= 5 5 5 5 5 5 5 ... [Ramanujan] Berndt Ch. 22
.621449624235813357639... )1(22
1cos1sin)1(1cos)1(1sin)1(1cos
2sicicici
= sin arctanxdx
x
xdx
ex10 0
= dx
e xx 20 1
.62182053074983197934... ( )
( )
1
2 12
k
kk k
.622008467928146215588... 2 3
6
CFG G1
2 .62205755429211981046...
4
1
22
1 2
, lemniscate constant
1 .623067836619624382082... sinh sinh sinh32 4 6
311
43 3 1
1 3 3
8
e e e
e
= ( )( ( ) )
!
3 3 1 1
81
k k
k k
J883
= 1
4
3 3
2 1
2 1
1
k
k k
( )!
.623225240140230513394... 2
1arctanh
2
11arcsinh
2
121log
2
1
= 11
4 1
1
4 11
( ) ( )k k
k k k
= 1
2 2 11k
k k( )
GR 1.513.1
= ( ) ( )!!
( )!!
1 2
2 10
k
k
k
k J129
=
0
2/)1(
12
)1(
k
k
k Prud. 5.1.4.4
=
2
1
2
1
1k
kkLi Adamchik-Srivastava 4.12
= x dx
x x( )1 12 20
1
=
02 142 xx
dx
=
4/
0 cossin
xx
dx
= sin
cos
/ x
xdx
2 20
2
= dx
x22 2
.623263879436410617495... 1
2 330 kk
.623658674932063698052...
0 12
3
4
33/2xe
dx
.623810716364871399208... 21 3
2log
.623887006992315787... pr k
kk
( )
2 11
104 .624000000000000000000 = 13078
1251
21
6
1
( )k kk
k
k F
.624063589774511570688... 1
6 4 3
3
2
1
4 320
coth
kk
.624270016496295506006... 42
5
3
2)3(50)5(24
24
=
11
)4()14(2
1)14()24(
kk
kkkk
=
2234
2 1
k kkkk
kk
3 .624270016496295506006...
2
424
1)()1(12
5
3
2)3(50)5(24
k
k kk
=
25
234
)1(
151116
k kk
kkkk
1 .624838898635177482811... )1(2)1()1( 102 KKK
=
0
12 2
dxex x
.624841238258061424793... ( )3
.625000000000000000000 = 5
8
.62519689192003630759... 2
3 2
8 3
27 2
2
20
2
sin
( sin )
/ x
xdx
.6252442188407330907...
1 3
)2(
3csc
3log
kk k
k
1 .62535489744912819254...
122
11
kk k
3 .625609908221908311931... 1
4
AS 6.1.10
1 .625789575714297741048... 2 1
12
2
2
2
k
k
k
k
k
ke
k
( )
!
.626020165626073811544...
2 2 1
2
0
sech
e
e
x
xdx
/ cos
cosh GR 3.981.3
=
dxee
xxx
cos
.626059428206128022755... ( )( )
12 1 11
21
22
k
k k
k
kLi
k
.62640943473787862544...
1 122/3
2
32
)1(
log
)32(16)3(2116
k xx
dxx
k
k
.626503677833347983808... 6 1
5 4 1 221
sin
( cos )
sin
k kk
k
.626534929111853784164...
( ) ( )2 2 11
k kk
.626575479378116911542... S k k
k
k
kk
22
1
2 2
2
( , )
( )
1 .626576561697785743211... 71 4/
.626657068657750125604... 1
2 2
3 .626860407847018767668... sinh( )! ( )!
22
2
2 1
4
2
2 2 2 1
0 0
e e
k
k
k
k
k
k
k
AS 4.5.62
= e x dxx2
0
cos sin
GR 3.915.1
.6269531324505805969...
1
331 )2(
1
12
)3(
kkk k
k
.627027495541456760072... 3 3 4 2 31
3 221
log log/
k kk
.627591004863777296758... ( ) )2 6
3 .627598728468435701188... 2
3
1
3
2
3
1
3
2
3
321
,k
k k
kk
= dx
x x
x
xdx2
3
301
1
log( )
= dx
x10
2
cos
= e
edx
x
x
/ 3
1
= log log( )
1 14
23
0 0
x dxx x
dx
.627716860485152532642... ( )( )
log
13 1 1
111
1 23
k
k k
k
kk
k
.627815041275931813278... 4
0 122
1cosh
1sinh
)12(!
112
4
1
x
dx
xxkkerfie
k
.6278364236143983844... 4
5
4 5
25
1
2 arcsinh
= 2 10
111
2
1
4
12
Fk
k
k
k
, , ,( )
.628094784476264027477... sin( )cos log( sin ) sin
1
22 1 4 1 12
= sin
( )
2
1 1
k
k kk
4 .62815959554093703767...
1
32
)1(24
19
36
11)3(2
k
kk
kk
HH
.628318530717958647693... 5 1
3 2
50
x dx
x
/
7 .628383212379538395441... sin ( ) sin/
1
2
3
21 3 i
= cosh( )
3
21
3
2 1 20
kk
1 .628473712901584447056... 1
11 k k
k
.628507443651989153506... Lik
k
k3 3
1
( )
.628527924724310085412... ( ) cot ( ) cot ( )/ / / 14
1 11
21 4 3 4 1 4
i
= ( )( ) ( )
// /
1
42 1 2 1
1 43 4 3 4
( )
( ) ( )/
/ /1
42 1 2 1
3 41 4 1 4
= ( ) ( ) Im( )
1 4 2 11
21
1
2
42 1
k
k kk
k
kk
k
k
i
14 .628784140560320753162... 2 35
64 1 1
9 11
1
22
2
2
32
( ) ( ) ( )( )
k kk k
k kk k
2 .62880133541162176449... ( )/
1 11
41
kk
k
e
k
.628952086030043489235... 0
1 112
1
22 2 2
( )kk
kk j
kj
106 .629190515800789928381... 24 21 164 2
0
dx
x( / )
5 .629258381564478619403... 2 1 2 112
1
e eEik k
kk
( )( )
!
1 .6293779129886049518...
2 12
22
2
2 2 2
20
loglog log xdx
e x
= 2 2
2 2
012
2
12
( log )
loge x dxx
1 .629629629629629629629 = 44
27 4
3
1
kk
k
.63017177563315160855... G
.630330700753906311477... 1
22
12
log( )
( )
k
k kk
= ( ) log2 2 11
11
21
kk kk
.63038440599136615460... 11
2
p
p prime
.6304246388313639332... 112
136
2 4 ( )
.630628331731022856828... ( )34
7
3 .630824551655960931498... 2
e
.6309297535714574371... log3 2
.631010924888487788268...
2 1
)(log
k k
k
.631103238605921226958... 3 1
4
3
3
3
0
sin sin
k
kk
Berndt ch. 31
.631526898981705152108... 1
2 11k
k k ( )
.631578947368421052 =12
19
11 .631728396567448929144... 105
16
9
2
1 .6318696084180513481... e dxxsin
0
1
.6319660112501051518... 9
4
1
3 21
F kk
GKP p. 302
.631966197838167906662...
1
2
2
22log
12)3(
kk
k
k
H Berndt 9.12.1
.632120558828557678405... 11 1 1
11
21 0
1
e k xx e xdx
k e
x( )
!( ) log GR 4.351.1
=
13
1
0
1coshcosh
x
dx
xdxxx
1 .632526919438152844773... 22
.6328153188403884421... ( )
logk
k k kkk k
1
3
11
1
31 1
.632843018043786287416...
2 1
2
2
2
12
2
log
k kk k
k
1096 .633158428458599263720... e7 6 .63324958071079969823... 44
.633255651314820034552... cos( )e
2 .6338893027985365459...
1 3
2
24
2
)3(72log
k
k
kk
k
.633974596215561353236... 3
1 3
.634028701498111859376... 1
11
0
( )k k
k
.634340223870549172483...
2
53log
2
53log
5
11
= 1)(2
kk
F
k
k
2 .634415941277498655611...
loglog sin1
2 1
2 2
20
e x
xdx
6 .63446599048208274358... Ei ee
k k
k
k
( )!
11
.634861099338284456921...
1
2
1
4/1
)1(
2csch2
k
k
k
.63486687113357064562... arcsec15 33
6
Associated with a sailing curve. Mel1 p.153
1 .634967033424113218236...
2
1
2
2 2212
13
161 2 1
2 1
1
( ) ( )( )
k kk
kk k
.635181422730739085012... )()1(22
2log
1
kk
k
=
xx
dx e xdxxlog log log1
0
12
0
GR 4.325.8
.635469911917901624599... 2
12
142
1
4
4
4 1
Gk k
kkk k
( )
( )
.6356874793986674953...
2 )(
)2(1
k k
k
.63575432737726430215... 2 2 2 31
42
1 32
( ) ( )( )
k
kk
= ( ) ( )( ( ) )
1 1 12
k
k
k k k
.635861728156068555366... I k
k kk
1
21
2
2 2
( !)
1 .635886323999068092858... 1
41
1
2 21
1
2 2
1
2
1
2 2
1
2
1
2 2
= ( )
/
1
1 2
1
21
k
k
k
k
.636014527491066581475... 1
2 2
1
120
csch
( )k
k k
02
2
cosh
cosdx
x
x
.63636363636363636363 = 7
11 63 1
1
F kk
k
6 .636467172351030914783... 2 1
1 222
0
logsinx x dx
.636534159241417807499... 2
3112 16
1
96
2
sin /k
kk
GR 1.443.5
= iLi e Li ei i
2 32
32 / /
.636584789466303609633... ( ) ( )2 7
.6366197723675813431...
0
2
)22(16
122
kk kk
k
.6366197723675813431...
124
11
k k GR 1.431
= cos
2 11
kk
GR 1.439.1
= 0
1 2/
= i eilog
.636823470047540403172... 1
5
1
51
1
43 5 2
5 5
24 5
( ) / i
1
51
1
43 5 2
5 5
21
1
43 5 2 5 51 5 2 5( ) ( ) (/ / i i
( )(
/1
5
1
43 5 2 5 5
1 5
i
= ( ) ( )
1 5 3 11
1
1
3
52
k
k k
kk
k
.637160267117967246437...
2 2 2
1
2
1
2 11
2
221
1
1
coth ( )( )
k
k
k
kk
k
.637464900519471816199... 6 1
4 2 5 2
2 2
1
sin
cos
sin
kk
k
.637752497381454492659... 3
2
2
12 2 2
e
x
xdx
cos
( )
.637915805122426308543... 1
2 4 222
4
0
cosh( )!
k
k k
.638000580642192011016... ( ) ( )k k
kk
1 1
2
1 .6382270745053706475... 1
2 dkk
.638251240331360040453... 26
442
21
cos k
kk
GR 1.443.3
.6387045287798183656... 3 9 3
24
1
6 421
log
k kk
= dxx
x
1
05
6 )1log(
.638961276313634801150... sin log Im2 2 i
.63900012153645849366... 1
3
5
3
5
3 1
4
0
3
x dx
ex
.639343615032532580146... i
i i
i i
k
kk
k2
21
22
1
2
21
22
1
2
14 2 1
2 11
1
log ( )( )
= arctan1
22 kk
.639432937474838301769...
1cos
4
5log1sin)21)(cos1csc21(cotcot
1cos45
1arc
=
1 2
sin
kk
k kH
.639446070022506040443... ( ) ( ) ( )3 4 2
.6397766840608015768... k
kk 3 11
.640000000000000000000 =
1
1
425
16
kkkk FF
.640022929731504395634... 3
8 83 3 43
43i
Li e Li ei
Li e Li ei i i i( ) ( ( ) ( )
= sin3
41
k
kk
.640186152773388023916... 4
32 log
1 .640186152773388023916... 7
32 log
.64031785796091597380... dxx
xx
1
0
2
1
log
27
502)3(16
.640402269535703102846... H
kk
kk
( )
( )
3
1 2 2 1
.64057590922154613385... 1
2 2 2 1 2
21 3 1 3 2 3 1 3
3
32 ( ) ( )/ / / /
k
kk
.64085908577047738232... 22
e
.640995703458790509... 3/1)3(
3
33
2)3(1227 H
2 .641188148861605230604...
4
1
4
3
256
1
8
3
4
36 )3()3(
22
GG
= 2/
03
3
sin
dxx
x
.641274915080932047772...
2 6 2 3 3 220
20
dx
x
dx
x
.6413018960103079026... dxx
xLi
1
0
2
3 3
log
3
12
.641398092702653551782... 3
24
5
5
6
1
6
hg hg
1 .641632560655153866294... 1
2
1 1 2
1 1 21 2
2
2 111
k k
k
kkk
( ) /
/
/
[Gauss] I Berndt 303
5 .641895835477562869481... 10
.642024000514891118449... ( )
1
2
1
1
k
kk k
.642051832501655779767... ( )( )
( )!sin
14 2 1
2 1
11
12
2
k
k k
k
k k
.642092615934330703006...
122 1
1211cot
k k GKP eq. 6.88
=
0
2
)!2(
4)1(
k
kk
k
k
B
.642092615934330703006... cot1
1 .642188435222121136874...
5225
5
1 , Lebesgue constant
7 .642208445927342710179... 7
G
.64276126883997887911...
4
32Li
.6428571428571428571 = 9
14
.643107628915133244616...
2
1 )(
11 dx
x
1 .64322179613925445451... 11
32 6
4 2 3 6
21 3
1 3 5 6 1 3
( )/
/ / /i
1
3
4 2 3 6
2
1 3 5 6 1 3
i / / /
= 6 3 1 16
613
2
k
k k
kk k
( )( )
.643318387340724531847... ( )
/
k
kLi
k kk k
1 1 1
21 2
2
.643501108793284386803... 2log4
3arctan
5
3arcsin
22arctan2 gd
9 .6436507609929549958... 93
.6436776435894211956... e
x x dxesinlog sin log
1 1
2 1
1 .64371804071094635289... e ee
kk
k
k
sin ( )( )!
12 1
1
1
.643767332889268748742... Gi
Lii
Lii
log2
8 2
1
2
1
22 2
= log( tan )/
10
4
x dx
GR 4.227.11
=
log( )1
1 20
1 x
xdx GR 4.291.10
= arctan
( )
x
x xdx
10
1
GR 4.531.3
.643796887509858981341...
( )1 e
eHe
.643907357064919657769... 1
6 4 2
7
2
1
4 4 621
tan
k kk
.64399516584922827668...
0
2
3 2
sin
2log42log
1x
x
1 .64429465690499542033... 2 22
( )k
k
.644329968197302957804... log jj
kjk
2 112
.644756611586665798596... G5
1 .644801170454997028679...
1 )4(
113sin
34
k kk
.6448805377924315...
1 log)32(
)1(
k
k
kkk
.644934066848226436472...
2
2)1(
2 1)2()2(1
6 k k
J272, J348
=
12
12 )1(
1
1
1
kk kkkk
= ( ) ( ) ( ) ( )k k k kk k
1 2 2 22 1
= ( ) ( ) ( ) ( )
1 1 2 12 1
k
k k
k k k k k
= 1
21
21k
dx
xk
=
x x
xdx
log
10
1
GR 4.231.3
= logx
x xdx3 2
1
= 1
1 1 20
1
x
x x
xdx
log
( ) GR 4.236.2
= xdx
e ex x
10
GR 3.411.9
1 .644934066848226436472...
2
12
12
162 1
1 2 1
1
( ) ( )/
( )( )
k
k
kk k
= H
k kk
k ( )
11
=
1 22
13
k kk
k CFG F17, K156
=
1 3
32
821
kk
k
k
k
= k k k k kk k
( ) ( ) ( )
1 1 11 2
=
logx
xdx
10
1
GR 4.231.2
=
0 )1(
)1log(
xx
x
= log
( )
log
( )
log
( )
log
( )
2
20
1 2
21
2
30 11 1 1 1
x
xdx
x
x
x
xdx
x
x xdx
= 1
03
2 )1log(dx
x
x
= log( )1 2 2
0
1
x x
xdx GR 4.296.1
=
log 10
e dxx GR 4.223.2
=
0 1dx
e
xx
=dx
e x
10
2 .644934066848226436472...
2
261 1
k kk
( )
3 .644934066848226436472...
2
262 1 2
k k kk
( ) ( )
.64527561023483500704... 3
62 2
3 3
2 log
log
=
12
1
3
)1(1
3
2
2
1
k
k
kk
= log( )( )
11
1 20
x xdx
.64572471826697712891...
1
)(
k kF
k
2 .645751311064590590502... 7
1 .645902515225396119354... 3
31
1
31
sinh
kk
.645903590137755193632...
0
2
1
sin)21()21(
4
1
2 xe
xii
.64593247422930033624...
2
)1(cot
2
)1(coth2)1(coth25
16
1 iii
=
8
1
16
1
81 2 1 1 2 13 4 1 4 3 4 1 4coth ( ) ( ) ( ) ( )/ / / /
1
81 2 1 1 2 11 4 3 4 1 4 3 4( ) ( ) ( ) ( )/ / / /
=
1262
1)68(1
kk
kkk
.646393061176817551567...
2
1)()(
k
kk
k
.646435324626649699803...
04 14
cos
2
1sin
2
1cos
4 x
dxx
e
.6464466094067262378... 22
11
.646458473588902984359...
1
33
18
3 4
3
2 2
3
1
3
1
42
2 2
2Li Li( )log log log
= x
edxx
30
.646596291662721938667... 11
25
24
25 5
1
21
1
21
1
arcsinh ( )!( )!
( )!k
k
k k
k
.646612001608765845265... 1
45 1 3 1 1
2 2
2
1
sin cos ( )( )!
k
k
k
k
.6469934025023697224... ( )
log( )6 4 1
11
4 6
2
k
kk k
k k
.64704901239611528732... 1
2
1 3 1
22 11
1
1k k
k
k
k
kk
( ) ( )
1 .64705860880273600822...
1 2
sin1
kk
k
.6470588235294117 = 11
17
1 .647182343401733581306... dxx
x
1 )(
log
3 .647562611124159771980... 36 6
22
( )
1 .647620736401059521105...
0
2cosh)1(4
2
dxxee x
.647719422669750427618... 1
2
1
2
1
21
1
2 21
coshcos
sinhcos
sinsin sin
!
k k
k kk
.647793574696319037017...
dxx
x
1
3
arcsinh
2
21log
2
1
2
1
.647859344852456910440... cos3
2
1 .647918433002164537093... 3 3
2
22
1
log log( )
x
xdx
.648054273663885399575... sech12 1
212
0
e e i
E
kk
kcos ( )! AS 4.5.66
.648231038121442835697... 3
5
2
5
5
20
2
0
arctan log log cosx x
edxx
.648275480106659634482...
1
222
2 cos2
2
1
3 k
ii
k
keLieLi
32 .648388556215921310693...
03 1
9991log36 dx
x
1 .648721270700128146849... ek k
ikk k
i
1
2
1
20 0! ( )!!/
.648793417991217423864...
2
15
2
15log
4
3
12 22
2
Berndt Ch. 9
.64887655168558007488...
1 )24(
)4(
k k
k
.64907364028134509045... 4
1
2
3tanh
32
= k
kk
k k
4
62 11
6 4 1
( )
.649185972973479437802...
0 )2(3
13
2
3log9
kk k
.64919269265578293912... ( )
log( )5 3 1
11
3 5
2
k
kk k
k k
.649481824343282643015... 4/
04cos6
1
3
2log
3
x
dxx
.64963693908006244413... 2
1sin
2 .649970605053701527022... dxx
xxG
1
12
223 arctanarccos
4
2log
16
.65000000000000000000 = 20
13
1 .650015797211168549834...
1 !)1(
1log
k
kk
ke
kk
Be
e
e Berndt 5.8.5
.65017457857712101250...
1
2
)6(!15894320
k kk
ke
3 .65023786847473254748...
2
3 23
41
3
4 31 1
log ( )( )k
k
k
kk k
.650311156858960038983...
3/1
3/23/2
3/1
3/13/1
3/2
2
)1(1log)1(
2
)1(1log)1(
3
2
3/1
3/2
2
11log
3
2
=
0123/1 )23(2
11,
3
5,
3
2,
3
2
2
1
3
2,1,
2
1
3
1
kk k
F
.6503861069291288806...
1
21
222
sin)1()(Im)()(
2 k
kiii
k
keLieLieLi
i
1 .650425758797542876025...
0 )12(!
121
k kkerfi
.650604594983248538006... 1
21kH
kk
.65064514228428650428...
2
cosarcsin122
=
13 )12(2)12(27
)1(4
k
k
kk
=
2/
02
22/
02
2
sin1
cos
cos1
sin
dxx
xdx
x
x
= 1
0
22
2
)arctan(1
dxxx
x GR 4.538.2
=
02 14
11log dx
x
.650923199301856338885...
22
1
1arctanh
)12(
1)24(sinhlog
2
1
kk kk
k
2 .6513166081688198157...
1
2
3
428 3
12 33
43
0
( ) ( )( )kk
.651473539678535823016... 15
32
5
4
35
64 5
2
1
log
k Hkk
k
1 .651496129472318798043... 2log
.651585210821386425593...
2
2 2
2 21
4 2
29 4
1 4 1 4
1 4 1 41
1/
/ /
/ /
sin sinh
cos cosh( )
( )
kk
k
k
= 1
2 2 21 k kk
2 .651707423218151760587... )5(168)3(120)3(89648
1 42
=
13
k
kk
k
HH
1 .651942792704498623963...
1
3
2
)3(62
)5(7
k
kk
k
HH
1 .652163247071347030652... 2/
0
2 tanlog16
)3(7 dxxx
1 .653164280286206248139...
22
2 )(
k k
k
23 .65322619113844249836... 31
12601
6
0
1 log( )
logx
x
xdx
2 .653283344230149263312... )7()2(
.653301013632933874628...
1
34
34
3
2 4sin
23
16
3
24
k
ii
k
keLieLi
i
GR 1.443.5
.653344711643348759976...
2
2 1)2(k
k
40 .65348145876833758993...
1
3
!!2
1
211711
k k
kerf
ee
.65353731412265158046...
2
55
24
59
2
552514
20
1 ii
2
55
24
59
2
55251
20
1 ii
2
55
24
59
2
55251
20
1 ii
2
55
24
59
2
55251
20
1 ii
=
1232
1)35(1
kk
kkk
.653643620863611914639...
0
1
)!2(
16)1(4cos
k
kk
k
1 .653643620863611914639...
1
12
)!2(
16)1(2sin2
k
kk
k
.653743338243228533787... ( )( )!
15 3
2
1
212 3
1
kk
k k
k
k k
.65395324036928789113... ( )
log( )4 2 1
11
2 4
2
k
kk k
k k
1 .654028242523005382213... coshcos sinhcos sin( sin )sin
!1 1 1 1
1
k k
kk
.6541138063191885708... 2 37
41 1 1
2 3 3 1
11
3
2
2 32
( ) ( ) ( ) ( )( )
( )
k
k k
k k kk k
k k
5 .65451711207746119382... /1e
.65465261560532808892... ( )3 1 1 1
21
2 22
k
kk Li
kk k
.654653670707977143798... 3
7
1
3
2
0
( )k
kk
k
k
.65496649322273310466...
1 )6(
1110sin
3
104
k kk
6 .655258980645659749465... )3(
8
14 .65544950683550424087... 163 1 1
42
1
Gk
kk
kk
( )( )( ) Adamchik (27)
.655831600867491587281...
1
1 !)1(
kk
k
k
k
.655878071520253881077... ( )2
2 1
c Patterson Ex. A.4.2
.65593982609897974377... ( )2 1 1
31
3 22
k
kLi
kk k
.656002513632980683235...
5
33Li
1 .65648420247337889565...
2
/12
2
2
)1(1
!
1)(
k
k
k
kekkk
kk
.656517642749665651818...
1
22
1
2
1
2
1coth
k k J951
.65660443319200020682... 4 2
3
2 3
31
42 1
1
log log
H kk
k
9 .65662747460460224761... 5log6
.6568108991872036472... dxx
x
1
06
2
1
)1(log
32
32log
2
32log34
1 .65685424949238019521...
0 8
112424
kkk
k
5 .65685424949238019521... 32 4 2
2 .656973685873328869567...
1
13
33 arcsin
82log3 dx
x
x
.656987523063179381245... ( ) logk k
kk 22
.65777271445891870069...
1
21
2
i i
.657926395608213776872...
2
3
2
3
22592
5533
4 iLi
iLi
i
=
13
6/sin
k k
k GR 1.443.5
.65797362673929057459... )3()4()2(
)4(
15
2
=
primep kk
k
primepk
k
pp
p
k 022
2
12
)( )1(
1
)1(
HW Thm. 299
=
12
)(
k k
k HW Thm. 300
1 .6583741831710831673...
0
22
22 )2(log
2log2log26 xe
dxx
.65847232569963413649... 2 2
20
12
4
7 3
8 1
log ( ) arccos
x x
xdx
= dxx
x1
0
2arcsin
8 .658585869689105532128... 45
4)4(8
4
2 .658680776358274040947...
2
3
62
52
2
3
= dxx
xex x
2
2/
0
tan
cos
4sin2
GR 3.963.3
1. 658730029128491878509...
1 2
)1(
kk
k
.658759815490980624984...
24
14
111)1(
kk kLi
kk
k
.658951075727201947430... 1
21
1
21
Eik kk
( )!
.65905796753218002746... 4
3172
31
( )( )k
kk
3 .659370435304991975484... )1sinh(sinh)1cosh(cosh1)2sinh2(cosh eeee
=
02 )2(!
1)1(
k
ke
kk
e
e
ee
1 .659688960215841425343...
1 21
1k k
k
k
.659815254349999514864...
2 1
/1 11
!
)()1(
k k
kk
ke
k
k
.659863237109041309297...
k
k
k
k
8
)1(2232
3
240
.65998309173598797224...
2
/1111
11
1logk
kkk
kk
=
2 2
11
k nkk n
Lin
Li
.660161815846869573928...
3
23
23
2 )/11(
/21
)1(
)2(
)1(
11
primepprimepprimep p
p
p
pp
p
= Twin primes constant 2C
8 .6602540378443864676... 75 5 3
.660349361999116217163... 11
3 121
k kk
.660398163397448309616...
1
2
2
22
2 )2/(sin
4
1
128
1
4 k
ii
k
keLieLi
.66040364132111511419... 4
21
8
1
421
coth
kk
J124
=
0 1
cossinxe
dxxx
.660438500147765439828...
044/9 22 x
dx
.660653199838824821037...
05
2
15
3csc
555
2
5
2dx
x
x
1 .660990476320476376040... G
1 .6611554434943963640...
22 1
log
k kk
k
.66130311266153410544...
12 !
)1(
)1(
11
k
kk
k
kB
e
809 .661456252670475428540...
0 !
2cosh2sincosh2coshsinh2coshcosh
k k
k
.661566129312475701703... log cos log ( )( )/ /21
4
1
21 12 2
e ei i
= ( )cos /
121
1
k
k
k
k
.661707182267176235156... 7)32log(42)21log(21362174105
1
Robbins' constant, the avg. distance between points in the unit cube
.66197960825054113427... 1
4
3 3
8
23
1
log log( )x
xdx
2 .662277128832675576187... )6()2(
2 .662448289304721483984...
1 )!(
1
k k
.662743419349181580975... Laplace limit. Root of 111 2
1 2
x
ex x
6 .662895311501290705419...
1
3
k
k
k
.66333702373429058707...
4
1
8 14 2 1
2
42 1
coth ( )
k
kk
k k
.663502138933028197136...
1 3
1
k kF
.663590714044308562872...
1333
4/cos
3
1
3
1
2
1
k k
kiLi
iLi
1 .663814745161414937366... )3(
2
1 .6638623767088760602...
1
24
12 )(
)1(24
k
k
k
kG
3 .6638623767088760602... 41
12
120
Gk k
k
k
( )
( )( )
.663869968768460166669...
13
41
)2(3
)(
164
4
1
2
164)3(28
k k
64 .663869968768460166669...
03
41
)2(3
)(
1
4
1
2
1)3(28
k k
.66428571428571428571 =
1 )1(4
12
140
93
kk
kk
4 .66453259190400037192...
0
2)12(
21
2cosh
k k
.664602740333427442793...
02010 )!(
)1(21;1;
kk
k
ekeJ
eF
.66467019408956851024... 3
84
0
2
x e dxx
= dxxx
xe x
2/
06
2tan
cotcos
cos212
GR 3.964.3
.66488023368106598117...
2
061 1 2
1
( ) ( )logx x
e edx
x x
.66489428584555810021... ( )
log( )3 1 1
11
3
2
k
kk k
k k
2 .66514414269022518865... 2 2 Gelfond Schneider constant, known to be transcendental
1 .66535148216552671854...
5
212
55
55log55log5
10
3
5
6
=
2 2 )65(5
36
5
1)(6
k kk
k
kk
k
.665393011603856317617...
3
22
3
123log32
18
1 )1()1(2
=
1 2k
k
kk
kH
2 .665423390461749394547... 28
.66544663410202726678...
arctanh1
1 e
.665738986368222001365...
12
2 )(
k k
k
.665773750028353863591... 4
arctan
1 .665889619038593371592... 25
.666003775286042277660...
20
12
222
)!(
1)2(
kk kkI
kk
k
1 .666081101809387342631...
0 22
)21arctan(2
24
3xx ee
dx
1 .666090786005217177154...
primepprimep ppppB
)1(
1111log2
HW Thm. 430
.666169509211720802743...
23
3 )(
k k
k
8 .666196488469696472590...
12
/2
0 !
)2(2
k
k
k
k
k
e
k
k
.666355847615437348637... 22
.666366745392880526338... 1sincos
.66653061575479140978... sin ( ) cos ( ) cos1 1 1 1 1si ci
= log sin( )x x dx10
1
.666666666666666666666 =
k
k
2
)1(
3
2
=
1
1
0 )2/1)(2/3(
)1(
)2)(1(4
12
k
k
kk kkkkk
k
=
1
34
8kkkF
=
1
2)2(
11
k k
=
23 1
21
k k J1034, Mardsen p. 358
= 2/
0
cossin
dxxx
=
2/
02sin1
1
dxx
= dxe
xx
02
cosh
= cosh log xdx
x31
=
0
8
3xe
dxx
1 .666666666666666666666 = 3
5
=
1
14
8kkkF
2 .666666666666666666666 = 3
8
=
1
4
8kkkF
2 .666691468254732970341...
0 )!!!(
1
k k
.666822076192281325682... 21
.666995438304630068247...
0 )!12(
3
32
3sinh
2
3cosh
k
k
k
k
.66724523794284173732... 21 2
2
1 1 12
1
log( ) ( )!
!
k
k
k
k k
.667457216028383771872... 4
arccos
.66749054338776451533...
2
1
42 22 2 e ei i
( ) sink kk
1 11
2
.6676914571896091767...
0 12
)1(
4
3,
2
1
4
1,
2
1
2
1
k
k
k
.66774488357928988201... ( )2 1 1
21
2 22
k
kLi
kk k
1 .66828336396657121205...
2
1
6
2log
6
2log)2( 3
32
3 LiLi
=
0
2
13
21
0
2
22
log8
21
logxe
dxx
xx
dxx
x
dxx
.66942898781460986875...
122
)(
kko
k
k
.669583085926878588096...
1 )1(12
1
kk k
96 .67012655640149635441...
0
333
1)3(122log82log2
xe
dxx
1 .6701907046196043386... k
kk 2 11
1 .67040681796633972124...
1
1
kke
15 .67071684818578136541... 55
20 .67085112019988011698... 3
2 3
.670894551991274910198...
0 )2(
111log1
kk kee
ee
.671204652601083... ( )
/
12 3 2
2
k
k k k
2 .671533636866672060559...
15
/11)1(
k
kk
k
e
1 .67158702286436895171... 2
12sin
!
11cossinh1coscosh1sin
2
1sin
0
k
kk
.671646710823367585219... e erf e
xdx
x ( )1 1
2 1
2
20
.671719601885874542354...
1
021
1loglog2log
6
5
6
1log
3
2
xx
dx
x
GR 4.325.6
=
01 1
log
ee
xx
.67177754230896957429...
033
1 )1(21
9
1
327
10
x
dx
k
kk
.671865985524009837878...
13
1)()(
2
1
k kkii
=
1 1
1 )14()14(2
11)12()1(
2
1
k k
k kkk
=
12 )(
1Re
k ikk
.672148922218710419133...
1 )2/5(
1
5
2log4
75
92
k kk
.672247528206935894883... dxx
xx
1
12
22 arctanarcsin
.672462966936363624928...
1
1
00 !!
)1()2(2)2(
2
1
k
kk
kk
HJY
.67270395832927634325...
1
22
6
log)()2()4(30
90
k k
kk
.672753015827581593636...
12 24
1
2cot
244
1
k k
.672942823879357247419...
0
13
13
)13()1(
3
31log
3
1
34 k
kk
k
2 .67301790986491692022...
2
3 1)(2)(k
kk
3 .67301790986491692022...
2
3 )()(k
kk
4 .67301790986491692022...
2
3 1)(k
k
1 .67302442726824453628... 2
2cos
4
1
4)1cosh()1cos(
2
2cosh2cos2
2
e
eii
=
0 )!4(
16
k
k
k
.6731729316883003802... 1
2
3
4 83 4
2 4
20
/ cossin
x
xdx
2 .67323814048303015041... 1sinh
21
ee
.67344457931634517503...
1 )(
1
k kq
.673456768265772964153...
1
2
2sin)2(2
22
1dx
x
xsi
.6736020436407296815... kk
k
2
1 4 1
.673917363376357541664...
1
2)1(
11
k kk
.673934654451819223753... 2
15log
8
3
20
2
Berndt 9.8
.674012554268124725048...
kLi
kk
k
kk
11
)1(
1)(
13
23
3 .674225975055874294347...
1
3
!8
))1(1)(33(1cosh
k
kk
k J844
.674600261330919653608...
2
31
22
)12(
)12(21)1(
26
42
)3(7
kkk k
kk
k
3 .674643966011328778996... pk
kk 21
pk the kth prime
1 .674707331811177969904...
2 )1)((2
11
kk k
.67475715901610705842...
1 2
1)2(
k kF
k
.67482388341871655835...
22)1(
log
k k
k
.675198437911114341901...
primep k k
kk
p 1 )!1(
)(
!
1
5 .675380154319224365966...
0 )6/1)(1(
)1()32log(32log
5
6
k
k
kk
.67546892106584099631...
0
2 2
sin
2log1
1x
x
.67551085885603996302...
02 12322
2
1arctan2
2
2arctan
xx
dx
=
0
4/
022 sin132
x
dx
xx
dx
14 .67591306671469341644...
1 1 1
3
222
1
02
)21(
42
22
)(
n m mm
kk
mnk
k
knmkk
.675947298512314616619...
2/1
04
4/
04 1
1
sin1
cos
24
5log2arctan
22
1dx
xdx
x
x
.675977245872051077662...
1
2/5)51/(1
2!1
5
1
kk
k
k
Fee
3 .676077910374977720696...
0
2
)!12(2
0sinh
k
k
k
ee
i
=
12
2
1 12 2
22
)2(
21
11
kk k kk
kk
kkk
.676565136147966640827...
21
1arctanh
1
12
1)2(
kk kkk
k
1 .67658760238543093264... ( )
!
k
kk
1
.676795322990996747892... ( )k
kkk 21
1 .676974972518977020305...
10
8/1
!)!2(
!)!1(
2!!
1
22
1
21
kkk k
k
kerfe
.67697719835070493892... ( ) ( )( )k kk
k
2
.677365284338832383751...
0 )2()!2(
11sinh141cosh1812
16212
k kkee
.67753296657588678176... 3
2 12
1
21
2
2
( ) ( )k k
k
.678097245096172464424...
1
3
3sin
2
1
8
3
k k
k
3 .678581614644620922501...
2
2
)!1(
)(
k k
k
17 .67887848183809818913... e erfk
k k
k
k
2
1
2 22 2
2 1
( )!!
( )!! !
2 .67893853470774763366... 1
3
.678953692032834648496... 2
1cot1
.67910608050053922751... 4
11
4 22
12 1 2
2
20
eK
xdx
x/ ( )sin
= 2/
0
2 )(tansin
dxx GR 3.716.9
719 .67924568118873483737...
1
02
6
77
7
1
log)()(360
256
61dx
x
xiLiiLii
GR 4.265
.679270890415917681932... 1
2 112j
kjk
2 .67953572363023946445...
0
2
2
3
43cschsinhcosh
3
4
k k
k
.679548341429408583377...
2
22 1)(k
k
.67957045711476130884...
03
01
2
)2(!4
1
)!2(4 xe
dx
kk
kex
kk
.679658857487206163727...
2
1)(
k
k
k
kH
2 .679845569858706852438...
1
log
k kF
k
11 .680000000000000000000 =
1
22
425
292
kk
kFk
.680531222042836769403...
22
1)(1)1(
)()1(
kk
k
k
kHkk
k
1 .680531222042836769403... 1
log1
1)(22
k
k
k
kkH
kkk
7 .6811457478686081758... 59
.68117691028158186735... 1
2 2
1
121
cot
kk
.681690113816209328462...
12
1
x
dx
1 .681792830507429086062... 4/18
.681942519346374694769...
12
2
1212 )1(21,2,1,1
4
11,1,1,1
2
3
kk k
kiF
iiF
.68202081730847307059... 4/
0
2232
cot192164
2log dxxxG
.682153502605238066761...
1 11
0
1 3
1
3
)(
13
1
j kkj
kk
kk
k
=
1 )13(3
132
kkk
k
Berndt 6.1.4
3 .6821541361836286282...
1 !
11
k k
5 .68219697698347550546... dxe
xdx
xx
xdx
x
xx
1
3
12
31
02
34
1
log
1
log
4
)4(21
120
7
GR 4.262.1
.682245131127166286554... 11
22 2 2 2 2 2 1 1
1
k
k
k( )
.68256945033085777154... )2/(sinh2
.682606194485985295135... 3log5
.6826378970268436894...
1 )7(
11
2
52cos
143
720
k kk
.682784063255295681467... 3/1
.682864049225884024308...
1
2 1)1(k
k
.68286953823459171815... 222 csc12coth36421192
1h
=
2242 /11
1
k kk
1 .682933835880662358359...
0
2)12(2
11
22cosh
k k
1 .68294196961579301331... )(1sin2 ii eei
1 .683134192795736841802... coshcosh sinhcosh sinhsinh1
2
1
2
1
2
J712
= 1
2
21
0
e ek
ke e
k
/ sinh( / )
!
.683150338229591228672...
12
)1(22
33
2
2
13log
8
93log
4
3
24
7
k
k
kk
H
.683853674822667022202...
1
1sin1log
1
k kk
3 .683871510540411993356...
1
0
2
1
1
!
22log)2( dx
x
e
kkEi
x
k
k
.683899300784123200209... ( )k
kk 21
.68390168402925265143...
4
1
4
32192
32
1 )3()3(42 G
= dxx
x1
02
4arcsin
.6840280390118235871... 2
22
12 2
162 2
2 1
2
2
4
log( )!!
( )!
( )!
( !)
k
k k
k
k kkk
k
=H
k kk
k ( )( )
1 2 11
=log ( )2
20
1 1
x
xdx
.684210526315789473 = 19
13
.6842280752259962142...
( )
( )
k
kk
1
22
21 .68464289811076878215... dxx
x
1
04
442
)1(
log)3(9
90
7
3
.6847247885631571233... log log( )
( )Kk
k jj
j
k
k0
1
1
2 1
1
22 1
11
K0 Khintchine's constant
= log ( )22 2 2
2
21
2
1
21
4
Li Lik
k
k
= ( ) log21
22
1
12
2 22
Likk
=
kkk
11log
11log
2
Bailey, Borwein, Crandall, Math. Comp. 66, 217 (1997) 417-431
.68497723153155817271...
112 1
)(
)1( kk
kk
kk
1 .685290077416171813048... 2 1
1
2
0
1k
kk
x
kx dx
Prud. 2.3.18.1
1 .68534515180093406468...
12 13
11
2
5sec
2 k kk
2 .68545200106530644531... 11
2
2
k k
k
( )
log
Khintchine's constant
1 .685750354812596042871...
4
1K
3 .685757453295764112754...
1
22
)2()3(
84
5
k
kk
kk
HH
.686320834104453573562...
2
11)(k
kk
.68650334233862388596...
2
3531
2
35312
6
1 ii
ii
=
1
1
13
1)3()1(2
1
1
1
k
k
k
kk
=
1212
1)13(1
kk
kkk
.686827337720053882161...
12 2
1
2
1
2
7tanh
7 k kk
.686889658948339591657... 7
)3(4
.6869741956329142613... 34G
1 .687023683370069268005... dxx
xG
1
12
22 arctan
2
2log
82
.687223392972767270914... dxx
xx
0
3
32 sincos
32
7
.687438441825156070767...
22 3
1)(
k k
k
1 .687832499411264734869... )4()2(3
.687858934422874767598... 8
32
3
2
3
tanh tanhsin
x
e edxx x
.688498165926576719332... ( )
cos( )/
2 2
29 42
0
2
e x dxx
.6885375371203397155... 11
41
kk
1 .688761186655448144358...
02/3
2/3
)12(
1
2
321
k k
.688948447698738204055...
02 )!2(!
1)2(
k kkI LY 6.114
.689164726331283279349...
2
/1
1
1!
1)2( 2
k
k
k
ek
k
3 .689357624731389088942... (log )log( )
2 25
1
2
20
arccschx
xdx
.689723263693965085215...
0
22
3
12log22log2
6 x
x
e
ex
.68976567855631552097...
2 5
)(4)1(
2
53log
5
15log
5
21
5
21
kk
kk k
.690107091374539952004... 2
arccsc
.69019422352157148739...
2 1 40
2
ee xdxx
/ cos
7 .690286020676767839767... 3log7
4 .69041575982342955457... 22
.69054892277090786489... cosh
sinh cosh2
0
11
2
1
2 x x dx
.690598923241496941963...
0 )2(6
12
3
43
kk kk
k
.69063902436834890724... 1
22 2 3
3
1
log coscos
k
kk
.690759038686907307796...
1
03
3
32 )2(
log
2
1
2
1
4
3
x
dxxLiLi
.69088664533801811203... 1
21 1
1
2 1
1
1
(cos sin )( )
( )!
k
k
k
k
2 .69101206331032637454...
02
12 2/12
1
2/1
1
2cot
21
kk kkk
=
112
)2(
kk
k
.691229843692084262883...
211
2
0
2
2
0
2
J x xdx x xdx( ) cos(sin ) cos cos(sin ) sin/ /
= sin(sin ) sin/
x xdx0
2
4 .69135802469135802469...
1
4
481
380
kk
k
.691367339036293350533...
0 34
141
4
31
k k
k
.69149167744632896047... 12 1 202
1
31
I
e
k
k
k
k
( ) ( ) ( )!
( !)
.691849430120897679...
sin( ) sinh( )
2 2
61
42 4
2 k
.691893583207303682497...
1
2
)2(22log42log24
kk
k
k
kH
.69203384606099223896... ( )
( )
2 1
21
k
kk
1 .692077137409748268193...
1
)2(3
23
2log
2
)3(3
kk
k HH
.692200627555346353865...
0
/1
!
)1(
kk
ke
eke
.692307692307692307 = 13
9
.6926058146742493275... 1
2 22 k kk log
.6929843466740731589...
8
5352
8
5352
4
14
5
1222 LiLiLi
=
12
31
4
)1(
kk
kk
k
FF
1 .6929924684136012447...
2/
0
3)3()3(
2
sin4
1
4
3
128
1
2
3
x
dxxG
.69314718055994530941...
1
1)1()1(2log
k
k
k J71
=
2
1
2
11
1
Likk
k J117
=
0
12 )12(3
12
3
1arctanh2
22
1arcsinh2
kk k
K148
=
2 2
)(
kk
k
=( )
log( )2 1
11
2
2
k
kk
k k
K Ex. 124(d)
=
112
112 42
)2()14(
2
)12()12(
kk
k
kk k
kkk
=
0 )22)(12(
1
k kk GR 8.373.1
=
11 1 )2(!)!2(
!)!12(
!)!2(
)!1(
)!2(
)!22(
kk k kk
k
k
k
k
k J628, K Ex. 108b
=
13 28
1
2
1
k kk [Ramanjuan] Berndt Ch. 2
=
1
3 28
)1(21
k
k
kk [Ramanujan] Berndt Ch. 2
=
1
3 327
)1(
2
3
4
3
k
k
kk [Ramanujan] Berndt Ch. 2
=
12 14k
k
k
H
=
1211
H nk
kkn
( )
=
1
cos
k k
k GR 1.448.2
= limn k
n
n k 1
1
[Ramanjuan] Berndt Ch. 2
=1
212 ( )s rsr
J1123
=
2 132 xx
dx
xx
dx
=dx
ex
10
=
1xe
x
e
dxe
=dx
x x
dx
x x2 3 1 3 120
20
= dxx
xdx
x
x
12
1
02 )1(
log
)1(
log GR 4.231.6
= dxx
x
122 )1(
log GR 4.234.1
= dxx
x
1
03
4 )1log(
=
4/
0
2/
0 cos)sin(cossin1
xxx
dxdx
x
x
=
03
4sindx
x
x
=
02cosh
dxx
x
= dxxxdxxx 2/
0
2/
0
tanlog)cos(tanlog)sin(
GR 4.393.1
= 1
0
2cos)log(sin dxxx GR 4.384.3
=
02 )2/cosh()1( xx
dx
= dxxx
x
0 cosh
)2/tanh( GR 3.527.15
= 1
0
)( dxxli GR 6.211
2 .69314718055994530941... 2 22 1
21
logk
kkk
=log( ) log2
21
x x
xdx
.693436788179183190098...
020 3)!(
)1(
3
2
kk
k
kJ
14 .69353284726928223312...
1
3
!
)(
k k
k
.69384607460674271193... )3(
5 .69398194001564144374...
23
2
2
)1(
21)()1()3(2
3 kk k
kkkk
=
1 1
1)()(i j
jiji
.694173022150715234759... 4/34/34/14/1 )1(1)1(1)1(1)1(14
i
=
1
1
14
1)14()1(2
1
1 k
k
k
kk
k
.694222402351994931745...
1
4/1
)!2(
!
2
1
4
3
4
1
k k
kkerf
e
.694627992246826153124... /1
1 .69476403337322634865...
1
1
)!2(
4)1()2(2log2
k
kk
kkci
.69515651214315706410...
22 2
log
k kk
k
7 .69529898097118457326... 6 9 .695359714832658028149... 94
8 .69558909395047469286...
1
2
2
)1(2log668
kk
kk
.69568423498636049904... ie ei i
22 2
( ) sink kk
1 11
1 .696310705168963032458...
1
2
)!2(2sinh2
k
k
k
ee
2 .696310705168963032458...
0 )!2(
coshk
k
k
ee
1 .696711837754173003159... u kk
k
( )
2 11
.69717488323506606877... Ei
e
k ek
k
k
k
( ) ( ) ( )
!
1 1 11
1
.697276598391672325965...
1 2
1
kk k
1 .697652726313550248201...
2/1
2
3
16
.6976557223096801684... 21
3 33
2
21
Lix
x xdx
log
.697712084144029080157... 1
0
00 arccos)1()1(12
dxxeLI x
.69777465796400798203... )2(
)2(
0
1
I
I Borwein-Devlin, p. 35
has continued fraction {0, 1, 2, 3, 4, 5, …}
.69778275792395754630... dxxx
xLi
1
0
2
2
)1)(2(
log
2
1
3
1
18
.69795791415052950187...
22 12
11
2csc
24 k k
.698035844212232131825...
2log5351125log55log53
2
625log
535
1
=
1
2
4kk
kk
k
FF
.698098558623443139815...
1
1
15
1)25()1(2
1
1
1
k
k
k
kk
7 .69834350925012958345... sinh
!( )
4
2
41
20
k
k k k
.6984559986366083598...
0 )!12(
2)1(
2
2sin
k
kk
k GR 1.411.1
=
0
1
)!2(
2)1(
k
kk
k
k
.699524263050534905034...
12 2/2
12log68
k kk
.699571673835753441499... | ( )| k
kk 2 11
.70000000000000000000 =
primep p
pp22
42
)1(
1
10
7
.700078628972919596975...
3/13/2 )1(44
2
1)1(1()1()1(
6
1 ii
3/13/213/23/1 )1(44
2
1)1(1)1(1)1(1
6
1
3/23/1 )1(1)1(16
1
=
1
1
133
1)36()1(2
11
k
k
k
kkk
.700170370211818344953...
122
12
11
)1(
)12(
kk kkLik
k
k
2 .700217494355001975743...
2
33 1)(k
kk
.700367730879139217733...
122
11
kk
.700946934031977688857...
1 )!2(
),2(2
k kk
kkS
1 .70143108440857635328... x dx
e xx
2
0
.7020569031595942854...
1
31
3
1
2
1)3(
x
dx
kk
.70240403097228566812... )1(4
1)1(
4
1
2
1i
ii
i
= 1
12 11 k k kk
.70248147310407263932...
02 552 x
dx
.702557458599743706303...
0 )2(2!
124
kk kk
e
.703156640645243187226... 2log23
8
2
5
1 .70321207674618230826...
2 12 69
4)(
3
23log
33 k k
k
kkk
.703414556873647626384...
0
2/)12( )12(
1
4
1tanhlog
2
11arctanh
kk kee
J945
.703585635137844663429...
4
1sin2log
))2/1cos(1(2
1log
2
1 GR 1.441.2
=
1
2/2/ )2/cos(1)(1(log
2
1
k
ii
k
kee
2 .703588859282703332127...
1
2!
22,2,2,2,1,1,12
k
k
kkHypPFQ
.703734447652443020827...
222 2
11log
2
)1(2
1)(
kk
kk kk
k
.703909203233587508431... 4G
.7041699604374744600... dx
xx1
.7044422...
primep pp )1(
11
48 .70454551700121861822... 2
4
.70521114010536776429...
( )
( ) ( )
( )
( )
k
k k
k
k kk k k
11
1
12 2 2
=
1
12
)(
k kk
k
=
1 2 )(
1
m kk km
1 .70521114010536776429...
( )k
kk 12
.7052301717918009651...
1 )(
1
k kp, p(k) = product of the first k primes
.70535942170114943096... dxx
2
1)1)(exp(
.70556922540701800526...
1 )8(
1117sin
1713
315
k kk
.70561856485887755343... ( )( )
log
11
2
11
1
21
1 1
kk
k k
k
k k k
.70566805723127543830...
0
2 )!2(!
)1(2)2(2
k
k
kkJ
2 .70580808427784547879...
1
202
4 )()2(
36 k k
k Titchmarsh 1.2.2
=
12
)2( )(
k k
k
.7058823529411764 = 17
12
.7060568368957990482... 3
21 1
1
2 120
log( )x
dx
1 .70611766843180047273... )6(60
137
.70615978322619246687... 42
)1)(2
kk
k
.70673879819458419449...
2
1)()1(k
kk
.7071067811865475244... 2
2
1
2 4 12 12 cos cos sin
= ( )
11
4
2 1
8
2
0 0
kk
kk
k
k
k
k
k
=( )!!
( )!!
2 1
2 21
k k
k kk
=
01 36
)1(1
36
)1(1
k
k
k
k
kk J1029
= 11
2 11
( )k
k k
1 .707291400824076944916...
1
31
3
20
3 )()(
)6(
)3(
kk k
klc
k
k
Titchmarsh 1.2.9
.707355302630645936751...
2
2
)2(16
12
9
20
kk kk
k
.70754636565543917208... 1
21 2
11
2
log ( )k
kk
k
15 .707963267948966192313... 5
.70796428934331696271...
kLi
kk
k
kk
111)1(2
212
.70796587673887025351... 7
21
7
11
2 1
70
k
kk
k
k
( )
.7080734182735711935... 2/1)2(
1
1212
)!2(
2)1(
2
1cos11sin H
kk
kk
GR 1.412.1
.70807785056045689679... 4 22
4
2
4
4
4
4
4log
i i i i
= ( )
( / )
1
1 4
1
21
k
k k k
6 .70820393249936908923... 45 3 5
.70828861415331154324...
0 13
)1(
3
2,
2
1
6
1,
2
1
6
1
k
k
k
.70849832150917037332... 2
12sin
)1(
2
1sin1cos22log
2
1(cos
2
1
1
1
k
kk
k
.70865675970942402651...
2
)(log)(k
kk
.70869912400316624812... dxx
x
1
0
2/11
11024
231 GR 3.226.2
.70871400293733645959...
2
11
kkk
.70875960579054778797... 2log2
92log
2
)8(324
82
2
G
=
124k
k
kk
H
4 .7093001693271033307... e
1 .70953870969937540242... 2 3
20 1
120
e k
kk
, ,( )!
.70978106809019450153... 8 2
1 1 21
1 2 1 22
40
( ) ( )cos( ) ( )
i e i ex
xdxi i
1 .709975946676696989353... 3/15
.710131866303547127055...
12)1(
2)2(24
k kk
= 1
0
2 )1log()log( dxxx
.7102153895707254988... sinh2
1201
164
3
kk
1 .710249833905893368351... 1
1 921 k kk
/
8 .71034436121440852200... dxx
x
0
2
cos12log4 GR 3.791.6
.71121190491339757872...
12/)3(2/)3(
122
2
1
2
1)1(
kkkkk
k Hall Thm. 4.1.3
=
1 2
111
kk
.711392167549233569697...
1
1)1(1
2
21
k
k
kk
3 .711471966428861955824...
51
222
2
53)537(
535
11
=
21 1)(
kk kF
.711566197550572432097... )3()2(22
)5(9)2,1,2(
MHS
.711662976265709412933... coth
42
3
.71181724366742530540... 2
13tan
133
4
1 .71206833444346651034... 1)()()(2
0
kkkk
1 .71219946584900048341...
22
)(
1 1 22
)(2
1
221
)4(
)2(
k
k
k s ks
k
kkk
k
1 .71231792754821907256...
1 3
)1(14log3log1
kk
k
k
.712319821093014476375...
22
1log
1
11)()1(
kkk
k
k
k
k
k
kkH
.71238898038468985769... 3
24
1 1
12
12
1
( )!( )!
( )!( )!
k k
k kk
4 .71238898038468985769...
0
3
3
0
2sin2
)1(2
2
3
x
x
k
kk
k
k
.712414374216044353028... 4log7
.71250706440044539642...
03 12
)1(
k
k
k
.712688574959647755609...
0
22
)!2(1
211
2coth
2 k
kk
k
B
e
J948
=
0 1
2/sindx
e
xx
1 .71268857495964775561...
12 14
21
2coth
2 k k
J948
3 .7128321092365460381...
3
2
3
1
2
3log33log3
2
ll Berndt 8.17.8
.71296548717191086579... cos ( ) sin ( ) sin1 1 1 1 1si ci
= log cos( )x x dx10
1
1 .713009166242679...
12 )(
11 dx
x
.71317412781265985501...
1 3
)1(1
)3/1()6/5(
)2/1(
k
k
k J1028
.71342386534805756839... 2
2
0
1
16
2
2
1
4
logarctanx xdx
1 .71359871118296149878...
1
)13(k
k
1 .714264112625455810924...
2 log)32)(1(
1
k kkk
.714285714285714285 = 7
5
1 .714907565066229321317... G2
.715249551433006868051...
12
2
2kk
k
k
H
.715636602030672568737...
2 )1(
11
2
)4(cos
k kk
.715884762286616949300...
1
02 )21(
arcsin
3
31log
2dx
xx
x
.71616617919084682703... 1
0
1
02
sinh1
cosh
4
1
4
3dx
e
xdx
e
x
e xx
.71637557202648803549...
222
11
2sin
22
k k
.71653131057378925043...
0
3/1
3!
)1(
kk
k
ke
.716586630228190877618... dxx )41log(22arctan5log1
0
2
.716814692820413523075... 27
1 .716814692820413523075...
1
22 )4/1(
)1(28
k
k
k
.716863707177261351536...
1
2
4125
256log
kkk
k
L
.716890415241513593513... dxee
eexx
x
1
0
2
2
1log
2
1
.717740886799541002535...
22
1)(
kkk
k
8 .7177978870813471045... 76 2 19
.718233512793083843006... 33
32 CFG G1
.7182818284590452354...
0 02 )2)(1(!
1
)3(!
1
!
12
k kk kkkkkke
=1
1 20 ( )! !k kk
1 .7182818284590452354...
0 1 2
2
)!2()2(!)!1(1
k k k
k
kk
H
kk
k
k
ke
2 .7182818284590452354...
0
/2
!
1
k
i
kie
= dxex
x
02)/1(
log
3 .7182818284590452354...
0
3
)!1(1
k k
ke
.718306293094622894468...
2
12
1log2
csch2
logii
=
1 12
1
2
11log
2
)2()1(
k kk
k
kk
k
.71837318344184047318... 22
1
3
1
2
1
4
3log
2
3log2log
12 22
22
LiLi
=
0 2xe
x
.718402016690736563302... dxe
xx
02
3
)1()4()3(6
.719238483455317260038...
13
1 133
)13(
kkk k
kk
.71967099502103768149...
2 !
)(
k k
k
.71994831644766095048... dxx
xxdx
x
x
0
5
5
052
cossin
)1(
log
48
11
.72032975998875729633...
12 244
1
2tanh
4 k kk
J950, GR 1.422.2
= dxx
x
0 2sinh
2sin GR 3.921.1
=
0
sinxx ee
x
.7203448568537890207... H
kk
kk
( )3
1 2
.720383533753944655666...
1
222 )1(
11
2
3cosh
2
5cos
1
k kk
2 .720416382101518554054... )2(2)3(5
2 .720699046351326775891...
0 )3/1)(1(
)1(
2
3
k
k
kk
.72072915625988586463...
02 5
1
2
19tanh
19 k kk
.721225488726779794821... 4
arcsinh
.72123414189575657124... 5
)3(3
.72134752044448170368... e4log4log
1 J153
8 .72146291064405601592... 22
k
k
k( )
.721631677641841900027...
1
1
23
)1(3
2arccsch
7
3
7
4
7
3
k
kk
k
k
1 .72194555075093303475... log
.722066688901...
12 1
)1(1
k
k
kk
.722222222222222222222 =13
181
1
1
2
40
log( )( )
xx
xdx
6 .72253360550709723390... )!1(
1
2
3
2
2
1
2/3
k
e
k
k
.72256362741482793148... 21 3
16
2
8
1
1
2
20
1 ( ) log log( ) log( )
x x
xdx
1 .72257092668332334308... 18
3
.72305625265516683539....
222 )13(
3
3
1)(
2
333log9
kkk kk
k
.72319350127750277100... dxx
x
0 sin2
sin
33
4
.72330131634698230862...
22 )(
1)(
k k
k
13 .72369128212511182729...
1
3
)!1(
3
3
12
k
k
k
ke
1 .724009710793808932286...
13
12 1
)1(4
)3(3
12 k
k
k
k
.72475312199827158952... ( ) log sin3 2
8 30
x x
xdx
1 .724756270009501831744...
1
K
4 .7247659703314011696... 105
16 3 , volume of the unit sphere in R7
.724778459007076331818...
1
1
1
1
2!
2)1(
!)!12(
)1(
2
1
2 k
kk
k
k
k
kk
kerfi
e
.72482218262595674557... 1
2
2
3 3
2
3
1
3 112
1
cot( )k
kkk k
1 .72500956916792667392... dxxx
x
0
22
)1)(2(
log
6
2log GR 4.237.3
.72520648300641149763... 150
97
40
, probability that three points in the unit square form an
obtuse triangle
.725613396022265329004...
11
)1log(
kkk
e
H
e
eee
1 .72569614761160133072... 2/
0
)tan3log(2
3log dxx GR 4.217.3
.72589193317292292137...
2
1
2
2
0
tanh sinsinh
x dx
x
.726032415032057488141... 3
1
8
.7264245534...
primep kkkpp 1
11
.72649594689438055318... log log2 23 2
.726760455264837313849...
0 2
2
2cosh
2sinh
)1log(4
2 dxx
x
x
GR 4.373.5
.7268191868...
primep pp 22 )1(
11
18 .726879886895150767705...
2
3 )1()()3(6)2(7k
kkk
=
232
23
)1(
1458
k kk
kkk
.72714605086327924743...
1
22
22
2
2sin
2 k
ii
k
keLieLi
i
2 .727257300559364627988... )4()2(
.72727272727272727272 = 11
8
.72732435670642042385... 1
2
2
42
0
2
sin
cos (cos ) sin/
x xdx
.727377349295216469724...
0
/1
!
)1(
kk
k
ke
.727586307716333389514...
5
32Li
.72760303948609931052...
2
312
7
23
1 2
( )( / )
k
kk
1 .72763245694004473929...
1 2!
11
kkk
.72784877051247490076...
2 !!
11
k kk
.72797094501786683705... sinh
1
6 .72801174749956538037... 2
.728102913225581885497... dxxx
x
124
2
14
3log
34
.728183137492543041149...
.72856981255742977004... log ( )212
21
2
3
23
2
3
Li
= log
( ) ( )
2
30
1
1 2
x
x xdx
1 .728647238998183618135... 2)( xofroot
22 .72878790793390202184... 7
30 1
4 4
21
log
( )
x
xdx
.7289104820416069624... 21
64 2 14 30
dx
x( )
.72898504860058165211...
cosh
1
.72908982783518197945... 45 5
64
4
31
( ) log
x
x xdx
.729329433526774616212... 221 e
.729439101865540537222...
1 )32(42log2
3
2log
3
4
k
k
kk
H
807 .729855042097228216666... e )2sinh2sinh(cosh)2sinh2cosh(cosh2
1
=
0 !
sinh
2
1 2
k
ke
k
keee
1 .7299512698867919550...
2 1
2
cosh(log ) cos( log )i
.73018105837655977384... 22
12
0 )12(2
1
2
22
28
2log
kk
kGk
k
k
Berndt 9.32.4
= 4/
0
4/
0
cottan
dxxxdxxx Borwein-Devlin, p. 138
= 4/
0 2cos
tan4/ dx
x
xx GR 3.797.3
= 4/
0
)sinlog(cos
dxxx GR 4.225.1
= 4/
0
)cottanlog(
dxxx GR 4.228.7
=
1
02 )1(
arctandx
xx
x GR 4.531.7
1 .730234433703700193420... dxe
eerf
ex
x
0
4/1
2
2
11
2
.730499243103159179079...
3/2
3/1
.731058578630004879251...
0
)1(2
2/1tanh1
1 k
kk ee
e J944
=
2
1
!
)(21
k
k
k
k
.73108180748810063843...
1
03
03
03
2
log1
1
1
log
1
log
27
2dxx
x
xdx
x
xxdx
x
x
= dxee
exxx
x
03 )1(
)1( GR 3.411.26
1 .73137330972753180577...
2 21
1
kk
1 .73164699470459858182... 3
.731771876673200892827...
1 2!2log
2
1
kk
k
k
HEie
.731796870029137225611...
2233
1 )1(
3)1)3(3
kk kkkk
2 .731901246276940125002...
1
7
/1
7 )!7(
)(
k
k
k k
e
k
k
.732050807568877293527... 3 11
2 1
2
0
( )
( )
k
kk k
k
k
1 .732050807568877293527...
0 0
2
6
2
6
1
3tan3
k kkk k
kk
k
k
3 .732050807568877293527... 12
5tan32
AS 4.3.36
21 .732261539348840427114...
04
442
)1(
log
45
7
3
2dx
x
x
.732454714600633473583... 1
1 .732454714600633473583... 1
1 .732560378041069813992...
2
1
2
12log
2
1
8
2log3 22
iLii
iiLiG
=
12
2
11
1log
x
dx
x
x GR 4.298.14
.732869201123059587436...
22 1
)(
k k
k
.733333333333333333333 = 11
15
=
1 1 )3(4
12
)3()!2(
!)!12(
k kk kk
k
kk
k
.733358251205666769682...
2232
1 )1(
11)13(
kk kkkk
3 .73345333387461085916...
1
,1
11csc
.73402021044145968963...
1 0 )(2
1
kk k
.7340226499671578742... 16 16 22
32 3
2 12
3 121
log ( )( )!
( )!
k
k kk
=
11
1
134 2
)3()1(
2
2
kk
k
k
k
kk
.73417442372548447512... 2 12
12
1
( )!
!
k
k kk
.7343469699579425786... e
x xdxecoslog coslog
1
2 1
.73462758513149342362... 2
2 21
1
42 41
sin sinh
kk
1 .734956638731129275406...
12 22
11csc2sinh
3
2
k kkh
.735105193895722732682... 3
4
CFG G1
1 .735143209673105397603...
1
3
)5(!11033000
k kk
ke
2 .73526160397004470158... ( )k
kk 21
.735689548825915723523...
1
22
12
/1
)1(1csch1
2 k
k
k
.73575888234288464319...
1 1
5123
!
)1(
)!1()1()1,2(
2
k k
kk
k
k
k
kk
e
=
0
1 2
dxex x
3 .7360043360892608938... 2 11
21 4
40
/ log
xdx
1 .736214725290954627894...
1
2 2
1
2 2
i i
1 .73623672732835559651...
1
2 )1)1(k
k kH
.73639985871871507791...
11 21
4!)!12(
!)!2(
3
1
27
32
kkk
k
kk
k CFG F17
=cos
( cos )
/ x
xdx
2 20
2
.73655009813423404267... ( )( ) ( )
11 1
2
k
k
k k
k
.736576852723235053198...
1
)(dx
x
xx
.736651970465936181741...
1 22
/1 2
)!1(
1)2(
k k
k
k
e
k
k
.736842105263157894 = 19
14
1 .736901061414085912424... )3(3
.73710586317587034347... ( ) ( )( ) ( )2 2i i
.73726611959445520569... ( ) /k
kk
kk k
1
2 3
1
2
3 2
2
arctanh
.73735096638620963334...
1 )1(
1
k k kkF
3 .73795615464616759679...
0
22
4/sinh
4/cosh)1log(12log
281624
x
xx
GR 4.373.6
1 .7380168094946939249...
1 1
22
3
)(
13k kkk
kk
.73806064483085791066...
3
23Li
4 .73863870268621999272... p
kk
k !
1
.73873854352439301664...
2 6
)(5)1(
3
2log5
4
3log5
34
51
kk
kk k
.73879844144149771531...
1
1
2
)1()1(
2
3log
3
1
kk
k
k
k
.7388167388167388167 =
2
1,6
693
512
.739085133215160641655... xxofroot arccos
.7391337000907798849...
13
1
33
2 sin)1()()(
212
1
k
kii
k
keLieLi
i Davis 3.34
19 .739208802178717237669... 22
.739947943495465512256... 1
1
1
1
1
12k kkk
j
j kjk
( )
.740267076581850782580...
12 3
1
6
13coth
6
3
k k
J124
7 .740444313946792661639...
1
4
21010 !!)2()2(3)2()2(2)2(2
k kk
kIIIII
.740480489693061041169... 23
CFG D10
.740520017895247015105...
1
2 1 kkk
e
F
ee
e
.740579177528952263276...
3
21
2
2
)()1(
24
52log
4
)3(7
kk
k kk
=
132
2
)12(2
21118
k kk
kk
.740726784269006256431...
2 )2,(1
1
K kS
.740740740740740740740 =
1
2
427
20
kk
k
1 .740802061377891359869... 31log3
.7410187508850556118...
3
1
3
)1(
3
1
6
3
2
3log3
112
hgk
kk kk
k
= dxx
x
1
02
3)1log(
.741027921523577355842...
1
21
1
1
!
)1(
!
)1(
2
5sinh
5
2
k
k
k
kk
k
k
k
F
e
.741089701006868123721... dxe
xx
1
0 1
sin
.741227065224583887196...
122
2
)2/1(
1
2coth2
32
k kk
1 .741433975699931636772...
22
3
)1(24
2log5
8
7
kk k
k
3 .741657386773941385584... 14
.741982753502604212262...
2
1)( 1)1(k
kk e
11 .743029813183056451016...
2
2 2)1()(k
kkk
.7431381432026369649... 22
2 10
1
Gx
x xdx
log arcsin
( )
=arccosx
xdx
10
1
GR 4.521.2
=x x
xdx
sin
cos
/
10
2
GR 3.791.12
=log( )1
1 20
1
x
xdx GR 4.292.1
.74314714161123874... pf k
ek
( )
0
.74343322147602010957...
2 log)1)(1(
1
k kkk
7 .743684425536057169182...
0
34
1
log)4(6611
90dx
e
xxx
.74391718786976797494...
1
21
)()2(log
)()2(
kk k
kk
k
k Titchmarsh 1.6.2
.744033888759488360480...
1 )12(2kkk
k
2 .744033888759488360480...
1
1
11 2
)(
222
1
12 kk
kkk
kk
kk
=
1 )12(2kkk
k
.744074106070984777872... 2
)3(G
.744150640327195684211...
1
3
3 5/sin
125
3
k k
k GR 1.443.5
.74430307976049287481... cos/
x dx0
4
.74432715277120323111...
1
1
12)1(
2
1arcsinh
55
8
5
2
k
k
k
k
2 .744396466297114626366...
log
5 .74456264653802865985... 33
1 .74471604990971988354... = 2
204 2 2 1
log xdx
x
1 .7449110729333335623... e
e
8
1
8
5)1sinh21cosh3(
4
1
=
1
2
)!12(k k
k
.74521882569020979229... )(log)(2
kkk
.7453559924999298988... 5
3
1
5
2
0
( )k
kk
k
k
.74562414166555788889... 1sinsin
1 .745804395794629191549... 16 2 241
1 4
1
2 21
( log )( )
( / )G
k k
k
k
.74590348439289584727... 1)2()(1
kkk
7 .7459666924148337704... 60 2 15 21 .74625462767236188288... e8
2 .746393504673284119151...
1
6
/1
6 )!6(
)(
k
k
k k
e
k
k
9 .7467943448089639068... 95
.7468241328114270254...
21
1
2 10
erfk k
k
k
( )
!( ) J973
= e dxx2
0
1
5 .74698546753472378545... 2
32 2 4
1
2
22
2
2
12
20
1
loglog
( )Li
x
xdx
.747119662029117550247...
( ) ( )
( )
k k
kk
1
2
2 .747238274932304333057... 1
22 5 15 6 5
= 5 5 5 5 5 5 5 ... [Ramanujan] Berndt Ch. 22
17 .7476761359958500014... 59535 7 62
24 1
6 7
30
1 ( ) log
( )
x x
x
.747727141203212136438...
2
1)1(
)2()(
k k
kk
2 .747896782531657045164... G3
9 .748110948000252951799...
1
2
!
2)2(2log
k
kk
k
HEie
1 .748493952693942358428...
1
2
)3(2
3
)3(
2
)5(11
k
k
k
H
.748540032992591012012... )2()2(2
1coth
22 ii
= )1()1(2
1iii
i
= 2)12()2()1(1
1
1
23
kkkk
k
k
k
k
.74867010071672042744...
2 1
128
1 3 1 1
4G
kk k
kk
( ) ( )( )
2 .748893571891069083655... 8
7
.748989900377804858183...
1
41
4
1
3
1)4(
x
dx
kk
.74912714295902753882... 3 3
3
1 21 12
1
( ) ( )!
!
k k
k
k
k
.749306001288449023606... 2/e
.749502656901677316351...
1 )1(2
1
kk k
.750000000000000000000 =
1 3
)2log(sin4
3
kk
kii
=
22
1 1
11)2(
kk kk J369, J605
=
12 2
1
k kk K133
=
13
2)22()2(
2
1
kk
kkkkk
= 11
3 21
( )kk
=
13
4log
x
dxx
1 .750000000000000000000 = 4
7
2 .750000000000000000000 =11
4
1
1
2 4 6 8
2 6 8
p p p p
p p pp prime
372 .750000000000000000000 =
1
6
3kk
k
.75000426807018511438... 4
33
1
21
20
1
arctan tancos
cos
d
1 .750021380536541733573...
1 )()()1(
k ekk
eehge
1 .75013834593848950211... 2
22
1
20sintan
kk
k
.7507511041240729095...
2
3
4)( log
)3(k
iv
k
k
.751125544464942482859... 4/1
.75128556447474642837...
1 1
21
)2(
)1(28
)3(5
k k
kkk
k
k
H
k
H
=
log( ) log( )1 1
0
1 x x
xdx
.751300296089510340494... dxx
02 13
11log12
3
7 .75156917007495504387... dtt
kk
k
0
2
13
13
4tanlog
)2/1(
)1(
4
Borwein & Borwein, Proc. AMS 123, 4 (1995) 1191-1198
.75159571455123294364...
1
22
1
22 1
2
121
log ( ) ( )! ( )e
Eik k k
kk
k
.75173418271380822855...
01010
4/10
4/10 )!2()!2(
)1(,1,1
2
1)1(2)1(2
2
1
k
k
kkiFiFJI
1 .751861371518082937122...
14
2/1
0 2!
)4(
k
k
kk k
e
k
k
1 .751938393884108661204 k
k
1
2
2
2
1 .75217601958756461842
3
1,3
6
1,3
108
1
3
1
27
34
36
1 )2(3
=
2
31
2
31
3
1
2
)3(33
iLi
iLi
2
31
2
31
3
333
iLi
iLi
i
= 1
06
2
1
log)1(
x
xx GR 4.261.8
.7522527780636750492643
4
.75267323992255463004...
0
4
4
042
sincos
)1(
log
96
23dx
x
xxdx
x
x
.753030298866585425452... )4(34
11 .753304951941822444427... )!1(
2
)!1(
122)2(
001
2
k
k
k
k
kk
kIe
kk
1 .75331496320289736814
122
11
k kk
.75338765437709039572... e Ik
k kk0
1
1
21
2 1
2
( )!!
( )! !
1. 75338765437709039572... e I e dxx0
0
11
2
2
cos
.75375485838430604734...
22
1
21 1 2 1 2coth i i
= 2 1
21 2 2 2 13
1
1
1
( )( ) ( ) ( )
k
k kk k
k
k
k
k
. 75405876648941912567...
2 )(
)2()1(
k
k
k
k
9 .75426251387257056568... 15
25 2 3
1
3
21
( ) ( ) ( )( )
H
kk
k
(conj.) 18 MI 4, p. 15
.754610025770972168663...
00
200 xarcsinh1
)1()1(2
dxexe
dxYH x
x
.754637885420670280961...
e
xe
dxxee
e
0
11
.75464783578494730955... 6
2
31
126
0
1
loglogx
xdx
4 .75488750216346854436... 3log3 2
1 .75529829246990261963...
2 21
1 120
log( )
( )( )
k
k k k
.75539561953174146939...
2
15
2
15log
10 22
2
Li
Berndt Ch. 9
=
2
53
30 2
2
Li
MIS
1 .75571200920378862263...
1
/1
1
1!
)5( 5
k
k
k
ek
k
3 .756426333323976072885...
2 2 2 4 2 2
1
1 2
2 1 2
2 21
csc cot csc( )
( / )
k
k
k
k
6 .756774401573431365862... sinh2
2 412
14
kk
.756943617774572695997...
0 )!2(k
k
k
B
.757342086122175953454...
0 )1(2)2log(2
x
dxEi
x
.75735931288071485359... 235
.7574464462934689241... 2 5 2 3 34
2 1
2
41
( ) ( ) ( ) ( )
( )( )
k H
kk
k
.75748739771880741669... 1
120 e
k
k
.757612458715149840955... 9
4)3(
.757672098556372684654...
1
22
)12()2(8
1
2
)3(
8 k
kkk
=
223
2
)1()1(
1
k kk
k
1 .757795988957853130277... 4
)3(3
32
3
= )()22()()22(4
1
4
)3(333 iLiiiLii
=
1
032
22
1
04 1
loglog
1
1dx
xxx
xdxx
x
x
.757872156141312106043...
0 111
xe
dxerfe
x
1 .758268564325381820406...
3
csc93
cot323
cot3372
2
=
1
2
)2(3k
kk
k
.758546992994776145344...
0
)1(
1 kk
k
.7587833022804503907... 11 2 2 1
0
1
eEi Ei e e x dxx( ) ( ) log log( )
.758981249114944388204...
3
2
2
3log3
6
3
2
3hg
.75917973947096213433... 2 3 2 ( ) ( )
1 .75917973947096213433...
1
2 1)2(1)2()3(2k
kk
=
12 )1(k
k
kk
H
.7596695583288265870... x dx
e ex x
1
.759747105885591946298...
.759835685651592547331... 4/13
.759967033424113218236... 2/2
2/2
2
2
1
16
1
12ii eLieLi
= 2
cos)1(
12
1 k
kk
k
.75997605244503147746...
1 12 )!(
1
!
)(2
k jj
k kk
ke
1 .760000000000000000000 =
1
3
425
44
kkkk FF
.7602445970756301513... cos( )
( )!
1
2
1
2 20
k
kk k
AS 4.3.66GR 1.411.3
.7603327958712324201...
1 5
11
kk
.760339706025444015735...
1 )!2(
),2(1
k k
kkS
.760345996300946347531...
1 16
)1(
16
)1(1)32log(
3
1
k
kk
kk J83
=
1
1
1 22
)1(!)!12(2
!)!2()1(
k
kk
kk
k
kk
kk
k J267
=
0 !)!12(
!)1(
k
k
k
k J86
=
22 3x
dx
1 .76040595997292398625... 16 232 2 1
16 1
2
4 30
log( )!
! ( )
G k
k kkk
17 .760473930229652590715...
1
2
5
10 )!()2(4)2(5
k k
kII
.761130385915375187862...
22
1
2 )1(
)(1)1()()1(
kk
k
kk
kkk
.76122287570839003256... cot2
.7612801797083721299...
13
1log
1
k k
k
k
.761310204001103486389...
4
335
4
33
4
35
4
33
32
11
iiiii
=
02 1
)1(
k
k
kk
1 .761365221094699778983... u k
kk
( )
!
1
.761378697273284672250...
2
/1
1
1
!
1)1(
k
k
k k
e
k
k
.761390022682049161058...
2 !
)(
k k
ke
.761393810948301507456...
0
2sinh)1(4
2
dxxee x
.761549782880894417818... )1(log
.76159415595576488812... iie
etan
1
11tanh
2
2
J148
=
1
2)1(21k
kk e
=
1
2
)!2(
)14(4
k
kkk
k
B AS 4.5.64
= 1
02cosh x
dx
1 .761634557784733194248... )5(3))3(1(6
)3(2)3(27222680
97 22
46
=
14
1
k
kk
k
HH
.761747999761543071113...
03
3/1
233
2
x
dx
.761759981416289230432... 2
3sin
.761964863942319850842...
1
1
!!
)1(
2
1
21
k
k
keerf
e
.7619973727342293746...
( ) ( ) ( )( / )
1 33 2
41
1
3
1
54
1
3
4
9 3
26
33
1 3
k
kk
.762054692886954765011...
020
12
1
2 2
xe
dxK
ex
2 .76207190622892413594... 2 3
30
12
3
2
6
1
2
1 2log log log( ) log
Li
x x
xdx
3 .7621956910836314596... cosh( )!
22
4
2
2 2
0
e e
k
k
k
AS 4.5.3, GR 1.411.2
=
0
22 )12(
161
k k J1079
2 .762557614159885046570...
1
71
2!)1(
128
583
kk
k
k
k
e
.76258964475273501375...
2
0
2
)()(
kk
kk
1 .762747174039086050465... dxx
x
e
dxx
0 1
2
arcsinh
11arcsinh221log2
19 .763312534850599760254...
44
)3(
4
)()3)(2)(1(
4
3
kk
kkkk
.763459046974903889864...
1
3
2)!1(4
32
kkk
ke
.76354658135207244811... 2 1 1 11
2 10
sin cos( )
( )!( )
k
k k k
5 .76374356163660007721...
4
3
4
12log72log6 2 ll Berndt 8.17.8
.76378480769468103146...
1 12k kk
k
H
H
.76393202250021030359... 53
.76403822629272751139... ( )( )
12
k
k
k
k
.76410186989382872062...
12
2
)4/3(
1
9
168
k kG
.764144651390776192862...
2
)(1)(k
kk
.764403182318295356985...
14
2 sin
6
1
k
s
k
k
.764499780348444209191... 1
2 1
1
2 11
4 1
2 4 11
1
1
12k
k
k
kk
kk
k k
( )
( )( )
( ) Berndt 4.6
=
1 )12(2
11
kkk
=
2 )2,(2
)1(
k
k
kS
6 .764520210694613696975...
1
2
20
24 )(
)4(
)2(
72
5
k k
k
Titchmarsh 1.2.10
1 .764667736296682530356... )4()3()2(
.7647058823529411 = 17
13
.7649865348150167057...
242
4/14/1 31
32
3sinh3sin
k
.76519768655796655145...
0 02200 )!)!2((
)1(
4)!(
)1()()1(
k k
k
k
k
kkiIJ
.765587078525921485814...
1
3
3 3/2sin
81
2
k k
k GR 1.443.5
.76586968364661172132...
1 )2)(1(
11
k kkk
4 .7659502374326661364... 4
3
6
31)3(180
2
)4(13)5(360
63
8 26
= 1)()1( 5 kkk
.7662384356489860248... 3
2
5
3 2 30
log/
dx
ex
.766245168853747101523...
1
02)1(
)2()1(1
21 dx
x
eEiEi
e
e x
.766652850345066212403... iii 1,3()1,3(cschcoth1 23
=
1
33 )(
1
)(
1
k ikiki
=
0 0
22
1
sin
2
1
1
sindx
ee
xxdx
e
xxxxx
8 .767328087571963189563...
1 )!1(1)1(12
k
k
k
kHEi
.767420291249223853968...
1
2)2(logk
k
1 .767766952966368811002... k
kkk
k
k
k
k
kk
8
2
2!)!2(
!)!12(
22
5 2
01
2
1 .76804762350015971349... 3/23
3/13
3
)1()1(3
2
381
8 LiLi
i
=
03
21
02
2
1
log
1
log
x
dxx
xx
dxx GR 4.261.3
5 .7681401910013092414...
12 1
213sinhcsc
3
1
k kh
.76816262143574939203... 61 3
86 6
21 3
23
3 3Gi
Li i Li i
( ) ( )
( )
=2
)3(21
86
3 G
=arcsin3
20
1 x
xdx
.7684029568860641506... 13
82
4
2
22
( ) coth( ) k
kk
= ( )2 11
16 2
2
kk kk k
1 .76842744993939215689...
1
/)1(12/ 12
22k
kk
.76844644669358040493...
0
33/2
cos3
1
26
13 3
dxxe x
4 .7684620580627434483... =
112
11
kk
.768488694016815547546... 3G
.7687004249195908632... =1
20 ( )!! !!k kk
3 .768802042217039515528... cos ( ) cosh ( )( )!
/ /2 1 2 12
41 4 1 4
6
0
k
k k
2 .768916786048680717672...
0
2
2
1
)2log(21log1arcsinh dx
x
x
6 .769191382058228939651...
2
0 cos24
cos
3
21 dx
x
x
.769238901363972126578... i2Re2logcos
1 .769461593584860406643... 2 2 4 6 2
0
1
log arccos logG x xdx
31 .769476621622465729670...
1
4
32
3log15
61
411
kk
kHk
.769837072045672480216... 22
222 csch4
2sinhcsch8
cschcoth4
1
=
1
1
2
1)2()1(4
1)2()()1(
k
k
k
k kkkkk
=
122
2
)1(k k
k
.769934066848226436472... 8
7)2(
.76998662174450356193... 2 2 11
1 211
0
Jk k
k
kk
( )!( )!
.770151152934069858700...
0
2
)!2(
)1(
2
1
2
1
2
1cos
k
k
k J777
.770747041268399142066...
0 2!)1(2
1
kk
k
k
B
e
.77101309425278560330... 4
11
4
2
22
2
14 1 2
2
20
eK
xdx
x/ ( )sin
= 2/
0
2 )tan2(sin
dxx GR 3.716.9
.771079989624934599824...
1
2
4
)2(
kk
k
1 .7711691814070372309...
1 3
2
23
)3(352log216
1
k
k
kk
kG
1 .771508654754528604291... )3(4)2(4
.771540317407621889239...
11 )2(
)1,2(1
!2
2
1cosh
kk k
ke
k
k
1 .7720172747445211300...
12 1k k
k
.772029054982133162950...
sinh2
1
.77213800480906783028...
2 2
1
2
1
2 11 2
40
ex
xdx
/ cos sincos
1 .772147608481020664137... 2log12
)3(2
1 .7724538509055160273...
1
2
=( )!
( )!
k
kk
1
2
1 1
.77258872223978123767... 4 2 22 1
2 1 21
log( )!!
( )! ( )
k k
k kk
=
22
0 2
1)(
)2(2
1
kk
kk
k
k
=1
1120 ( )( )k kk
= dx
e ex x
1 20 /
= 1
0
1
0 1dydx
xy
yx
1 .77258872223978123767...
1
1
212log4
kkkH
2 .77258872223978123767... )1(4
112
2
)1(2log4
01
kk
kHkk
kkk
k
=
0 )2/1)(1(
1
k kk
=
222
)(
kk
k
=
03
4 2sindx
x
x
.773126317094363179778... dxx
x
1
0
2/9
1256
63 GR 3.226.2
.773156669049795127864...
2 2 )1(
1)(
1
k primep primepkpk pp
kp
=
23
)(2
k kk
k
= )(log)(
2 1
skk
k
s k
1 .773241214308580771497...
1
2
)!2(
22sinh22cos2
4
1
k
k
k
k
.773485474588165713971... 7
3)3(
.773705614469083173741... 4log6
1 .773877583285132343802...
1 1
1
k kk FF
.773942685266708278258... )sin( e
40 .774227426885678530404...
0
4
!15
k k
ke GR 0.249
1 .77424549995894834427... 3
264
1
1 4
1
1 4
3 1
3
1
31
( )
( / )
( )
( / )
k k
k k k
5 .77436967862887418899... 1
2 0
sinh sin sinh
x x dx
.7745966692414833770... 3
5
1
6
2
0
( )k
kk
k
k
1 .774603255583817852727...
1 82
2224464
kk
kk
.774787660168805549421...
0 2
)1(
kk
k
k
2 .77489768853690375126...
2
1
12
5
6
3 1 1
62
( ) ( )( )( )
k
kk
kk
8 .7749643873921220604... 77
.776109220858764331948...
1
1
0 !!
)1()2(1
k
k
kkJ
.776194758601534772662...
21
1
kkk
k
3 .776373136163078927203... )3(
.77699006965153986787... dxx
x
20
1
2 .77749955322549135074...
215
/1
)!5(
)(
kk
k
k
k
k
e
.777504634112248276418... 2
1)1(
1)1log(1
6 22
2
eLi
eLie
=
02 )!1(
11
k
k
k
B
eLi
=
1
0 1xe
dxx
.77777777777777777777 = 9
7
17 .77777777777777777777 = mil/degree
14 .778112197861300454461...
0
12
!
22
k
k
ke
1 .778279410038922801225... 4/110
1 .77844719779253552618... pf k
e kk
( )
!
0
.778703862327451115254... tan ( ) ( )
( )!
1
2
1 2 2 1
2
1 2 1 22
1
k k k
k
k
B
k
.778758579552758415042...
11
1 1cos1
)!2(
)2()1(
kk
k
kk
k
.778800783071404868245...
0
4/1
4!
)1(
kk
k
ke
1 .778829983276235585707...
02
2
2 1
coslog
1
2log
x
x
e GR 4.322.3
1 .779272568061127146006... )3(
.77941262495788297845...
3 3
2
3
2
3 10
1
arctan
e
e
dx
e ex x
1 .78005080846154620742...
1 )3(
11
2
5cos
6
k kk
.780714435359267712856... 3
tanh
.7808004195655149546... k
ekk
11
.780833139853360038254...
1 2
)(
kk k
k
1 .781072417990197985237...
0 !k
k
ke
=nn
n
loglog
)(lim 1 HW Thm. 323
=
npn
pn
111log
1lim
.781212821300288716547... )1(1Y
.781302412896486296867...
122
)1()1(2 )13(
1
)23(
1
3
2
3
1
9
1
k kkg
J310
= dxxx
x
1
021
log GR 4.233.1
.78149414807558060039... 2
221
k
k
k
.781765584213411527598...
3
8
2
3log3
3210
21
6 .781793170552500031289... )2()(13
)3( 22
kkk
=
223
23
)1(
124
k kk
kkk
1 .78179743628067860948... 2 11
6 55 6
1
1
/ ( )
k
k k
.781838631839335317274...
1
2/1 12k
k
1538 .782144009188396022791...
4
1)3(
113 .782299427444799775042...
1
2/2
!
sinh2
2
1
k
kee
k
kee
6 .78232998312526813906... 46
1 .7824255962226047521... 1
0
2log]1},2,2,2{},1,1,1[{2
xe
dxxHypPFQ
.78245210647762896657426... F kkk
( )2 12
.78279749383106249726... 1
6
8 2
9
33
1
log log( )x
xdx
.7834305107121344070593...( )
1 1
1 0
1k
kk
x
kx dx Prud. 2.3.18.1
.783878618887227378020...
2
1)(
k kF
k
1 .783922996312878767846...
2
52/)51(
)!2(5253515
20
1
k
k
kk
Fee
4 .784000000000000000000 =
1
41
2)1(
125
598
kk
kk kF
2 .784163998415853922642... 2
2/3
.7853981633974483096...
0 12
)1(
2
1arcsin)1(
4 k
k
k
AS 23.2.21, K135
= )(Im1log(Imarctanh 1 iLiiii
=
3
1arctan
2
1arctan4 Borwein-Devlin p. 73
=
239
1arctan
5
1arctan4 Borwein-Devlin p. 73
=
1985
1arctan
20
1arctan
4
1arctan3 Borwein-Devlin p. 73
=
1 )14)(14(
121
k kk GR 0.232.1
= ( )
( )sin( )
1
2 12 1
1
21
k
k kk J523
=
1 12
)12sin(
k k
k GR 1.442.1
=
1
2/sin
k k
k
= sin3
1
k
kk
GR 3.828.3
=( ) cos( ( ))
1 2 1
2 10
k
k
k
k
/ 2 Berndt Ch. 4
= arctan( )
2
2 2 20 kk
[Ramanujan] Berndt Ch. 2, Eq. 7.3
= arctan1
2 21 kk
[Ramanujan] Berndt Ch. 2, Eq. 7.6
=
12 1
1arctan
k kk K Ex. 102, LY 6.8
= ( ) arctan
121
21
k
k k [Ramanujan] Berndt Ch. 2
=
1
2)12(
11
k k GR 0.261, J1059
=k k
kk
( )
( / )
1
1 2 21
J1061
=
02
02
02
022 14224)1( x
dx
xx
dx
x
dx
x
dx
=
041 x
dxx GR 2.145.4
=
03/24
023 21 x
dx
xxx
dx
= xdx
x1 40
1
=
log
( )
log
( )
x
xdx
x
xdx2 2
02 3
01 1
=
0xx ee
dx
=sin
sin
/ x
xdx
2 20
2
= dxx
xxdx
x
xxdx
x
xx
0
2
22
0
2
0
cossincossincossin
= dxx
xxdx
x
xx
0
3
3
02
2 cossincossin
=
02
4sindx
x
x
=
03
cossin
x
xxx
=
2/
02)cos(sin
xx
dxx
= dxx
xx
2/
02sin
cot1
=
0
sin)( dxxxsi
=
0
sin1 dx
x
xe x
= dxxx2/
0
2 tanlog)(sin
= 1
0
logsinlog
)1(dxx
x
xx GR 4.429
=
0
)arccot1( dxxx GR 4.533.1
= 1
0
cossin)( dxxxx GR 6/469/1
1 .7853981633974483096... 4
12
2 30
k
k k
k
2 .785562097456829308562...
1 !
)(
k k
kp , p(k) = number of partitions of k
.78569495838710218128... cos sin( )
16 161
1
8 41
k
k k J1029
.7857142857142857142 =14
11
.78620041769482291712... 8
9
8
27
1
2 2 arcsinh
= 2 10
111
2
1
8
12
2
Fk
k
k
kk
, , ,( )
1 .78628364173958499949...
212 2
log
)!1(
1
2
log
kk
n
nkk
k
n
kk
6 .78653338973407709094...
1
4 1)2(k
k
1 .78657645936592246346... 1
11
112 1
1 ( ) ( )k k p pkk k
kp prime
Silverman constant
1 .78657645936592246346... 1
11
112 1
1 ( ) ( )k k p pkk k
kp prime
.78668061788579802349... )2sinh2(coshsinh)2sinh2cosh(cosh2
1e
=
0
/1
!
sinh
2
1 2
kk
e
ek
kee
.786927755143369926331...
1
51
5
1
4
1)5(
x
dx
kk
.78693868057473315279... 22 1
1 20
e k
k
kk
( )( )!
.78704440142662875761... ( ) ( )log
( ) ( )2 2
1
2
3
23
1 23
2
20
1
Lix
x xdx
2 .787252017813457378711...
12/ 1
1
kke
Berndt 6.14.4
1 .787281880354185304548...
0
3
)!2(
3
9
4
k
k
k
e
4 .78749174278204599425... !5log
.787522150360869746141... csch
24
2log
2coth
42
1 2
2
12
12
1
2
1
4
1 iiii
=
1
21
)1(
1)1(
k
k
kk
k
.78765858104243429... 2/1)3()3(68 H
.78778045617246654606...
2 )(
1
k klfac , )...,,1()( kLCMklfac
3 .78792362044502928224...
1
3 1)2(k
k
.78797060627038829197... 15 5
25 4
2
( )( ) ( )
.78800770172316171118...
1222 4
log
222log
4
1dx
x
xiLi
iLii
1 .78820673932582791427...
3 122
0 )(log)(log)(log)(n kkk
knkkk
.788230216655105940660... log log sin( )
( )!( )2 2
1 4
2 2
12
2
1
k
kk
k
B
k k [Ramanujan] Berndt Ch. 5
= 1
0
2
x x
xdx
cos [Ramanujan] Berndt Ch. 5
.788337023734290587067... coth
4
.7885284515797971427...
( )( ) ( )
( )
3
2
3
161 2 1 1
2
12
3
2 32
k k kk
kk k
.78853056591150896106...
22
22
log11log
1
1
1)(
kkk kk
k
kkk
k
=
1 21 2
111)(
n kn
n kn kk
Lik
k
1 .78853056591150896106...
2
1)(1k
kk
k
1 .78885438199983175713... 5
4
8 .78889830934487753116... 3log8
.789473684210526315 = 19
15
4 .78966619854680943305... =
1
1 !!
2222
k kk
kI
.78969027051749582771... ( )( )
log
11
2
k
k
k
k
2 .789802883501667684222...
1
2
43
4log44
81
188
kk
kHk
.790701923562543475698...
1 2
1
k kk H
.7907069804229852813...
101 !!!)!2(
1
)!2(
!!
)1(!!
1
kkk kkk
k
kk
.7916115315243421172... H
kk
k
3
31 1( )
1 .791759469228055000813...
11 6
5
6
56log
kk
k
kLi
1 .791831656602118007467...
1
2
1
2/1
)1(1
2csc
2 k
k
k
1 .792363426894272110844... )1()1()(4
coth22
)1()1(22 iiiii
=
12 1k
k
k
H
.792481250360578090727...
1
4 222log2
3log3log
xx
dx
2 .79259596281002874558... 2
7
.792902785140609359763...
1 2
1
kk ke
1 .793209546954886070955...
2
2
.7939888543152131425...
1 4
)1(1
)8/3()8/9(
)2/1(
k
k
k J1028
.794023616383282894684... e2 Berndt 8.17.17
.79423354275931886558...
0
)1()1()1(
1
sin)1()1(
2)(Im dx
e
xxii
ii
x
1 .79426954519229214617... kk
Hk 21
1)1(
9 .79428970264856009459... 1
42
20 0 2
1
I e Ie
k k
kk
( )sinh cosh
( !)
25 .794350166618684018559... )3(
3
.794415416798359282517...
0
3
)2)(1(2202log30
kk kk
k
.795371500563939690401... 5/1
.7953764815472385856... HypPFQ
kk
k
k
k
[{ , , },{ , }, ]( )
( )
1111
22
1
4
1
120
.79569320156748087193... 2
sin
.795749711559096019168... Lie
Lie2 3
1 1
.7958135686991373424... tan tanh1 1
4 .7958315233127195416... 23
.795922775805343375904...
1
2
3kkkH
.79659959929705313428...
Eik k
e
xdx
k
k
x
( )( )
!1
1 11
1 0
1
J289
= HypPFQ[{ , },{ , }, ]1 1 2 2 1
= xe dxx e x
0
= log x dx
ex0
1
= ( )
!( )( )
( )! !
11
112
0
1
1
k
k
k
kk k k k
8 .7965995992970531343... 8 1 12 11
4
1
Eik
k kk
k
( ) ( )( )
! Berndt 2.9.8
.796718392705881508606...
1
3 1)3(k
k
.79690219140466371781...
2
1)2(
)(
k k
k
.797123679538449558285... dxxx
x
0
222
)2)(1(
log2log
9
2log GR 4.261.4
.797267445945917809011...
112
)1(1
3
2log
2
11
kk
k
k
1 .7977443276183680508...
2
2
)1(
log
k kk
k
.79788456080286535588... 2
2
.797943096840405714600...
22
3)2()()1(
1)3(2
k
k
k
kkk
k
9 .7979589711327123928... 96
.7981472805626901809... 133
32
31
32tanh
3 3
e
= )66()56()36()26(0
kkkkk
1 .7981472805626901809... 333
32
332tanh
3
e
= 13
3
3 3
2
3 3
2
i i i
=
02 1
1
k kk
3 .79824572977119450443...
1
52/51
)!1(55
)53(2
k
k
k
Fee
.798512132844508259187...
3
1 )(!
)1()1(
k
k kk
kk
.798632012366331278403... e
k k k
k
k
2
0
1
8
2
1 3
!( )( )
.79876560170621656642... )1(
1()1(2)2(1 2
=
2
2
2
1)()(
1
k
k
k
kkk
.798775991447889498103...
1
1
)1()!2(
4)1(2sin
2
2cos
2
3
k
kk
kk
.7989752939540047561... 2 1 1
1
2
120
sin ( )
( )!( )
k
k k k
.79917431580735514981... ( )
log2 1
2
11
1
21 12
k
k k kkk k
1 .799317360448272406365...
2 12 22
)(
2
3
k jk
j
kk j
kkk
.799386105379510436347... 2
2log
sinh44
1
2
1
2
1
21
21
8
1 iiii
=
023 1
)1(
k
k
kkk
.79978323254211101592...
41 24
0
2
e x dxcos( tan )/
GR 3.716.10
.800000000000000000000 =
1 4
)1(
5
4
kk
k
J352
= F k
kk
2
1 4
.800182014766769431681...
1112
1
2
1sin
1
2)!12(
)2()1(
kkk
k
kkk
k
1 .8007550560052829915... log( )k
kk
12
1
= dxx
xH
12
)(
6 .80087349079688196478...
1
)3( )(
k k
k
.80137126877306285693... 2 3
3
( )
.801799890264644999725... 7 3
8
1
4
2 1
4
2
1
( ) ( )
k kk
k
12 .80182748008146961121... log9!
.8020569031595942854... ( )32
5
.80257251734960565445... 1
5
14
5 52
12
1
1
1
arccsch( )k
k k
k
1 .802685399488733367356... ( )
!
41
1
1
4k
kek
k
.80269608533157794645... 3 3
2 31
1
3 22
sin
kk
.8027064884013474243... si
k k
k k
k
( ) ( )
( )!( )
2
2
1 4
2 1 2 10
1 .8030853547393914281... 3 3
22 3 2
1
1 30
( )( )
( )
( )
k
k k
=log2
0
1
1
x
xdx
GR 4.261.11
=x x
xdx
log
( )
2
20
1
1 GR 4.261.19
=log2
21
2
0 1
x
x xdx
x
edxx
= dydxxy
xy
1
0
1
0 1
)log(
.80325103994524191678... 4 12
24 1
2 2 2
0
2
Gx x
xdx
logsin
cos
/
.8033475762076458819... 1
2 211
kk
1 .8037920349100624... ( ) /
1 1
1
1k
k
ke
.80384757729336811942... 6 3 3
2 .80440660163403792825... 3
212
3
.80471895621705018730... logRe log
5
22 i
1 .804721402853249668... 11
21
kk k
2 .80480804893839018764... e
k
k
k
1
41
4/
1 .8049507064891153143... ( )
!
/3 1 1
1
1
1
3
k
k
e
kk
k
k
.8053154534982627922... 1
4 2
1
4 2
1
4
22
2 2
csch csc ( ( )h i
2 4 2 4
2 2
( )( ) ( ) ( )
ii i i
=H
k kk
k ( )21 1
.80568772816216494092... 11
61
kk
.80612672304285226132... Lkk
k1 2
1
1
2
1
2/
.8061330507707634892... 4
9 3
121
kk
kk
= 4
3
5
3
2
1 3 2 31
k k
k kk
( )
( / )( / ) J1061
=
023 )1(x
dx
1 .8061330507707634892... 14
9 3
121
kk
kk
=dx
x( )3 20 1
.8063709187317308602... 2 2 2 2 3 3 2 2 3 3 2
1
32 2
2log log log log log log Li
= log log( )x x dx 20
1
.80639561620732622518... dx
e x
xdx
e xx x
0 0
1 .806457594638064753844...
5
2
10
9
1 .80649706588366077562... k
kk !
10
.806700467600632558527... 2 2
2 21
1
2 2 21
sin
( )
kk
.80687748637583628319... 128 32 4 8 48 22 G log
=log( ) log
/
13 4
0
1
x x
xdx
2 .80699370501978937177... 21
21
/k
k k
.8070991138260185935...
22
2 )2(
k k
k
.80714942020070612009... 11
22
( )k
kk
2 .80735492205760410744... log2 7
.807511182139671452858... 4
3
1
3
1
31
3
0
1
, e dxx
.80768448317881541034... 1 3
2 .80777024202851936522... dx
x( )0
Fransén-Robinson constant
.80816039364547495593... ( )
( )
k
kk
1
2
2 .8082276126383771416... 4 3 2 ( )
= ( ) log1
1
2
0
1
x x
xdx
4 .8082276126383771416... 4 3 ( )
= log( ) log/1 1 2
0
1
x x
xdx
= dydxxy
yx
1
0
1
0
22
1
)log(
5 .8082276126383771416...
1
2 2)2()1(1)3(4k
kkk
.80901699437494742410... 2
1 5
4
.80907869621835775823... 2 21
21
log( )
k
kk
.8093149189514646786... 1
2
1
2,
.80939659736629010958... 1
3
3
2
1
3 1 11 3 2 3
cosh( ) ( )/ /
=1
5 3
2
5 3
2
i i
= 11 3 1
32 1
k
k
kk k
exp( )
1 .8098752090298617066... H kk
kk
( )
1
21
.81019298730410784673... 1
0
42 log)1log())6()5()4((244120 dxxx
7 .8102496759066543941... 61
.81040139440963627981... 1
22
2
1
2 120
csch ( )k
k k
11 .81044097243059865654... 4
1 k
k
k
4 .81047738096535165547... e e i i / log2
.81056946913870217... 8 4
3 2
2 2 1
2 12 21
21
( )
( ) ( )
( )
k
k
k
kk k
1 .81068745341566177051... 36 2 31
1 6
1
1 62
1
2
1
21
( )
( / )
( )
( / )
k k
k k k
.8108622914243010423... sin
1
11
22 kk
.810930216216328764...
0 )1(2
)1(1,1,
2
1)2log3(log2
kk
k
k
=dx
ex
1 20 /
.81114730330175217177... 15
15
2
1
420
tanh
k kk
551 .8112111771861827781... 2036
0
ek
kk
!
.81139857387224308909... log( / ( ))
( )
k k
kk
1
322
1 .81152627246085310702... arccosh
.8116126200701152567... 1
2
2
41
1 2
2 1
1
1
arcsinh( ) ( )!!
( )!!
k
k
k
k
=( )
1 42
1
1
k k
k k
k
.8116826944033783883... 4 2 2 2 12
22
2
log log
H
k kk
k
1 .8116826944033783883...
12
2
22log22log4
k
k
kk
H
=
21 1)2(2log
kk kH
1 .8117424252833536436...
1
22
1
4
)4(3)2,2(
jk jkMHS
.8117977862339265120...
H kk
kkk k
( ( ) )log
2 11
11
2
22
143 .81196393273824608939...
1
5
!
)(
k k
k
.81201169941967615136... 6
12
21
4
e
k
kk
k
( )!
1 .81218788563936349024... 2
3
e
.81251525878906250000... 1
2
1
2
1
2
1
22 2 2
2 2 2 ...
.8126984126984126984 =
2
1,5
315
256
.81320407336421530767... 2
3
2
31
1
9 121
sin
kk
J1064
.81327840526189165652...
1
4
2 3
4
3
4 8
1 4
3 4
/
/ sin
1 .81337649239160349961...
1 .8134302039235093838... sinh
( )!
2
2
4
2 10
k
k k GR 1.411.2
= 142 2
1
kk
GR 1.431.2
.81344319810303241352... 11
46 2 4 2
3
42 log log
( )
= ( ) log( )1 12
0
1
x x
xdx
.81354387406375956482... 4
9
768
18 42
1
log kH kk
k
1 .81379936423421785059...
3
2
3
1
2
1
2
1
3hghg
=3
2sin
3
2
=d
2 20 cos
= log( )
log( )1
1
1
1
0
3
30
x xdx
x
xdx
=sin
(sin ) /
sin/ x
xE
xdx
1 4 220
2
GR 6.154
.814748604226469748138...
1
22
1)1(2
2log26
24k
kk
k
.8149421467733263011...
1
3
2 2sin
3
2
3 k k
k GR 1.443.5
.8152842096899305458... 3 2 2 3
3
1
1
4
30
logx
xdx
47 .81557574770022707230... 5
32 1
5 2
20
log xdx
x
=
xx ee
xiLiiLii
4
55
5
)()(2416
5
=x dx
x
4
0 cosh
GR 3.523.7
.815690849684834190314...
1 2
1
k kE
.81595464948157421514... 3
38
.8160006632992495351...
2
2
1
1
tanh(log )
.816048939098262981077...
0
1
22
121
4
3
k k
k
2 .816190603250374170303...
1
2
)3(24
19
12)3(
k
kk
kk
HH
3 .816262076667919561... 1
2
220 0 2
0
Ie
I ek
kk
cosh
( !)
2 .81637833042278439185... 6
12
1
2 22 21
( )
( )f k
kk
Titchmarsh 1.2.15
.81638547489578... ( ) ( ) ( )3 1 3 1 3 11
k k kk
.8164215090218931437...
1
121 12
)12(
12
)2(
kk
kk
kk
=
0
22
2
12 2
1
2
log
)2(
1
kkk
kkk
k
.81649658092772603273... 2
3
1
8
2
0
( )k
kk
k
k J168
.81659478386385079894... 3 2
8
1
1 1
2 loglog
x
x
dx
x GR 4.298.12
1 .81704584026913895975... H
kk
k
( )
!
3
1
1 .81712059283213965889... 63
5 .8171817154095476464... 33 102
5
1
ek
kk ( )!
.817222646239917754960... dxxx
xxK
e
0
2/1)1(
sincos)1(
2
=
02 1
sincosdxx
x
xx Prud. 2.5.29.23
.81734306198444913971... ( )61
5
16
16
1
k
dx
xk
.81745491353646342122... 3
2
9
4 6
32
21
cos k
kk
GR 1.443.2
= 1
2 23
23Li e Li ei i
4 .81802909469872205712... e
.818181818181818181 =9
11
1 .81844645923206682348... arccsch1
3
4 .818486542041838256132...
0 )1(
1
k kK
1 .81859485365136339079... 2 2 14
2 11
1
sin ( )( )!
k
k
k
k
2 .81875242548180183413... 3
11
.8187801401720233053... dxx
x
0
2cosh
loglog2log2 GR 4.371.3
1 .81895849954020784402... ( ) ( )2 3
3 .819290222081750098649... 2 6 32 22 2
3
1
log( )
k
kkk
26 .819615475040601592527...
1
2
2
22
223cscsinh3
k kk
kk
1 .81963925367740924975... dxx
x
12
2
4
log
2
1,3,4
4
1
.82025951154241682326... 1
11 1e
k
ekk
kk
( ) Berndt 6.14.4
1 .82030105482706493402... ( )
( )!sinh
8 6
2 1
1
1
24
1
k
kk
kk k
1 .82101745149929239041... ( )21
kk
.8221473284524977102... 2 11
2
( )k
k
1 .82222607449402372567... 3 3 2 3 2 31
2
0
( ) ( ) ( )log
x x
edxx
1 .82234431433954947433... ( )
!
/k
k
e
kk
k
k
3 1
1
1
31
.82246703342411321824... 2
12
=( )
1 1
21
k
k k J306
=H
kk
kk 21
=
1 !
)1,(1
k kk
kS
= kk
k
( ) ( )
1
21
2
=x
edx
x
x xdx
x
xdxx1
1
02
1
2
0
1
log log( )
=x
edxx10
2
log
GR 3.411.5
= log( )log
11
0
1
0
x
xdx e dxx Andrews p.89, GR 4.291.1
=
log( )1 2
0
1 x
xdx GR 4.295.11
=logx
xdx
11
2
[Ramanujan] Berndt Ch. 9
=
log x
xdx
10
1
GR 4.231.1
=
log 10
1
e dxx
=log( )
( )
1
1
2
20
x
x xdx GR 4.295.18
= log1
1
2
2 20
x
x
x
xdx GR 4.295.16
= log 11
1
x
dx
x
=log( )
( )
x
x xdx
2
20
1
1
=x
edx
x
3
0
2
1
=e
edx
x
e x
2
0 1 GR 3.333.2
=xe
xdx
x
sin0
= dxx
x
02
2
cosh
1 .82303407111176773008... ( )
( )!sinh
4 2
2 1
12
11
k
k kkk
.82329568993650930401... ( )kk
k 2 12 11
.8235294117647058 = 14
17
.8236806608528793896...
22 2
062
x xdx
ex
log
1 .82378130556207988599... 18 3
22
( )
1 .82389862825293059856... 3
17
.82395921650108226855... 3 3
4
3
21
log sin
x
xk
.82413881935225377438... 1
2
2
0
2
cix
x( ) log
sin/
.82436063535006407342... e
k
k
k
k
kk kkk k2
1
2 2 2100 0
! ! ( )!!
.82437753892628282491...
1
22
1)3()3(23
25
k
kk
= k
k kk
1
12 32 ( )
.82447370907780915443... 1
42 2 1
164 4
1
sin sinh
kk
1 .824515157406924568142...
2
53,0,2
4
5 2etaEllipticTh
=
15
124
12
0 12k
k
k kF
.82468811844839402431... )1(4
3)1(
4
32log3
23
4
1
1
khgk
k
k
=
1 )34(
3
k kk
.8248015620896503745...
( )
!
2
1
k
kk
8 .8249778270762876239... 2
.825041972433790848534...
122
)1()1(
)4/1(2
2
2
2
2 k k
kiii
1 .82512195293745377856... J J J Jk
k
k
k3 2 1 0
5
20
2 6 2 7 2 21
( ) ( ) ( ) ( )( )
( !)
13 .82561975584874069795... ( )!
12 22
0
ek
k
k
LY 6.41
9 .82569657333515491309... 9
G
2 .82589021611626168568... k
kk !
12
.825951986065842738464...
10 )2/1(1
1
2
)()1(
kk
kk
k kp
.82606351573817904... ( )1
14 3 22
k
k k k k k
.82659663289696621390... k k k
kkk k
( )
( )
2 1
3
3
3 122 2
1
.8268218104318059573... sincos ( )
( )!2
1 4 1
1
21 2
2
1 2
2
k k
k k GR 1.412.1
3 .82691348635397815802...
2 3 2
2 242
2
2 log
log
= log sin2
0
2x xdx
x
GR 4.424.1
.8269933431326880743... 3 3
21
1
9 21
kk
GR 1.431
=
1 23cos
kk
= 0
1 3/
.8270569031595942854... ( )33
8
.82789710131633621783... 1
1
1
12
e ii /
4 .82831373730230112380... 3 5log
.828427124746190097603... 2 2 21
4 1
2
0
( )
( )
k
kk k
k
k
1 .828427124746190097603... 2 2 12 1
2 21
( )!!
( )!!
k
k kk
.82853544969022304438... 1
3 320log
x dx
x
5 .82857445396727733488...
121
2 1 24
3
0
Lix dx
ex /
.82864712767187850804... log !
!
k
kk
2
.8287966442343199956... 14 3 161
22
121
23
2
( )( )
( )
kk
= x
e edxx x
2
20 1/
16 .8287966442343199956... 14 31
2
121
23
0
( )( )
( )
kk
.8293650197022233205...
1 2
12
2
)(
k primeppk
k
.82945430337211926235... 1
21 k kk !log !
.82979085694614562146... ( ) ( )
( )
( )3 2
1
1 21
k
kk
1 .83048772171245191927...
)1sin(cos
1sin1cos1
1
1
1sin
1cos1
ii
ii
e
eii
i
.83067035427178011249... log ( ) kk
2
5 .83095189484530047087... 34
.83105865748950903... H ( )/
22 3
2 .83106601246967972516... 3
72
3
5
3
8
1
2
20
loglog( )
arccotx
xdx
.8313214416001597873... 1
2 2
2 1
122
1
e e
k
e ekkk k
cot( )
8 .8317608663278468548... 78
.83190737258070746868... 1
31
13
13
( )
( )
k
k pk p
.83193118835443803011... 2 10
Gx x
edxx
tanh
1 .83193118835443803011... )()(2 22 iLiiLiiG
=( !)
( )!( )
k
k k
k
k
2
20
4
2 2 1
Berndt 9
= x
xdx
sin
/
0
2
Adamchik (2)
=x
xdx
cosh0
=arcsin x
xdx2
0
1
1
= dxx
xh
1
02 1
arcsin
=arctan x
x xdx
1 20
GR 4.531.11
= K x dx( )0
1
GR 6.141.1
= K x dx( )2
0
1
Adamchik (16)
.83205029433784368302... 3
13
1
9
2
0
( )k
kk
k
k
3 .83222717267891019358... 1
49 3 3
1
1 1 30
log( )( / )
k kk
.83240650875238373999... dxxe x log2
2log34
1
0
2 2
GR 4.383.1
1 .83259571459404605577... 7
12
.83261846163379315125... 1 3/
5 .8327186226814558356... 45 5
81
0
1 3 ( )log( )
log x
x
xdx
.83274617727686715065...
log2
.8330405509046936713... 3
8 2 14 20
dx
x( )
= 5
4
7
4
1
1 4 3 41
k k
k kk
( )
( / )( / ) J1061
.83327188647738995744... Li22
3
.83333333333333333333 = 5
6
=( )
1
50
k
kk
=1
1 2 31 ( )( )( )k k kk
=
27
2345 1
k kk
kkkkk
=1
4 2
2 2 1
2 21 1k
k kk
k
k
k
k k( )
( )!!
( )! ( )
.8333953224586338242... | ( )|( ) | ( )| k kk
kk
k
kk
1
2 1
2 2
4 11 1
.83373002513114904888... cos cosh ( )( )!
1 1 14
40
kk
k k
1 .83377265168027139625... root of ( )x x
1 .83400113670662422217... ( )
( )!sinh
2
2 1
1 1
1 1
k
k k kk k
7 .8341682873053609132... 15 90
90
2 1
4 1
2
4
( )
( )
.83471946857721096222... 1
3
12
3
2
1
30ee
k
k
k
cos( )
( )!
3 .83482070063335874733... pf k
kk
( )
!
0
.83501418762228265843... ( ) ( )sin
31
4 32 2
2
31
Li e Li ek
ki i
k
.83504557817601534828... 2
3 213
8 2 81
2
log/k kk
= 81
2
1
3
( ) ( )
k
kk
k
=( )!
( )!
k
k kk
1
2
2 121
.835107636... h k
kk
( )2
1
100 AMM 296 (Mar 1993)
.8352422189577681... 1
1 F kkk
4 .8352755061892873386... 31 28350 5
84 1
6 6
30
1
( ) log
( )
x x dx
x
1 .835299163476728902487... 5
24 22 2 2 1 1
12
20
( log ) log( ) logG xx
dx
.83535353257772361697... 135
453 2 4
4
( )
.835542758210333500805... dxx
xxx
0
2/5
cossin
3
2 Prud. 2.5.29.24
.83564884826472105334...
3 3
2
3
1
6 5
1
6 21
log
k kk
J79, K135
=( )
1
3 10
k
k k K Ex. 113
= dx
x x
dx
x
dx
e ex x2 11
30
1
201
2 .83580364656519516508... 5
3 327
1
1 3
1
1 3
3 1
3
1
31
( )
( / )
( )
( / )
k k
k k k
1 .83593799333382666716... 1
1 ( !!)!!kk
.83599833270096432297... ( )k
kLi
k kk k2
22
1
1 1
.8362895669572625675... ( )
!
1
12
k
k k
.8366854409273997774...
0
/11
)!1(
)1(
kk
ke
ekee
6 .8367983046245809349...
0 123
33
82
k
k
k
k
.83772233983162066800... 4 10
4 .83780174825215167555... 4 21
22
0
csclog
/
x dx
x
.837866940980208240895... Cosh int1
.837877066409345483561... log( )( )
( )2 1
2
2 11
k
k kk
Wilton
Messenger Math. 52 (1922-1923) 90-93 1 .837877066409345483561... log2
.83791182769499313441... cos( )
( )!
1
3
1
2 30
k
kk k
AS 4.3.66, LY 6.110
.83800747784594108582... 3
37
1 .838038955187488860348...
0
4
)!14()2()2(
1
2
sinh
k
k
kii
=
12
2
22 12
2211
kk kk
kk
k
.83856063842880436639... log2
1
2
2
e
e J157
=
1 !
2
k
kk
kk
B [Ramanujan] Berndt Ch. 5
.83861630300787491228...
1
3
)5(
)3(124
125
k k
k
.83899296971642587682... G2
95 .83932260689807153299... ek
k
k
k
( )( )
!
11
0
GR 1.212
.83939720585721160798... 5
11
1
1 6
1e
k
k
k
k
( )
( )!
1 .83939720585721160798... 531
e ,
2 .83945179140231608634... log arctan log3 2 21
21
22
0
1
x
dx
.84006187044843982621... 2
1 2
2 1 20
1
logarcsin
( )
x
x xdx GR 4.521.6
1 .8403023690212202299...
02
2
02
2
2
21log
sin1
sin22 dx
xdx
x
x
.84042408656431894883... 7 45
8
2
7
2
0
e
k k
k
k
!( )
.84062506302375727160... 2 2 1 11
14
40
logx
dx
.840695833076274062...
sin sinh/ /
2 2
21
21 4 1 4
2 42 k
.84075154011728336936... 23
5
( )
( )
.84084882503400147649... ( )
1
3 2
1
1
k
k kk
.84100146843744567255... 2
8
1 .84107706809181431409... 11
2
kk !
6 .84108846385711654485...
22
0
2
24
logcos ( )
sin
/
x x x x
xdx GR 3.789
2 .84109848849173775273... dxx
x
0
2
44
cosh240
7
1
0
1
0
1
0 1
logdzdydx
xyz
xyz
3 .84112255850390000063...
1
23
2
216
341
36
111)3(2
k
k
k
H
5 .84144846701247819072...
logcos
sin2 4
210
Gx x
xdx GR 3.791.4
= log sin10
x dx
GR 4.4334.10
.8414709848078965067...
0
3/2
)!12(
)1(ImIm1sin
k
ki
kie AS 4.3.65
= 0
1 /
= 2
1
211J ,
= 112 2
1
kk
= cos1
21k
k
GR 1.439.1
= 14
3
1
32
1
sin kk
GR 1.439.2
= coscoslog
xdxx
xdx
e
0
1
1
.84168261071525616017... ( )71
6
17
17
1
k
dx
xk
.841722868303139881174... 1)()1(
22log21
2
1
2
kkkk
k
.84178721447693292514... 2 3 11
320
logx
dx
1 .84193575527020599668... 1
22 2
2
1 20
Eik k
k
k
( ) log!( )
.842048876481416666671...
020
1
cos)1(2 dx
x
xK
.842105263157894736 = 16
19
119 .842226666212576253... 1
4096
1
4
3
4 15 5
5
20
1
( ) ( ) log
( )
x
xdx
.84233963038644474542...
0101 3)!1(!
)1(
3
1;2;
3
23
kk
k
kkFJ
.84240657224478804816...
1 2
1
k kk F
.8425754910523763415... 1
3 3 331e
B
kk
kk
.84270079294971486934... erfk k
k
k
( )( )
!( )1
2 1
2 10
AS 7.1.5
= 2 2
0
1
e dxx
6 .84311396670851712778... 3 33 1
3 1
11
65
60
log( )
( )( )
k
k k k
10 .84311984239192954237... 3 3 4 23
212 42 2 log log log log
=
6
5
6
1ll Berndt 8.17.8
.8432441918208866651... kk
k
2
1 4 1
.8435118416850346340... Gxdx
x 2 2
20
1
16
2
4
log arctan
=x dx
x
2
20
4
sin
/
GR 3.837.2
.8437932862030881776... cos ( )( )
( )!3
0
1 21
3
1
41
1 3
2
k
k
k
k GR 1.412.4
16 .84398368125898806741... e Ik
k kk
20
0
22 1
( )!
.8440563052346255265... 21
2
1 2
2
1 2
2 1 12
1
1 0
sin( )
( )!
( )
( )!( )
k k
k
k k
kk k k
.84416242155207814332... 483 2
2
261 2
2
917
4
2log log
=k H
k k kk
kk
4
1 2 1 2 3( )( )( )
1185 .84441907811295583887... 120 6 360 5 39 4 180 3 31 2 1 ( ) ( ) ( ) ( ) ( )
= k kk
5
2
1 ( )
.84442580886220444850... Li33
4
9666 .84456304416576336457... cosh( )!
2
4
0 2
k
k k
.844657620095592823802...
152csc)575151(
55
2
11
2
11
4arc
512csc5235
)55(22/3
harc
=Fk
k
k
k 21
350 .84467200000000000000 =5481948
15625 4
4 2
1
k Fkk
k
.84483859475710240076...
0 )14(!
)1(1,
4
1
4
1
4
51,0,
4
1
4
1
k
k
kk
.84502223280969297766... cos( )
( )!
1 1
20
k
kk k
AS 4.3.66, LY 6.110
.8451542547285165775... 5
7
1
10
2
0
( )k
kk
k
k
.84515451462286429392... 9 32
3
1
ek
kk ( )!
.84528192903489711771... ( )
( )!sinh
2
2 1 2
1 1
22 11 1
k
k k kkk k
.84529085018832183660... 22
2
208
1
21 2
1 4
2 2 1
log( ) ( !)
( )!( )
k k
k
k
k k Berndt Ch. 9, 32
= x x
xdx
sin
sin
/
2 20
2
1 .84556867019693427879... 3 22
0
x x
edxx
log
.846153846153846153 =11
13
.84633519370869490299...
( )
( )
( )6
3 31
k
kk
1 .84633831078181334102... 7
621 3
2 1
70
k
kkk
.84642193400371817729... ( ) ( )
( )! /
1
2
1
22 1
1
k
kk
k
k
k k e
.84657359027997265471... log log
cosh(log( ))2
2
1
212
23
1
x
xx
dx
x
.84662165026149600620... 2( ) e
.84699097000782072187... ( ) ( ) ( ) ( )2 3 2 22
k kk
=2 1
25 4 31
k
k k kk
2 .84699097000782072187... ( ) ( )2 3
.8470090659508953726... ( )
1
15 4 3 20
k
k k k k k k
.8472130847939790866... dxxx
2/
0
2 cossin4
31
.847406572244788... 1
21k
kk F
.8474800638725324646... ( ) ( ) ( )( )!
1 1 111
e Ei eEiH
kk
k
=
( )( )!
111
k
kk
=1
11
111
1
11m k
e
m km
k
k
k
m
k
k!( )!( )!( )!
AMM 101,7 p.682
3 .8474804859040022123... 1 2
31 ( !)k
k
kk
1 .84757851036301103063... log log2 22 2
1
k
k
kk
Prud. 5.5.1.15
= 2
0
)( dxxH
.84760282551739384823... 4 3
9
1
2 2
12
12
1
( )!( )!
( )!
k k
kk
2 .84778400685138213164... ( )
!
/k
k
e
kk
k
k
4
0
1
41
.84782787797694820792... 7
256
1
4
3
31
k
k
k
sin GR 1.443.5
= 1
4 41 21 4
34
34
i
Li e Li ei i( ) / / /
.84830241699385145616... 3 3 13
313
2
k
k k
kk
( )
3 .848311877703679270993... 2
sin1
)()(2 1
22/
22/
2
k
keLieLi
i
k
ii
1 .84839248149318749178... 8 2
3
32
1
log log( )
x
xdx
.84863386796488363268...
22 )1(
)(??1
)2(
)(
kk kk
k
k
k
= sum of inverse non-trivial powers of products of distinct primes
5 .84865445990480510746... 16
27 1
2
3 20
log
/
x dx
x
.8488263631567751241... 8
31
1
4 22
kk
9 .8488578017961047217... 97
.84887276700404459187...
eerfi
k
k k
k
k
k
kk
1 40 0
1
2
1
2 1
1
2 1 2/
( ) !
( )!
( )
( )!!
.84919444474737692368...
13
)2(
)23(
1
)3(
1
3
1
)3(54
1
k k
.849320468840464412633... ( )k
kk 31
18 .84955592153875943078... 6
.849581509850335443955... ( )
( !!)
1 1
21
k
k k
.849732991384718766...
14
0 )()(
k k
kk
.84985193128795296818... H kkk
2 11
2 1
( )
=1
2 11
11
11
12
22( )
log log logk
kk
kk kk
21
1arctan
1
4
2log
12
1)2(
4
2log
kk kh
kk
k
.8504060682786322487... 33 1
1 330
e k
k
kk
( )( )!
.85068187878733366234... 8 2 12 2
0
1
G x
xdx
log
( )
.85069754062437546892... 2 4 21
2 4
1
2 4
log
i i
1 .85081571768092561791... sec ( )( )!
1 12
2 k kE
k
.8509181282393215451... csch12
1
i
i e esin J132
= 21
2 10 e k
k
GR 1.232.3
= 12 2
2
2 12
1
( )
( )!
kk
k
B
k AS 4.5.65
.851441200981879906279...
4
222
2
1
2
1
2
1)31(
4
)2(2)31(
23
1 3/13/23/1
3/2 ii
=
02 4/1)12()12(
1
k kk
.85149720152646256755...
He k eke
k1
1
11 1
1/ ( )
= ( )k
ekk
12
.85153355273320083373... e e ee k
k
kk
log( )
( )1
1
10
.85168225614364627497... i i i i i
4
1
4
1
4
3
4
3
4
=sin
cosh
x
xdx
0
1 .851937051982466170361...
0
sin)( dx
x
xsi , Wilbraham-Gibbs constant
4 .85203026391961716592... 7 2log
.85212343939396411969... 22 2
1
1 221
i i
k kk ( / )
.85225550915074454010...
12/1
)12/(12log2
1
)2(
)12(2
kk
k
k
k Berndt 8.17.8
1 .85236428911566370893... dxx
x
12
2
3
log
2
1,3,3
4
1
.85247896683354111790... 3 2 4 3 1 32
( ) ( ) ( ) ( ) ( )
k
k
k k k
=2 2 1
1
4 3
4 22
k k k
k kk
( )
.853060062307436082511...
1
33/23/1 8
11
2
)1(
2
)1(
4
k k
.85358153703118403189... 4
3
4
32
1
3 21
log( / )k kk
= xe e dxx x
10
GR 3.451.1
.85369546139223027354... 2 24
31
2
2
12
log( ( ) )
k kk
k
=4 1
2 1 22
k
k kk
( )
1 .85386291731316255056... 1
21k
k k
7 .85398163397448309616... 5
2
1 .85407467730137191843... 1
4
1
42
1
4
1
2
5
41
12
2 1 40
Fdx
x, , ,
= K1
2
.85410196624968454461... 3 5 5
2
1
5 1
2
0
( )
( )
k
kk k
k
k
4 .854318108069644090155... dxx
x
02
161loglog12log24 GR 4.222.3
1 .85450481290441694628... ( )
( )!sinh/k
kk
kk k2 3
1
2
3 2
1
.85459370928947691247... 2 8
2
4 0
log log cosx x
edxx
6 .85479719023474902805... e e
ek
k
e e
k
1
1
121
/cosh sinh(sinh )
sinh
! GR 1.471.1
.85562439189214880317... 2
1
2
1
2 2 10
erfk k
k
kk
( )
! ( )
2 .85564270285481672333... bp k
kk k
k
( )
2
1
1 202
0
6 .85565460040104412494... 47
1 .85622511836152235066... ( )
!( )
11
3
2
k
k
k
kk
=k e k k
k e
k
kk
2 1 2
3 12
3 1/
/
8 .856277053111707602292... 2 2
52
0
ee xdxx
sin
.85646931665632420671...
11
11
0
ek k
k
kk
/ ( )
! ( ) )
.85659704311393584941... ( ) coth24
.857142857142857142 =6
7
1
60
( )k
kk
.85745378253311466066...
4G
.85755321584639341574... coscos1
.85758012275100904702... ( )k
kk 2 12
.857842886877026832381... 4 53
21
22
0
( ) ( )
x Li x dx
.85788966542159767886... 3
23 2
1
2 1 23
2
0
1
( )log
( )( )
Lix
x xdx
.858007616357088565688...
3 2
)3,(1
kk
kS
.85840734641020676154... 41
2
20
cos
( sin )
x
x
= dxx
x
1
0 1
arccos
= log( )cosh sinh
cosh1 2
20
2
xx x x
x
dx
x
GR 4.376.11
.85902924121595908864... 35
128 1
7 2
0
1
x dx
x
/
.85930736722073099427... 24 e
7 .85959984818146412704... 2 3
0
2
31
k
k
k
k
k
k
e
k
( )
!
.859672000699493639687...
1
0 1)2()(k
kk
5 .85987448204883847382... e Not known to be transcendental
.86009548753481444032... 2
1
2 1
2
( )
( )!
k
kk
.86013600393341676239... 96 354
2
1
ek
k kk !( )
.860525175709529372495...
3/1
3/13/2
3/1 2
112
2
1131
2
)2(231
23
1 ii
=
11
1
13 2
)3()1(
2/1
1
kk
k
k
k
k
1 .8608165667814527048... 4
3
1
31 3
0
ek
k kk
/
!
.86081788192800807778... 4
5
1
2
4
5
1 5
2 21
arcsinh
logF
kk
kk
=
22
0 12)12(
)1(
xx
dx
k
kkk
k
.8610281005737279228...
2 2
21
4
1
221
coth
kk
J124
.861183557332564183...
12 2
log)
2
1,2,2(
4
1
x
dxx
.8612202133408014822... ( )81
7
18
18
1
k
dx
xk
.8612854633416616715... 3
36
1 .86148002026189245859...
1
0
1
0
1
0
1
0
2
14
3)3(32log8 dzdydxdw
wxyz
zyxw
.86152770679629637239...
1
1
)12(!
)1(1,0,
2
1
2
11
111
k
k
kkeerf
.86213147853485134492...
2/
02
22
)cos(sin4
2log
8
xx
dxxG
.86236065559322214... 1
21k
k k ( )
20 .8625096400513696180... 1
21 2
0ee e
k k
ke e
k
/ cosh
!
.8630462173553427823... 34
3
1
3 1
1
1 30 0
log( )
( ) /
k
kk
xk e
.8632068016894392378... k
k
kkkH
kkk
1log
11)1(
6 12
1
2
4 .8634168148322130293... 48 8
22
( )
3 .86370330515627314699... 2 1 312
csc
AS 4.3.46
.86390380998765775594...
1
3
)3(
)3(26
27
k k
k
.86393797973719314058... 11
40
6
60
sin x
xdx GR 3.827.15
.86403519948264378410... 1
83 4 2 42 4
4
1
e e ek
kk
cos coscossin cossinsin
!
1 .864387735234080589739... 6 9 12
9
4
9 93
1 31 3 1 3 1 3
// / /sec sec sec
[Ramanujan] Berndt Ch. 22
.8645026534612020404...
02/3)12(
)1(
4
3,
2
3
4
1,
2
3
8
1
k
k
k
.8646647167633873081... 11 22
1
1
ek
k k
k
( )
!
2 .86478897565411604384... 9
.864988826349078353... 1
3 12k
k k( ( ) )
.86525597943226508721... e
1 .8653930155985462163... 7 2 12 4 18 3 1 ( ) ( ) ( )
= ( ) ( ) ( )
1 13
2
k
k
k k k
= 8 3 1
1
4 3 2
2 42
k k k k
k kk
( )
.86558911417184854991... ( )
( )!cosh
8 6
2
11
1
6
14
k
kk
kk k
.86562170856366484625... 1 2
.86576948323965862429... 12
arctan2 gde
=
k
k
k )2/1(
1arctan)1()1(sinharctan 1
[Ramanujan] Berndt Ch. 2, Eq. 11.4 = )1(tanharcsin
.86602540378443864676... = 3
2 3 6 sin cos
=( )
1
12
2
0
k
kk
k
k
.86683109649187783728... 1
2
5
10 5
1
5 120
csch
( )k
k k
.86697298733991103757... 1
4 22 1 2
1
4 10
log( )k
k k K135
=1
8 7
1
8 31 k kk
J82, K ex. 113
= 16
1
4 22 2 2 2 2 2
2 2
2 2
arctan arctan log
= 16
4 27
8
3
84 2
8
3
8
cot cot logsin logsin
=dx
x x
dx
x2 21
40
1
1
4 .86698382463297672998... 2
36 2
1
1 1 40
( log )( )( / )
k kk
.86718905113631807520... 21
2
1 22
1
21
arcsinh
( )k k
kk
k
k
5 .8673346208932640333... 1
128
1
4
3
463 3
3
0
( ) ( ) tanh
x x
edxx
11 .8673346208932640333... 1
128
1
4
3
43 3
3
0
( ) ( )
cosh
x dx
x
1 .86739232503269523903...
1
24
24
2
222csc1sin1sin
1
k kk
kkhii
5 .86768692676152924896...
1
)(
k kF
k
.86768885868496331882...
1 2 ),2(
)1(
k
k
kkS
1 .86800216804463044688...
2 1 23 4 40
/ /
dx
x
.86815220783748476168...
6
5
6
1
432
1
108
)3(13
18
)3(13 )2()2(
= 3 3 30
1
3 1
1
3 2
( )
( )
( )
( )
k k
k k k J312
.86872356982626095207... ( )31
3
1 .86883374092715470879... )2,4(2268
)3(6
2 MHS
.86887665265855499815... ( )
( )log
1
2 11
1
21 1
k
kk
kkk
.86900199196290899881...
11
11
cosh)!2(
)2(
kk kk
k
114 .86936465607933273489... 1
22
22 2
1
e ek
ke e
k
k
/ cos
!
3 .8695192413979994957...
2
/1
1
1
k
k
k
k
.869602181868503186151... 2
16
1
2
1
2
1
2
1
2
2 2
1 1
( ) ( )
4
2 2
16 2cot
= H
kk
k 4 221
1 .86960440108935861883...
122
2
4/1
18
k k
9 .86960440108935861883... 2 2
0
1
1
logsin
sin
x
x
dx
x GR 4.324.1
7 .8697017727613086918... 21
1
2
1
3k
k
k
k
k
ke
( )
!/
.87005772672831550673... 4
2
2
2
0
log
loge xdxx GR 4.333
3 .87022215697339633082... I Ik
k
k
k
k
kk k0 1
3
21
3
0
2 22
2( ) ( )
( !) ( )!
.870282055991212145966... 2
5cos
2
321cos
2
321cos
13
ii
=
1
33 )1(
11
k kk
.87041975136710319747... 21
32
1
2arcsin arctan
=( )
( )
( )!!
( )!!
1
2 2 1
2
2 1 30 1
k
kk
kkk
k
k k
.87060229149253986726...
4
23
1724 2 2 3
1
2 1
log ( )( )
( )
H k
k kk
k
.87163473191790146006... 1 4/
1 .871660178197549229156... 23 e
.87235802495485994177... 2
21
28 21
1
4 1
1
4 1
( )
( )
( )
( )
k
k
k
k k J327
= 1
64
9
8
7
8
5
8
3
8
=
02
2/)1(
)12(
)1(
k
k
k Prud. 5.1.4.4
=
logx dx
x40 1
.87245553646666678115... 5
16
1
16 2
3
1
e
e
k
kk
( )!
.87247296676432182087...
( )
( )
k
kk
1
12
.87278433509846713939... 29
207 1 ( )
1 .8729310037130083205... x dx
e ex x
2
1
.87298334620741688518...
k
k
kkk
k 2
)1(6
)1(315
0
3 .87298334620741688518... 15
.87300668467724777193... ( ) ( ) ( ) ( )
1 3 1 3 3 1 31
1
k
k
k k k
=
13 1
2
2
1
k k
2 .87312731383618094144... 4 82)
3
1
ek
k kk
!(
9 .87312731383618094144... 4 11
4
1
ek
kk
( )!
10 .87312731383618094144... 4e
1 .8732028500772299332... 11 2 1
2 2
( )
( )
( )
( )
k
k
k
kk k
.87356852683023186835... 1
log J153
1 .87391454266861876559...
1 21
1k
k
k
2 .87400584363103583797... 3 2 2
20
2
20
2
8 2 2 1 2
log log logx
xdx
x
xdx
7 .8740078740118110197... 62
.87401918476403993682... 1
6 2
1
42 3 2
0
2
3 2
0
2
sin cos/
/
/
/
x dx xdx GR 3.621.2
.87408196435930900580... ( )
( )!cosh
k
k k k kk k2 2
1 1 1
2 1
.87427001649629550601... iii
i
24
12
4
1
2
.87435832830683304713... 1
43
3
23
3 3
3 3 32
1
log log
H kk
k
.87446436840494486669...
2 2
powerintegernontrivialaω2
1)(1)()(
k nk
k n
kkk
.87468271209245634042... 3G
.87500000000000000000 = 7
8
1
72 1 2 1 1
0 1
( )( ) ( )
k
kk k
k k
=
1 )3)(1(k
k
kk
H
1 .87500000000000000000 =
1 )2/5(
11
8
15
k kk
3 .87500000000000000000 =
1
9
)1(8
31
kkke
k Prud. 5.3.1.1
1 .87578458503747752193... 3 2
082
x x
edxx
tanh
3 .87578458503747752193... 3 2
20
2
08 1
log
cosh
x
xdx
x
xdx GR 3.523.5
= 3
3 3
2
42
i Li i Li ix
e ex x( ) ( )
.87758256189037271612... 6/1)1(Re2
1cos
=
0 4)!2(
)1(
kk
k
k AS 4.3.66, LY 6.110
2 .87759078160816115178... G
.87764914623495130981... 2
21
2
1
21
log( )k
k k k
=
2
)()(k
kk iLiiLi
= log 1 2
0
1
xdx
x GR 4.295.2
= log( )
11
1 20
xdx
= log1
1
2
20
1
x
xdx
=x dx
x10
2
cos
/
Prud. 2.5.16.18
= dydxyx
yx
1
0
1
0221
1 .87784260817365902037... coth
1 .8780463498072068965... 109
81 3
1 3 4
1
e k
k kk
/
!
5 .8782379806992663077... 3
82 4
1
2 2
3
21
, ,log x
xdx
.878429128033657480583... 2 21
24
1
24
1
2 2 10
sin cos( )
( )! ( )
k
kk k k
.879103174146665828812... ( )( )
log
12 1 1
111
12
1
k
k k
k
k k k
.87914690810034325118... logk
kk2
1 1
.87917903628698039171...
1 )1(!
1
k kkk Related to Gram’s series
.87919980032218190636... 2
2 10
k
k kk
( )
.87985386217448944091... k
k kk
!
1
.8799519802963893219...
0
1 113 1
1
3 1
( )kk
kj k
jk
= 1
3111ijk
kji
.88010117148986703192... 2 11
1 410
Jk k
k
kk
( )
!( )!
1 .880770870194...
0
1 11
11
1( )
( )
k
k k k kk k
=1
111 jk jkkj ( )
4 .880792585865024085611... I0 3( )
.88079707797788244406... e
e e
k
kk
2 201
1 1
2
1
tanh ( ) J994
5 .88097711846511480802...
1
2
12 )!22(
)!(
4
1,
2
1,2,2
k k
kF
.88107945069109218989...
0
2
10 2)!3(!
)1(6
2
1,4;
kk
k
kkF
1 .88108248662437028022... 9 3 12 2 3 31
1 5 60
log log( )( / )
k kk
= ( )k
kk 6 2
2
1 .88109784554181572978... cosh2
2
2 .88131903995502918532...
tanh/2
1
1 220
k kk
=
2
1
2
1 iii
.881373587019543025232... log(1 + 2) = arcsinh11
4 2 1
2
0
( )
( )
k
kk k
k
k J85
= 8
sinlog8
3sinlog
8tanlog
= arctanh1
2 i iarcsin
.88195553680044057157...
k
kk
12
21 2 2 1( )
1 .88203974588235075456... 1
121 k kk log( )
.88208139076242168... 2
21 2
2 1
2 1
0
erfk k
k k
k
( )
!( )
.8823529411764705 =15
17
.88261288770494821175... e k
k
1
2
1/ !
.88261910877658153974...
1
3
3
63
1
322
csc
( )k
k k
10 .88279618540530710356...
02 1
631log32 dx
x
.883093003564743525207... 2
4 11
k
kk
3 .88322207745093315469... 5 1 14
120
logx
dx
.88350690717842253363... ( ) ( )2 1 2 11
k kk
3 .8836640437859815948... e EiH
kk
k
1 111
( )( )!
.88383880441620186129...
12
1
)!(
)1(1,2,2,2,1,1
k
k
kkHypPFQ
.88402381175007985674... 4
81 3
3
3
g J310
1 .88410387938990024135... 64
10395
5volume of the unit sphere in R5
1 .88416938536372010990... e log 2
1 .8841765026477007091... 11
4
3
27
0
2
x e x dxx log
.88418748809595071501... H k k
kk
k
! !
( )!21
=
)3log2(33
2
3
1
27
2 )1()1(
=
2
13
2
31
3
3log
3
2
33
222
iLii
iiLi
.88468759253939275201... 16
17
16
17 17
1
41
2 40
arcsinh ( )! !
( )!k
kk
k k
k
3 .88473079679243020332... 22
2
2
2
2
1
GH k
kk
k
k
log ( !)
( )!
1 .88478816701605060798... 2
31 3 3 3
3
2 10
cosh sinh( )
k
kk k
1 .88491174043658839554...
1
022
22
)1(
log
32 x
dxxG
1 .8849555921538759431... 35
.88496703342411321824... 2 2
2 2212
1
16 1
k
kk ( )
= k kk
( )2 11
1 .885160573807327148806...
02
222
)2(
log
2
2log
6 x
dxx
25 .88527799493115613083... e EikH
k
kk
k
2
1
1 2 2 2 2 12
( ) log!
1 .88538890738563... 3
2
2 )( dxx
2 .885397258254306968762... 612
4 2 22
2 2
0
1
log log arccos logx xdx
.88575432737726430215... 2 2 3 ( ) ( )
=2
1 21
H
k kk
k ( )
=
1
02
2
12
2
)1(
log
)1(
logdx
x
xxdx
xx
x
=
0
2
02
2
21xxxx eee
dxx
e
dxx
8 .88576587631673249403... 8
1 .88580651860617413628...
2 4
1)(5
8
5
4
2log15
4
5
kk
k k
.8858936194371377193... 3
35
5 .8860710587430771455... 16
4 1e
( , )
2 .88607832450766430303... 5
.88620734825952123389... 2
31 3
2 1
1 2 1
0
erfk k
k k
k
( )
!( )
.88622692545275801365... 2
AS 6.4.2
= 3
2
AS 6.1.9
=( ) ( )!
!
1 31 12
1
k k
k
k
k
=
0 2/3
2/11
k k
k
= sin2 2
20
x
xdx
= e dxx
2
0
LY 6.29
=
02
tan
cos
sin2
dxxx
xe x GR 3.963.1
.88626612344087823195... 24 5 11
4
0
( )
x
e edxx x
24 .88626612344087823195... 24 5 11
44
0
1
( )log( )
x
xdx
.88629436111989061883... 2 21
2log
1 .886379184598759834894... 5
2
2
22
2
22 2 12
2 1
cot ( )
kk
k
k
k
.88642971056031277996... ( )
11
1
k
kk k
2 .88675134594812882255... 5
3
.88679069171991648737... 1
2
2
3 3
1
3 120
csch ( )k
k k
.8868188839700739087... sech1
2
2
2 41 2 1 22
0
e e
E
kk
kk
/ / ( )! AS 4.5.66
.887146038222546250877...
log log
1 3
2
1 3
2
i i
= log ( )( )
11
13
31
1
1
k
k
kk
k
k
24 .8872241141319937809... log
( )
4
2 1
k
k kk
9 .88751059801298722256... 9 3log
.88760180330217531549... ( )
1
3 2
1
1
k
kk
2 .88762978583276585795... F
kk
k
2
1 !
.8876841582354967259... 2 21
2
1
1 2 10
log ( )( )
( )! ( )Ei
k k
k
kk
38 .88769219039634478432... k kk
5 2
2
1 ( )
.88807383397711526216... 3 3
6
CFG D14
8 .8881944173155888501... 79
.88831357265178863804... 52456log5)53885(32
5199445log
5
224
80
51
=
0 15
)1(
10
1
5
3
10
1
k
k
k
.88888888888888888888 =8
9
1
80
( )k
kk
1 .88893178779049819496... 253
5
1
1 5
1
1 52
2 1
2
1
21
( )
( / )
( )
( / )
k k
k k k
7 .8892749112949219724... 1
1 11 k kk ! ( )
.8894086363241... 11
2 41
k k kk ( )( )
1 .89039137863558749847...
41
2 1 23
2
0
Lix dx
ex /
.89045439704411550353... 2 3 3
81 13 20
log
( )
x
xdx
5 .89048622548086232212... 15
8
.890729412672261240643...
2 2
3
23
5
6log log
1 .89082115433686314276... dxx
x
12
2
2
log
2
1,3,2
4
1
.890898718140339304740... 2 11
6 11 6
1
/ ( )k
k k
.8912127981113023761... 1
0
2log
2
1]1},2,2,2{},1,1,1[{
xe
dxxHypPFQ
=
12
1
03 !
)1(
)1(!
)1(
k
k
k
k
kkkk
.89164659564565004368... 3
8
5
2 1
4
0
2
x dx
ex
.89169024629435739173...
41 2 2
0
2
e dxcos (tan)/
GR 3.716.10
= dxx
x
02
2
1
cos
.891809...
1 )1(
)(
k kk
k
.89183040111256296686... dx
x x21 1log( )
.89257420525683902307...
010 4/15
14
)1(4
)1(
4
5log4
xk
kk
k
k
e
dx
kk
.89265685271018665906... 1 3
1 2
/
/
1 .8927892607143723113... 3 23log
3 .89284757490956280439...
0
2/
2! k
ke
k
ee
.892894571451266090457... 11
2 kpp
p primep primek
( )
.892934116955422937458...
primep ppB
111log1 in Mertens’ Theorem HW 22.7
=
2
)(log)(
k k
kk
.89296626185090504491... ii eLieLi 24
24
.89297951156924921122... 4
3
1
3
1
3
3
0
e dxx
.89324374097502616834... HypPFQk
k
k
13
2
3
2
1
4
1
2 1 20
, , ,( )
(( )!!)
=
210
0
2
H x dx( ) sin(cos )/
2 .89359525517877983691... x x
edxx
2
0
1
1
log( )
1 .89406565899449183515...
1
2
)2(4
360
7
k
k
k
H
= dxx
xx
1
0
2log)1log(
1 .89431799614475676131... ii ii iie
)(2/ )(
2cos2
.894374836885259781174... 82
1
96
1
414 3
1 42
33
2
41
Gk
kk
( ) ( )( / )
.8944271909999158786... 2
5
1
16
2
0
( )k
kk
k
k
.894736842105263157 =17
19
.89493406684822643647...
2
22 26
3
4
2 1
11 1
k
k kk k
k
k
k( )( ) ( ( ) )
= 1 122 3 2
2 2k k k kk k
k k
( ) ( )
= 2
2
1
1 13 31
2
22k k
k k
k k kk k
( ) ( )
1 .89493406684822643647... 2 2
0
1
6
1
4
1
1
( ) logx x x
xdx
.895105505200094223399... ( )
( )!
/k
k
e
kk
k
k
1 1
11
1
22
1 .89511781635593675547... Ei li ek kk
( ) ( )!
11
1
AS 5.1.10
.89533362517016905663... si
k k
k k
k
( ) ( )
( )!( )
2
2
1 2
2 1 2 10
.89580523867937996258... iLi e Li e
k
ki i
k2 4 4 41
( ) ( )sin
.89591582830105900373... 2
4 2
1
2csc
= ( )
1
2 1
1
21
k
k k
.89601893592680657945... sin
2 21
1
16 121
kk
J1064
.89603955949396541657... 3
39
1
1
2 2
20
loglog( )
x
xdx
.896488781929623341300... x dxx 2
0
1
.89682841384729240489...
1,2,
2
1
2
12 2Li
= log log log log2 222 2 2 3 3 2
1
3
Li
=( )
( )
1
2 1 20
120
k
kk
xk
x
e
1 .89693992379675385782... ( )
!
/k
k
e
kkk
k
k
3
20
1 2
31
.89723676373539623808... 2
11
1 .89724269164081069031... 7 3 1
6 2 1 12 70
dx
x /
.89843750000000000000 =
1
3
50,3,
5
1
128
115
kk
k
.89887216380918402542... 1
2 1
2
1
log
( )
k
k kk
.8989794855663561964... 2 6 41
8 1
2
0
( )
( )
k
kk k
k
k
4 .8989794855663561964... 24 2 6
.89917234483108405356... 3
2
1
3
1
3 2 10
erf
k k
k
kk
( )
! ( )
.89918040670820836708... 22 2 2
1
2 1 421
cot
/kk
.89920527550843900193... 3
4 17
17
2
1
3 221
tan
k kk
9 .8994949366116653416... 98
5 .899876869190837124551...
1
22
24
4
1
24
5
360
17)3(3
k
kk
k
HH
11 .89988779249402195865... 1
1 G
.90000000000000000000 = 9
10
1
90
( )k
kk
.9003163161571060696... 2 2
11
16 21
kk
GR 1.431
=0
1 4/
=
122
cosk
k
GR 1.439.2
2 .900562693409322466279... 3 3
16 24 22
22
0
1 ( )log log arccot x xdx
1 .90071498596850182787... 2
1
2
2
( )kk
k
2 .90102238777699398997... 1
1 21 kk ! /
30 .90110011304949758896... 11 11
5
1
ek
kk
( !
.90128629936044729873...
0
2
32
)1(2
2
12
kk
k
k
kK
.90135324420067371214... 1
2
2
42
0
1
si
x xdx( )
log cos
.90154267736969571405...
13
1)1(0,3,1)3(
4
)3(3
k
k
k
AS 23.2.19
= dxx
xx
1
0
log)1log( GR 4.315.1
= x dx
ex
5
0
2
1
.901644258527509671814...
012 )13(2
)1(
2
1,
3
4,
3
1,1
kk
k
kF
1 .902160577783278... Brun’s constant, the sum of the reciprocals of the twin primes: primeatwinp p
1
Proven by Brun in 1919 to converge even though it is still unknown whether the number of non-zero terms is finite.
.902378221045831516912... ( )( )
11
21
2
1
kk
k
k k
.90268536193307106616... cos 2
.9027452929509336113...
3
2
3
2
3
5
2 .90282933102514164426... ( ) ( )
1 13
2
k
k
k
1 .902909894538287347105...
0 !
)1(
k
k
k
I
1 .9036493924...
2222
1)(log
log21dxxx
kk
k
.9036746237763955366... sin Ime
ie2
.90372628629864941069...
( )
( )
k
kk
3
12
.90377177374877204684... 6
3
62 3
1
6 10
log( )k
k k
.90406326728086180804... 1
3 10k
k
1 .904148792675624929571...
3
2
3
2
3
2
3
4,2,
2
3F
3
210
3/211 IIe
=
k
k
k
k
kk
2
3!0
6 .90429687500000000000 = 6463
512
1
55 0
5
5
1
, ,k
kk
.9043548893192741731... 3
82
7 3
4
16 6 1
2 2 1
2 2
3
log( ) k k
k kk
=( ) ( )
1
2
2
2
k
kk
k k
.9045242379002720815... ( )
( )!( )
1
2 4 10
k
k k k J974
=2
12
1cos
2
2 x
dx
xC
9 .905063057727520713501...
1
31)1(
2
2)3(8
k
k
kk
.90514825364486643824... e
e
3
3
1
1
J148
16 .90534979423868312757...
1
5
40,5,
4
1
243
4108
kk
k
.90542562608908937546... ( )k
Fkk
1
12
1 .90547226473017993689... e
.90575727880447613109... si
k k k
k
k
( ) ( )
!( )( )
2 1
2 1120
.90584134720168919417... 2 2 22 1
11
1 22
1
1
log log( )
( )
( )( )
kH
k
H
k kk
kk
kk
=
12 24k
k
kk
H
.90587260444153744936... 1
2 2 11k
k k (
6 .90589420856805831818... 3
53 3 3 4 2
1
1 1 60
log log( )( / )k kk
.90609394281968174512... e
3
12 .906323358973207156143...
1
22
1)1(23
)3(8k
k kk
.90640247705547707798...
4
1
4
1
4
5
=
0
4
dxe x
.90642880001713865550... 1
1 k kk !
.9064939198463331845....
2
012
2
)1(16
)1(1,2,
2
1,
2
1
k
k
kF
kk
k
.9066882461958017498...
2 0
log sinx x
xdx
.9068996821171089253...
0 56
1
16
1
3sin
332 k kk
J84, K ex. 109e
= maximum packing density of disks CFG D10
= ( )
( )( )
1
1 3 10
k
k k k
= ( )
( )
1
3 2 10
k
kk k
J273
=dx
x
dx
x x
dx
x20
4 20
203 1 3 1
= x dx
x x
2
4 20 1
= xx
dx2
0
1
311
log
1 .90709580309488459886...
1
2)12(
31
2
3cosh
4
1
k k
8 .907474904294977213681...
0
2
2
2/1
2/3
2
3sinh
2csc3
k k
kh
.90749909976283678172... b k
k
k
kk
k k
( )( )
( )log ( )2
1 1
2 2
Titchmarsh 1.6.3
.90751894316228530929... ( ) logk kk
12
23 .90778787385011353615... 5
64
5 4
0
x
e edxx x
= log tan/
x dx4
0
2
GR 4.227.3
=log4
20
1
1
x
xdx
GR 4.263.2
.907970538300591166629...
2
1
2
1
4
2log
48
522
22 iLi
iLi
.90812931549667023319... 4 2
4190 48 96
1
768
2
cos( / )k
kk
GR 1.443.6
= 1
2 42
42Li e Li ei i/ /
.90819273744789092436... 1
32
5 3
2
5 3
2
i i
= k
kk k
k k
1
13 1 3 23
2 1
( ) ( )
=
14
1
6
5
k kk
.90857672552682615638... ( ) ( )
( / )1 2
31
4
3
1
6
4
3 1 3
k
kk
.90861473697907307078... 0
1 1 112
1 2 1
2
( ) log( )k
k k jkkk
k
kjk
kj
.90862626717044620367...
2
)( 22)1(k
kk
.90909090909090909090 = 10
11
1
100
( )k
kk
.90917842589178437832... 1
2 k kk !log
.9092974268256816954... sin( )
( )!
sin( / )
!2
1 2
2 1
2 22 1
0 1
k k
k
k
kk
k
k
AS 4.3.65
.90933067363147861703... 1 1 1
2 1
exdx
e(cos sin )sinlog
1 .90954250488443845535... 3 3 2 2 12
0
1
log log log
x
dx
1 .90983005625052575898...
k
k
kkk
22
)1(5
1
5cos153
2
5
0
1 .90985931710274402923... 6
1 .91002687845029058832... 1 2 35
44 1
31
( ) ( ) ( )H
kk
k
.91011092585744407479... ( )2
2
1
221
2 21
k
kLi
kkk k
.91023922662683739361... 1
3 30log
dx
x J153
.91040364132111511419... 1
8 42
1
420
cothkk
.91059849921261470706... sin(log ) Im i
.91060585540584174737... Li34
5
1 .910686134642447997691... 1
2 2
1
2
2
21
eerf
k k
k
k
k
!
( )!
244 .91078944598913500831... 8
396 64 2
33 2
22
0
log ( )/x Li x dx
.911324776360696926457...
2 2 2
)(1)()(
k k nkn
kkk
.91189065278103994299... 9
2
.91194931412646529928... 3
34
1 .911952134811735459360...
12 2
11
2
7sech
2
11cosh
3
2
k kk
23 .9121636761437509037... 65
5 1e
( , )
.91232270129516167141... 1
11219 14 2 2 2 csc( )
1 .9129311827723891012... 73
.91339126049357314062...
121 k
kk
.9134490707088278551...
( )32
3 .91421356237309504880... 5
22 CGF D4
.9142425426232080819... 7
2
7
1
7 220
arctandx
x x
.9142857142857142857 =32
354 1 2
1
4 40
( , / )( )k
k k
2 .91457744017592816073... cosh( )!
33
20
k
k k
.91524386085622595963... 2
1cot
2
1
=
i e
e
i i
i
B
k
i
i
kk
k
1
2 1 2
1 1 1
1 1 1
1
22
0
cos sin
cos sin
( )
( )!
= limsin
aa
k
k
k
0 1
.91547952683760158139... ( )
1
7 10
k
k k
4 .91569309392969918879... 120 46 1255
4
1
( )!( )
ek
k kk
3 .91585245904074342359...
32 3
1
1 320
tanh/
k kk
.91596559417721901505... G Li i ( ) Im ( )2 2 , Catalan’s constant LY 6.75
=
2
1
2
1
28
2log22
iLi
iLi
i
= ( )
( )
1
2 1 20
k
k k
= 1
4 3
1
4 12 21 ( ) ( )k kk
= 1
2
4
2 2 1
2
20
( !)
( )!( )
k
k k
k
k
Adamchik (23)
= 1
16
3 1 1
42
1
( )( )( )
k
kk
kk
Adamchik (27)
= 1
8 21
3
41
kkk
k
, Adamchik (28)
= 8
2 33
8
12
2 1 20
log
( )
k
kk
k
= x
e edxx x
0
=
log logx
xdx
x
xdx2
0
1
211 1
= log1
1 10
1
2
x
x
dx
x GR 4.297.4
=
log
1
120
1 x
x
dx
x Adamchik (12)
=
log1
2 1
2
20
1 x dx
x Adamchik (13)
= log tan/
xdx0
4
GR 4.227.4
= 1
2 0
2 x
xdx
sin
/
Adamchik (2)
= 1
2 0
x xdxsech
Adamchik (4)
= arcsinh(sin )/
x dx0
2
Adamchik (18)
= 2 2 2 20
4
0
4
log( sin ) log( cos )/ /
x dx x d
Adamchik (5), (6)
= arctan arctanx
xdx
x
xdx
0
1
1
GR 4.531.1
=
arctan x
xdx
1 20
= 1
22
0
1
K x dx( ) Adamchik (16)
= 1
22
0
1
E x dx( ) Adamchik (17)
= 1
1 10
1
0
1
( )x y x ydxdy
Adamchik (18)
.9159981926890739016... ( log ) log( )
( )
8 3 18 2
12 2
1
1
4
4 40
x
x xdx
5 .91607978309961604256... 35
.91629073187415506518...
11 5
3
5
3
2
5log
kk
k
kLi
.91666666666666666666 =
0 11
)1(
12
11
kk
k
=
23
27
234
245
1
11
)1)((
1
kkk k
k
kkkkk
kkkk
.9167658563152321288... cos cosh( )
( )!2 2
1 2
44 4
0
k k
k k
.91681543444030774529... HypPFQk
k
k
k
k
, , , ,( )
( )!
1
4
3
41
1
64
1
4
2
0
.91767676628886180848...
2
45
3 1)4(2
k kk
kk
.91775980474025243402...
93 3 1 3
1
1
3
3 30
(log )log( )
( )
x
x xdx
.91816874239976061064... 6
5
.91872536986556843778... 21
2 2
1
2
1
2 1 21 4 1 20
sin( )
( )!/ /
J
k
k
kk
.91874093588132784272... 7
16
13
32
1
2 21 4
2
0
e erfk k
kk
/ !
( )!
.91893853320467274178... )1(!
lim2log2/1
n
n
n n
en
= Stirling’s constant , GKP 9.99
the constant in ...12
1
2
loglog!log
n
nnnnn
= log ( ) x dx0
1
GR 6.441.2
=1
1 20
1
log logx
x
x
x dx
x x
GR 4.283.3
.9190194775937444302... sinh
exp( )
41
1 4 14
2 1
k
k
kk
= 1
2 2 2 i i
.9190625268488832338... 7
4
.91917581667117981829... ( ) ( )( ) ( )1 1i i
2 .91935543953838864415... H
kk
k
2
1 !
.9193953882637205652... 2 2 1 41
2
1
2 12
1
cos sin( )
( )!
k
k k k
.92015118451061011495...
02
02
2
12
11log
sin1
sin
2
2dx
xdx
x
x
.920673594207792318945...
02
0,1,1
)1( kke
k
ee
e
=( )k
ekk
11
.92068856523896973321...
0
8/1 2)!12(
!)1(
22
12
kk
k
k
kerfi
e
2 .92072423350623909536... log log( )2
22
1
1 20
1
G
x
xdx GR 4.292.1
= arccosx
xdx
10
1
GR 4.521.2
= x x
xdx
sin
cos
/
10
2
GR 3.791.9
12 .92111120529372434463... 2 82
1
1 420
csch ( )
/
k
k k
.92152242033286316168... 2
2 2
2 2 2 2
sin sinh
cosh cos
= 11
11 1 4 14
2
1
1
k
kk
k
k
( ) ( )
1 .92181205567280569867... 4 21 2
2
1
log( / )
H
k kk
k
7 .9219918406294940337... e I
k
k
kk
20
0
2
2
1
2
1
1
2 1( )
( )!
.92203590345077812686... 2
216 4
1
16
2
cos( / )k
kk
= 1
2 22
22Li e Li ei i/ /
.92256201282558489751... erfk kk
k
1
2
1
4 2 1
1
0
( )
! ( )
.92274595068063060514... ( )x dx 10
1
.92278433509846713939... 3
23 ( )
1 .92303552576131315974... 2
1
2 1( )kk
.923076923076923076 =12
13
1
120
( )k
kk
.92311670684442125248... ( ) ( )
( )!sin
1 4
2 1
1 11
2 211
k
kk
k
k k k
.92370103378742659168...
4 1
4 1
2
12
0
tanhk kk
.923879532511286756128... 8
3sin
8cos
2
22
.92393840292159016702... 90 1
44 41
( )
( )k
kk
.9241388730... point at which Dawson’s integral, e e dtx tx
2 2
0
, attains its
maximum of 0.5410442246. AS 7.1.17
.92419624074659374589... 4 2
31 2
1
27 331
log ( )
k
k k k Berndt 2.5.4
= Hk
kk
/ 2
1 2
.924299897222937... ( )
log
1
2
k
k k
.924442728488380072202... 2
5025 5 5 1
cot csclog
x
xdx
.92465170577553802366... ( )
1
8 10
k
k k
5 .924696858532122437317...
2 3
1)(5
36
5
2
3log5
3
5
kk
k k
.92491492293323294696... 1
3 23 2 2
2
3 log( )
log
= ( )
( )( )
1
1 4 10
k
k k k Prud. 5.1.8.9
.92527541260212737052...
1
32
22
1
22
)14(
)14(4
4
)2(
32
3
kkk k
kkkk
= dxx
x
121
arctan
.925303491814363217608... erfik
k kk
( )( )!
!( )!1 2
12
121
.92540785884046280896... sec1
2
1
e ei i
.925459064119660772567... 12
12
2
1csch12
2
e
ee
=( )
1
12 20
k
k k
.9260001932455275726... 11 12
2 2 21
sinh ( ) ( )k i k ik
=2 2
2 1
2
4 21
k
k kk
.92614448971326014734... 3 2 5 ( )
.92655089611797091088... 12
2
cosh
1 .92684773069611513677... 2
2128
22
4 1
2 2 1
log( )
( )
k k k
k kkkk
8 .92697404116274799947... 1
122
2 12 2
3
0
loglog
( )( )
x
x xdx GR 4.262.3
.92703733865068595922... 1
8
1
4
1
4
1
2
5
41
12
2 1 41
Fdx
x, , ,
.927099379463425416951... 3 3
26 2 3 6 2 3
1
4
22
2
loglog log log
Li
=( )
( )
1
3 1 20
130
k
kk
xk
x
e
.92727990590304527791...
1
5 3 5
10
5 5
2
5 3 5
10
5 5
2
= 2 1
11 12
21
2
k
k k kF k
k
kk
k
( )
( ) ( )
.927295218001612232429... 7
1arctan
45
4arcsin
2
1arctan2
=
0 )12(4
)1(
21log
21log
kk
k
k
ii
ii
.92753296657588678176... 7
4
2
2 ( )
.927695310470230431223... 1
2
23
23
23
1
Li e Li ek
ki i
k
/ / cos( / )
.92770562889526130701...
4
41105
8
4
( )
( )
( )k
kk
HW Thm. 300
2 .92795159370697612701... e2 1 2 1 2 1 2 1 2 1cos sin( sin ) sin( sin ) cosh( cos ) sinh( cos )
=
ie e
k
ke e
k
k
i i
2
22 2
1
sin
!
1 .92801312657238221592... 2 1 2 14
20
( log ) log log
xxdx GR 4.222.3
3 .928082040192025996... pd k
kk
( )
!
1
.92820323027550917411... 4 3 61
12 1
2
0
( )
( )
k
kk k
k
k
6 .92820323027550917411... 48 4 32
61
2
k
k
k
kk
.9285714285714285714 =13
14
1
130
( )k
kk
1 .92875702258534176183... 2 3
0
1
3
49
36
1
1
( ) logx x
xdx
.92875889011460955439... 1
22 2
2
220
Ik k
k
k
( )!( )!
.92920367320510338077... 5
2 2
1
1
55
51
i e
e
k
k
i
ik
logsin
4 .92926836742289789153... e
.92931420371895891339... ( ) ( ) ( )2 3 4 312
42
k k
kk
3 .92931420371895891339... ( ) ( ) ( )2 3 412
41
k k
kk
1 .92935550865482645193... ( )
!/3
11
1
1
3k
ke
k
k
k
.929398924900228268914... 3 3 1 19
20
log log log
xx
dx GR 4.222.3
1 .929518610520789484999... 2 2
2 213
3
4 1 9
k
kk ( / )
.92958455909241192604... 2 10
11
3
4
3
1
3
1
3 3 1F
k
k
kk
, , ,( )
( )
.93002785739908278355... 263
2310 2 1515
csc
.930191367102632858668... e
.93052667763586197073... ( )
!( )!
k
k k kI
k kk k
1
12
1 1
21
1
.93122985945271217726...
0
2
)12(8
)1(
2
1arcsinh232log
2
1
kk
k
k
k
k
.93137635638313358185... | ( )| k
kk 21
6 .93147180559945309417... 10 2log
.9314760947318097375...
( )( )
3360 1
4
31
k H
kk
k
1 .9316012105732472119... e e k
k e
e
kk
2
2 0
cosh
!
.93194870235104284677... 3 4
32 2 14 20
log
( )
x
xdx
.93203042415025124362... ( )
1
9 10
k
k k
3 .932239737431101510706...
3
2 3 21
log/
k
kk
.9326813147863510178... 4
21
1
162 41
sinh
kk
.932831259045789401392...
1
02
22
123
12
)(2
)1(2log2
12 x
dxxLi
kkk
k
1 .932910130264518932837... 37
163
12
2
22
0
( ) ( )x Li x dx
.93309207559820856354... cos( )
( )!
1 1
2 20e k e
k
kk
AS 4.3.66, LY 6.110
2383 .93316355858267141097... 8777
1
ek
kk
!
.93333333333333333333 =14
15
1
140
( )k
kk
5 .9336673104466320167... 1
1256
1
4
3
4 13 3
3
21
( ) ( ) log
xdx
x
1 .93388441384851971997...
1
1
.93401196415468746292...
1
1
!
)1(sinhcosh11
k
kke
k
eeee
= e
xdxe0
151 .93420920380567881895... 413 1 10
0e
k
k
k
k
( )
!
.93478807021696950745...
1 61
31
21
))((
)(
3
4
6
72
k kk
kk
J1061
.93480220054467930941...
2
3
)12(!)!12(
!)!2(4
2)1(
12
2
k kkk
k
=1
1 2
1 1
2
4
2 121
22
21( / )
( ) ( ) ( )
( )k
k k
kk
k
kk k
1 .934802200544679309417... dzdydxxyz
zyx
1
0
1
0
1
0
2
13
2
2 .934802200544679309417... 2 2
022
1
x xdx
x
sin
cos
4 .934802200544679309417...
2
1
2
1
2
( ) logi i
= 3 21
21
212
20
( )( )
,
kk
=
211 212 )!2(
4)!1()!1(2
4
!)!12(
!)!2(
k
k
k
k
k k
kk
kk
kkk
k
= k k kk k
k kk k
2
2
2
2 22
14 1
1 ( ) ( )
( )
= 2 12arcsin
= dxx
x
04
3
)1(
log
= log( )
2
02
1
1 1
x
x
dx
x x GR 4.297.5
= logtan
tan2
0
1
1
x
dx
x GR 4.323.3
8 .93480220054467930941...
2
1
24
1
2
( )
.93534106617011942477... 7
7291
1
3 1
1
3 1
4
4 41
4
( )
( )
( )
( )
k k
k k k
.93548928378863903321...
1225
11
5/1
0
5sin
2
155
22
5
k k
.93594294416994206293... ii eLieLi 23
23
.93615021799926654078... log( )
( )( )2
3 2
1
2 1 3 10
k
k k k
.93615800314842911627... 11
12
( ) ( )k kk
.936342294367604449... sin1
11
12 2 2
2
kk
5 .936784413887151464... x dx
e ex x
3
1
.93709560427462468743...
81 2 2 2
0
2
( log ) log(sin )cos/
x xdx GR 4.384.10
= 1 2
0
1
x xdxlog GR 4.241.9
= xe e dxx x
1 2
0
GR 3.431.2
7 .9372539331937717715... 63 3 7
1 .93738770119278726439... 3
2 2 6
2
2 11
2 2
1
log( )
k
kk
k
Adamchick-Srivastava 2.23
.9375000000000000000000= 15
16
1
150
( )k
kk
.93750000000013113727... ( )
1 1
1
k
kk k
k
.93754825431584375370...
( )log
2 21
k
kk
Berndt 8.23.8
= 1
1 311 ( )!
log
n
k
k
n
kn
.93755080503059465056... ( )
1 1
12
k
kk k
1 .93789229251873876097... 3 2
2 20
2
2016 1
log
( )
xdx
x
x dx
e ex x
= )()( 33 iLiiLii
= log log2
20
1 2
211 1
xdx
x
xdx
x
GR 4.261.6
=
log2
40 1
x dx
x
= log tan/
x dx2
0
4
GR 4.227.7
= dydxyx
yx
1
0
1
022
22
1
)log(
.93795458450141751816...
24
222
1)()(315
24
1
k
k
k
Hiii
.93809428703288482665... ( )
1
10 10
k
k k
.93836264953979373002... 81
24
1
28
1
2 4 10
cos sin( )
( )! ( )
k
kk k k
.93846980724081290423... Jk
k
kk
1 20
1
2
1
16
( )
( !)
.93852171797024950374... ( ) ( )
( )! /
1 2
1
11
12 1
12
k
kk
k
k
k k e
6 .938535628628181848... k Hk
kk
2 3
1 2
( )
2 .93855532107125917158... 12
1 1( ) ( )k kk
1 .93872924635600957846...
22 6
21
1 2
2 1
2 21
csc( )
( / )
k
k k k
.93892130408682951018...
2
coth
1 .93896450902302799392... 5 5
4
5 5
5 5
25 5
4
5
21
2
5log
log
=5
5 1
1
1 4 5 51 02
2k k k k
k
k kk
k( ) ( )( / )
( )
4 .939030025676363443... 3 24
256 2 23040
12 4
41
( )
( )
( )
a kk
where is the nearest integer toa k k( ) 3 . AMM 101, 6, p. 579
1 .939375420766708953077...
1
0
2 3/1
log)3( dx
x
xLi
.93958414182726727804... 3
33
1 .9395976608404063362... log log21
2
2 2
2 2 22
1
H kk
k
4 .9395976608404063362... 3 21
2
2 2
2 2 22 1
1
log logkH k
kk
.93982139966885912103...
primekthp
k
k
pk 1)()(
.9399623239132722494...
primep pp
p42
22
1
1
)4(
)6(
21
2
.94002568113... g4 J310
.94015085153961392796... ( )
( ) !
k
k kk
12
1 .94028213339253981177... ( )
!
/2 1 1
1
1
1
2
k
k
e
kk
k
k
.94084154990319328756... 115
384
5
50
sin x
x GR 3.827.11
1 .94112549695428117124... 24 2 321
8
2 2
0
kk
k
k
.9411764705882352 =16
17
1
160
( )k
kk
.94226428525106763631... 89
8
1
8 10
log( )
( )
k
kk k
4 .9428090415820633659... 42 2
3 CFG D4
.94286923678411146019... 2
316 4
1
12
sink
kk
Davis 3.32
= iLi e Li ei i
2 3 3
.942989417989417 = 7129
7560
1
3 31
k kk ( / )
.94308256800936130684... ]1},2,2,2,2{},1,1,1,1[{ HypPFQ
=( )
!( )
1
1 40
k
k k k
.94318959229937991745... 5
4
2
42
1
222
cot
kk
= 2 2 11
1
k
k
k
( )
1 .943596436820759205057...
1 )(
)(
)6(
)3()2(
k kk
k
=
primepprimep pp
p
pp )1)(1(
1
)1(
11
32
6
5 .94366218996591215818... 3 2 66
1
1 620
csch ( )
/
k
k k
.94409328404076973180... dxx
xx
1
0
32 arcsin2arcsin
4
)3(
= dxxx2/
0
2 )cos2log(
Latter result due to Peter J.C. Moses and Seiichi Kirikami (A193712)
.9442719099991587856... 4 5 81
16 1
2
0
( )
( )
k
kk k
k
k
8 .9442719099991587856... 80 4 5
.94444444444444444444 =17
18
1
170
( )k
kk
.94495694631473766439... cos( )
( )!
1
3
1
2 9
k
kk AS 4.3.66, LY 6.110
.9451956893177937492...
21
2
1
2 13 30
Lik
k
kk
( )
( )
=log2
21
2
02 2 1
x
x xdx
x
edxx
= log2
0
1
2
x
xdx
.94529965343349825190... 0
1 1
( )
( )!
k
kk
.94530872048294188123...
2
5
15
8
1 .9455890776221095451... ( )
( !)
2 212
10
1
k
kI
kk k
1 .94591014905531330511... log76
71
Li
.94608307036718301494... 1
0
sin)1( dx
x
xsi
=( )
( )!( )
1
2 1 2 10
k
k k k AS 5.2.14
= sin1
1 x
dx
x
= 1
0
coslog dxxx
4 .94616381210038444055... 320
epf k k
kk
( )
.946190799031120722026...
2
1)(2
)1(226logk
kk k
k
1 .9467218571726023989... sink
kk 21
.946738452432832464562...
1 !
)(
k k
k
5 .94678123535278519168...
52 5
1
1 520
tan/
k kk
.94684639467237257934... 1
43 1 31 3
3
1
e ek
kk
cos cossin(sin ) sin(sin )sin
!
= ie e e ee e e ei i i i
83 3
3 3
.94703282949724591758... 7
7204
7 4
8
14 1
41
( )( ) ( )k
k k J306
.947123885654706166761...
2
1
42
1
23 41 4
1 4//
/cot ( ) coth( )
i i
= k
kk
2
41 2
.94716928635947485958... 11
11
2
1
321
tanh
k kk
.947368421052631578947... 18
19
1
180
( )k
kk
1 .94752218030078159758...
162 8 2 8 22 2 2 log log
= e xdxx
2 2
0
log GR 4.335.2
.947786324594343627...
sin1
113 2
1
kk
.94805944896851993568... 8
1 21
8 7
1
8 11
k kk
J78, J264
1 .94838823193110849310... Q k
kk
( )
!
1
.94899699000260690729... ( ) ( )2 3
3
4 .949100893673262820934... 1
3 1 ( )
.94939747687277389162... arctan(sinh csc ) ( ) arctan1 1 11
11
k
k k
[Ramanujan] Berndt Ch. 2, Eq. 11.4
.9494332401401025766... k k
kk
kk3 1 31
1
1
( )
.949481711114981524546... 2
6
.94949832897257497482... 2 1 1
4120
sin ( )
!( )
k
kk k k
.9497031262940093953... 2
2 216 3
11
6 1
1
6 1
( )
( )
( )
( )
k k
k k k J329
=
log x
xdx6
0 1
9 .9498743710661995473... 99
.95000000000000000000 =19
20
1
190
( )k
kk
.95023960511664325898... 2
22
206
31 5
2
1
2 2 1
log( ) ( !)
( )!( )
k
k
k
k k Berndt Ch. 9
.95038404818064742915...
6
3
2
1
3
1
3 221
coth/
kk
.95075128294937996421...
1
3
)2(
)3(7
8
k k
k
.95105651629515357212... 10
cos5
2sin2010
4
1
.9511130618309... ( )( )
1 14
1
k
k
k
k
1 .95136312812584743609... 2 21 8
22
1
1
sin( )
( )!
k k
k k
.9513679876336687216... 15
8
2
6
2
0
e
k k
k
k !( )
.95151771341641504187... 1
36
1
6
2
3
1
3 11 1
20
( ) ( ) ( )
( )
k
k k
=
log logx
xdx
x x
x
xdx
e ex x1 130
1
31
20
.9520569031595942854... ( )31
4
1 12 5 3
2
k k k kk
= ( ) ( )k kk
2 12
.95217131705368432233... 2 2
2 2 0
4
logcos log(sin )
/
x x dx
.952380952380952380 = 20
21
1
200
( )k
kk
.95247416912953901173...
02)12(2
)1(
2
1,2,
2
1
4
1
kk
k
k
2 .95249244201255975651... ( )! !
ej kkj
112
11
.95257412682243321912... e
eek k
3
33
1
1
21
3
21
tanh ( ) J944
.9527361323650899684...
2
1
1 220e t
tdt
cos
5 .952777785411260053338... dxx
x
0
224
2cos2log
1
2
)4(11
180
11
Borwein & Borwein, Proc. AMS 123, 4 (1995) 1191-1198
.953530125551415563988...
2
1)4(
)(
k k
k
.95353532563643122861... 551
2101
3
5
0
ek
kk ( )!
.954085357876006213145...
2 1 )!1(
)(
!
)(
k k k
kk
k
k, where
n
k
kn1
)()(
4 .95423435600189016338... Eik k
k
k
( ) log!
2 22
1
AS 5.1.10
.95460452143223173482... 1 4 5 ( ) ( )
3 .954608700594592236394... )3()2(23
)3(2
.95477125244221922768... 3
23 2 1
1
20
1
log log log
x
dx
7 .954926521012845274513... 2 100
2
I e dxx( ) sin
.95492965855137201461... 3
11
36 21
kk
GR 1.431
= cos
6 21
kk
GR 1.439.1
=0
1 6/
2 .95555688550730281453... !
1)(2
2 k
k
k
k
k
.95623017736969571405... 7
128
3 3
42 1 13
1
( )( )k k
k
=k k k
kk
( )
( )
4 2
2 42
4 1
1
2 .95657573850025396056... 2931587
2 3
6
0
ek
kk
( )!
.956786081736227750226... dxe x
0
sinhcosh1
.95698384815740185727... 5
162
33
31
sin /k
kk
GR 1.443.5
.95706091455937067489... G
6 .9572873234262029614... dxxx
2
2 1)(
.957536385054047800178...
2 1
)(logs n
ns
.957657554360285763750... 1
22
1
2
1
2 11
cot tank kk
Berndt ch. 31
1 .95791983595628290328...
0
2
2
2
)1log(
21log22
1arcsinh
x
x
2980 .95798704172827474359... e8
.958168713149316111695...
22 2
2
1
21 1
1
81 1 ( ) ( ) ( ) ( )( ) ( )i i i i
5
81 1 2 31 1i
i i ( ) ( )( ) ( ) ( )
= ( ) ( ) ( )
1 2 12
2
k
k
k k k
.95833608906520793715... ( )
( )!
1 1
21
k
k k
.95835813283300701621... cos cosh( )
( )!
1
2
1
2
1
40
k
k k
.95838045456309456205... 1
1024
5
8
3
82
1
82
8 82 2 2 3 2
( ) ( ) ( ) cot csc
= 11
4 1
1
4 13 31
( )
( )
( )
( )
k k
k k k
=
03
2/)1(
)12(
)1(
k
k
k
.95853147061909644370... 1
3
3
2 3
1 1
3 10
cos( )
( )!
e k
k
k
.95857616783363717319... 1 2
21
2
2 20
tanh( / )( )
/
/ /
e
e eek k
k
J944
2 .95867511918863889231...
4
3
4
1 1
.9588510772084060005... 21
2
1
2 1 41
1
402 2
1
sin( )
( )!
k
kk kk k
GR 1.431
= 0
1 2/
.95924696800225890378... 4 11
2 1
1
21
cik k
k
k
( )( )
( )!
.95951737566747185975... 1
2 1 2 11 2
0sinh( / )( / )
e
ee k
k
J942
.95974233961488293932...
03
3/2
233
2
x
dxx
.95985740794367609075... 2
12sin
1
2
1sin1cos22log
2
1cos)1(
2
1
1
k
kk
.9599364907945658556...
1
1)1()(k
kk
1 .96029978618557568714... k k
kk
( ( ) )
!
2
2
1
.96031092683032321590... 16
17
1
4
1
2 1 4
1 2
0
arcsinh
( ) ( !)
( )!
k
kk
k
k
.9603271579367892680... 4
4 21
2
1
1 4 2 11 40e
erfk k
k
kk
/
( )
( )! ( )
.96033740390091314086... 32 2
151
1
16 22
kk
1 .9604066165109950105... 2 22 2
0
2
48 8
2
4
2
4
log loglogsin cos
/
x xdx
.96090602783640284933... 2 22 1
2
1
log( )
H
k kk
k
.960984475257857133608...
1
1
12
)()1(
kk
k k
.96120393268995345712... 2 21
2 2
1
8 2 10
arctan( )
( )
k
kk k
1 .96123664263055641973... 3 3 2 ( ) ( )
.961533506612144893495... ii eLieLi 42
42
2
6
1188
.96164552252767542832... 4 3
5
( )
2 .96179751544137251057... ( )
( )!
2 1
113
1
2
k
k
e
kk
k
k
.96201623074638544179...
0 )1)(12(4
)1(
2
1arctan
5
4log4
kk
k
kk
.962423650119206895... 21
22
1 5
2
1
16 2 1
2
0
arcsinh
log( )
( )
k
kk k
k
k
.96257889944434482607... log 7 22 3
1
33 3
arccot
= 11
3
1
6 1
1
6 13 11
( )k
kk k k
1 .962858173209645782869...
1
1
k kL
.96307224485663007367... 2 5 ( )
.96325656175755909737... 5 1 1 1 11 5 2 5 3 5 4 5 1 ( ) ( ) ( ) ( )/ / / /
= 11
52
kk
1 .96349540849362077404...
12 63232
1
8
5
k kk
.963510026021423479441...
2 2 1
1
2
1 2
1 21log
( / )
( / )
=
3
21
.96400656328619110796... ( log )log( )
( )1 2
1
1
2
2 20
x
x xdx
.9640275800758168839... tanh22 2
2 2
e e
e e
3 .9640474536922083937... ( ) ( ) ( ) ( )i i i i 1 1
.9641198407218830887... 26
4 23 3
4
1
2
2 1
4 31
log( ) ( )k
k k k
.96438734042926245913... 1
5 51
( )
( )
k
kk
.96534649609163603621...
( )
( )
( )10
5 51
k
kk
HW Thm. 300
.96558322350755543793... 1
2 20k
k
.9656623982056352867...
( )2
2 12 11
kk
k
.9659258262890682867... 12
5sin
26
131
4
2
AS 4.3.46
.96595219711110745375... 6 5 3
108 16 20
log
( )
x
xdx
.96610514647531072707... erfk k
k k
k
3
2
2 1 3
2 2 1
2 1
2 1
( )
! ( )
4 .96624195886232883972... ( ) ( ) ( ) ( )2 3 4 5
1 .9663693785883453904...
222
2
1,3,
2
1
4
133
iLi
iLii
= dxx
x
12
2
2/1
log
.9667107481003567015... 1
21 1 1 1
0
1
cosh sin cos sinh cos cosh x x dx
4 .96672281024822179388... 11
1
kk !!
.96676638530855214796... 7
71
1
49 21
sin
kk
GR 1.431
=0
1 7/
1 .96722050079850408134...
1
0
1
0
1
0
1
02222
3
12log24
8dzdydxdw
zyxw
zyxwG
.96740110027233965471...
2
22
12
)1()()1(
2
3
4 k
k kkk
.96761269589907614401... ( )4 3
2 14 31
kk
k
.9681192775644117... 5
.96854609799918165608... 1
64
1
8
5
8
1
4 11 1
20
( ) ( ) ( )
( )
k
k k
=
log logx
xdx
x x
xdx
1 140
1 2
41
=
1 12213
loglogdx
xx
xdx
xx
xx
3 .96874800690391485217...
2arccsch51551log552arcsinh2log5252515
2
= F Hk k
kk 21
4 .96888875288688326456... 2
2 2
( )
( )!
k
kk
.96891242171064478414...
0
12/1
16)!2(
)1()1(Re
4
1cos
kk
k
k
.96894614625936938048...
3
3032
31
2 1
( )( )
( )
k
k k AS 23.2.21
= Im ( ) ( ) ( )Li ii
Li i Li i3 3 32
= sin /k
kk
23
1
=
1
0
1
0
1
02221 zyx
dzdydx
5 .96912563469688101600... 2
2 11
k
k k k! ( )
Titchmarsh 14.32.3
.969246344554363083157...
2
1)4(
)(
k k
k
.96936462317800478923...
2 2 2 121
2
csc
=( )
1
2
1
4 21
k
k k k
.96944058923515320145... 4
729 3
5
5
g J310
2 .9706864235520193362... 2
9 .9709255847316963903... 2 2
.97116506982589819634... G1 3/
54 .9711779448621553884 =43867
798 9 B
.97201214975728492545...
2
0
2
0 sin2
sin
cos2
cos1
3
22
x
dxxdx
x
x
3 .972022361698548727395... k k kk
( ) ( )
1 12
379 .97207708399179641258... 276 150 22
4
1
logk Hk
kk
.97211977044690930594... 15 5
165
11 5 0
1
51
( )( )
( )( , , )
k
k k AS 23.2.19
.97211977044690930594... 15 5
16
1
1 50
( ) ( )
( )
k
k k
.97218069963151458411... 2
3 2
2 2
31 2
2
3 log( )
log
=1
1 2 1 4 10 ( )( )( )k k kk
.97245278724951431491... 61
6
1
6 2 10
sin( )
( )!
k
kk k
= 11
6 2 21
kk
.97295507452765665255... log7
2
= 11
3 6 1
1
3 6 13 1 31
( )
( )
( )
( )
k
k
k
kk k k
K ex. 108
1 .97298485638739759731...
21 1)(
kk kH
1 .97318197252508458260... 8
)3(13
381
5 3
= 1
216
2
3
1
62 2
2
20
( ) ( )
x dx
e ex x
= ( )( ) ( ) ( ) (/ / / /3
2
2
31 1 1 11 3
32 3 2 3
3
1 3 Li Li
= log log log2
30
1 2
2 11
2
311 1
x
xdx
x
x xdx
x x
xdx
.97336024835078271547...
1
3
1
2
2
3
2
32 3( ) /
2 .97356555791412288969... 4 2 3 3 ( ) ( )
1 .973733516712056609118...
1
2
)3)(2)(1(48
752
k
kk
kkk
HkH
.97402386878476711062... S k k
kk
2
1
2
2
( , )
( )!
.97407698418010668087... log( )1 e
.974428952452842215934... dxexx
x
e
0
22
))(1(
log
33
1
.9744447165890447142...
22 22
iLi
iLii
=
0 )12)(12(4
)1(
2
1,2,
4
1
4
1
kk
k
kk
.97449535840443264512...
132
cos2282
8sin
8
kk
GR 1.439.1
.97499098879872209672... 1
22 1 2
1
3 40
log( )
/
k
k k
= dx
x x( ) cosh /1 420
GR 3.527.10
1 .97532397706265487214... 3
64
2
16
2
16 4
4 2 2 2
2 20
log log log x
x xdx
.97536797208363138516... sin Im ( )
2
2i
1 .975416977098902409461... 2/
02
32
sin8
)3(21
4
2log3 dx
x
x
1 .97553189198547105414...
1
1
)!2(
16)1(2sin2
k
kk
k
25 .97575760906731659638... 4
1524 4
2
4 4
0
( )x
e ex x
1 .97630906368989922434... )1()1( 00 LI
= 1
0
)exp(sin dxx
2 .9763888927056300267... 11
360
3
24
1
22
1
42
2
21
( ) ( )( )
H
kk
k
18 MI 4, p. 15
.97645773647066825248... sin( )
( )!( )
1
2
1
2 1 2 11
1
2 11
kk
k
kk k
.976525256762151736913...
2
2 2 211
1
21
1
2 1
cos( )kk
.976628016120607871084...
6
5
6
1
36
1 )1()1(2 h J314
1 .9773043502972961182... ( ) ( )
( ) ( ) ( )2 3
3
6
21
31
12
1
k
k
k
kk k
HW Thm. 290
.977354628657892959404...
12
212
3
2log)2(
2
)3(7
k k
k
.97745178 ...
1
31 )(
)1(k
k
k
k
3 .97746326050642263726...
I e dx e dxx x0
0
2
0
1( ) cos cos
2 .9775435572485657664...
12 24
2 22
2
2 2 2
log log
log
= log sin2 2
20
x x
xdx
.977741067446923797632...
0
1
32
221
4
12
k k
k
4 .9777855916963031499...
k
k
k
kIIe
kk
2
2!)2,2,
2
3(F)1()1(
01110
109 .9778714378213816731... 5535
2
22
3
0
k
k
kk
k
1 .977939789810133276346... 7
4
3
8
1
1 4
1 5
2 31
G k
k
k
k
( )
( / )
.97797089479890401141... 1
100
1
10
3
5
1
5 11 1
20
( ) ( ) ( )
( )
k
k k
= dxx
xdx
xx
xx
1
1
0523 1
loglog
1 .9781119906559451108... 2
2 2
06
log xdx
ex GR 4.355.1
.97829490163210534585... G1 4/
40400 .97839874763488532782... 40320 9 18 ( ) ( )( )
.97846939293030610374... Li23
4
.97895991797814145164... )21cot(221cot(2)223(log24
arcarci
=
2/
0 cossin
xx
dxx
.97917286680253558830... cos cosh( )
( )!/ /
1
2
1
2
1
4 23 4 3 40
k
kk k
.97929648814722060913... 2 21
2 2
1
2 10
sin( )
( )!8
k
kk k
.97933950487701827107... 161
4
1
2 1 4 12
0
sin( )
( )! ( )
k
kk k k
.97987958188123309882... loglog( )1 3
2
1
1
2
20
1
x
xdx GR 4.295.38
.98025814346854719171...
0 )12(32
)1(22log2
kk
k
kk
k
2 .981329471220721431406... 1)1(1
)3(21
3
kk
k
k
.98158409038845673252... 31
3
1
9 2 10
sin( )
( )!
k
kk k
= 11
9 2 21
kk
.98174770424681038702... 5
16 1
5 2
x
xdx
/
GR 3.226.2
.98185090468537672707... ( )
!
( )
( )!!/2
2
2
21
1 1
1 2
1
2k
k
k
kek
k k
k
k
.98187215105020335672... iLi Li
21 12
3 42
1 4 ( ) ( )/ /
= sin / k
kk
42
1
1 .98195959951647446311... 6
5
2
5 5
3
5
3
5 1 5 30
csc /
dx
x
.98201379003790844197... 1 2
21
2
2 24
0
tanh( )
e
e eek k
k
J944
.982097632467988927553... 19
27 3
1 3 3
0
e k
k kk
/
!
1 .9823463240711475715... 22
9
2 2
3
2
3
2 2 12
2
30
log e
x x
e
x
dx
x
x x
GR 3.438.2
.98265693801555086029... 2 6 ( )
.98268427774219251832... 1 3
12
1
6 2
3
21
12 2
26
2
coshsin
ikk
.982686076702760592745...
0 3)12(2
)1(
4
1,
2
3,
2
3,
2
3,1,1,
2
1,
2
1
k
k
kk
kHypPFQ
.98295259226458041980... 945 1
66 61
( )
( )k
kk
.983118479820648366...
sin1
114 2
1
kk
.98318468929417269520...
1
02
2
2
log
2
1,3,
2
1
8
1
x
x
=i
Lii
Lii
2 2 23 3
.983194483680076021738... 691
675675
1
1
6
6
pp prime
5 .98358502084677797943... ( ) ( ) ( ) ( ) ( )2 3 4 5 6
.98361025039925204067... 2 16
2 21
2 1
3 1
31
Gk k
k
k
log( )
( )
.98372133768990415044... 1
144
1
12
7
12
1
6 11 1
20
( ) ( ) ( )
( )
k
k k
= x x
x xdx
log3 3
1
10 .98385007371209431669... 2 4 2 2 3 23 3 1
1
2
32
( ) ( )( )
k k
k kk
= ( ) ( )k k kk
2
2
1
.98419916936149897912... 403
393660
6
6
J312
.9843262438190419103... 2
636 5
3 ( ) ( )
.9845422648017221585... 2
22
20
1
62 2
3
4
1 1
log
( ) ( ) log ( )x x
xdx
.98481748453515847115... ( )
14
1
k
kk k
.98517143100941603869... 2 2 1 21 4 / , from elliptic integrals
.98542064692776706919... log1
3
.9855342964496961782... 96 16
15 4
24 4
1
( )
( )k
kk
.9855510912974351041... 31
302406
31 6
32
16 1
61
( )( ) ( )k
k k J306
4 .98580192112184410715... 4 23
8 8
11
43
40
logsin logsin( )
( )( )
k
k k k
1 .98610304049995312993... 1
512
5
8
1
8 12 2
2
40
1
( ) ( ) log
x
xdx
=
142
2logdx
xx
x
.98621483608613337832... 21
2
1
2 1 4 2 10
sik k
k
kk
( )
( )! ( )
6 .98632012366331278404... 5 9
4
2
3
2 2
0
e k
k k
k
k
!( )
.9865428606939705039... 11
768 21
1
4 1
1
4 1
4
4 41
( )
( )
( )
( )
k k
k k k J327, J344
=
04
2/)1(
)12(
)1(
k
k
k Prud. 5.1.4.4
.98659098626254229130...
6
1
3
2
432
1
2
)3(13
3162
5 )2()2(3
= ( )
( )
1
3 1 30
k
k k
12 .98673982624599908550... 28402
2187
1
2
1 10
1
( )k
kk
k
.98696044010893586188... 2
10
8 .98747968146102986535... 1856
243
380
81
4
3 4
4
1
logk Hk
kk
.98776594599273552707... sin 2
.98787218579881572198... 4
105
7 15
42015 15
44 4
csc csch
= ( )
1
1542
k
k k
6 .98787880453365829819... 2
156
1
1
4 3
0
1
( ) logx x
xdx
.98829948773456306974...
2
3 12018 3 1 2 1
log x
xdx
.98861592946536921938... 3 23 1 1
1
144 21
kk
GR 1.431
=
1223
cosk
k
=0
1 12/
.98889770576286509638... sin sinh ( )( )!
1 1 12
4 2
2 1
0
kk
k k GR 1.413.1
= 114 4
1
kk
.98894455174110533611... 1
1536
1
4
3
43 3 ( ) ( )
= i
Li i Li i2 4 4( ) ( )
=( )
( )
sin /
1
2 1
24
04
1
k
k kk
k
k
1 .98905357643767967767... 3 2
609 3 1
log x dx
x
.9890559953279725554...
0
2
222 )2(log
122 xe
dxx
1 .98928023429890102342...
( )log
22
21
k
kk
1 .989300641244953954543... )2()2(5
4)3()3( iiii
.98943827421728483862... 0
1 2( )kk
kk
.98958488339991993644... cos cosh( )
( )!
1
2
1
2
1
4 40
k
kk k
.98961583701809171839... 41
4
1
2 1 160
sin( )
( )!
k
kk k
= 11
16 2 21
kk
.98986584618905381178... 41
4
2 1
64 2 10
arcsinh
k
k k
k
kk
( )
( )
1 .98999249660044545727... 1 3 23
2
1 9
22
1
1
cos sin( )
( )!
k k
k k
.99004001943815994979... 1
3456
5
12
7
122
1
12
7 4 3
4322 2 2
3
( ) ( ) ( ) ( )
= 11
6 1
1
6 13 31
( )
( )
( )
( )
k k
k k k
.990079321507338635294...
16
15
16
7214204
256
1 )1()1(2
=
162
log
xx
dxx
4 .990384604362124949925... 10
3
74
81 3 2
3
1
kk
kk
78 .99044152834577217825... 1212 1
1
10
0
e
k
k
k
k
( )
( )!
.99076316985337943267... k kk
( )
12
2
.99101186342305209843... 1
41 1( ) ( ) ( ) ( )i i i i
21 .99114857512855266924... 7
5 .991155403803739770668...
8
7
8
32176
4096
1 )3()3(4
=
122
3log
xx
dxx
4 .9914442354842448121... 6
3 ( )
.99166639109270971002... HypPFQkk
, , , , ,( )!
1
5
2
5
3
5
4
5
1
3125
1
50
.99171985583844431043... 1
7 71
( )
( )
k
kk
6 .99193429822870080627... ( )kk
2
7
2 .9919718574637504583... 323 Borwein-Devlin p. 35
.99213995900836... 7
1 .992173853105526785479...
0
)1(k
kI
.99220085376959424562... 4
125
2 53
31
sin /k
kk
GR 1.443.5
.99223652952251116935... 56
94815 3
7
7
g J310
1 .992294767124987392926...
1
2
30,2,
1
)1(
)1(
kke
k
ee
ee
1 .992315656154176128013... )()(768 55
5
iLiiLii
.99259381992283028267... 63 7
647
1 1
71
( )( )
( )
k
k k AS 23.2.19
.99293265189943576028... 2
1 1
0
( ) sin(tan )
e xdx
x GR 3.881.2
.99297974679457358056... ( )3 2
2 13 21
kk
k
.99305152024997656497... 1
1024
5
8
1
82 2 ( ) ( )
.99331717348313360398... ii eLieLi 22
22
2
322
.993317230688294041051...
12 )4/1(
1
21
2122
k kk
ii
.993703033793554457744...
12 )1(
)3(34
)4(17
k
kk
kk
HH
.9944020304296468447... 6 2 2 26 2 1
22 3
21
log log( )
k
kkk
.99452678821883983884... 3
318 3 h J314
1 .9947114020071633897... 5
2
.99532226501895273416... erfk k
k k
k
22 1 2
2 1
2 1
0
( )
!( )
1 .9954559575001380004... dx
xx0
1 .99548129134760281506...
2
20
( log ) log sin
x xdx
x GR 4.421.1
.99551082651341250945... ( ) ( )
( )
( )
( )
91 45 5
187501
1
5 1
1
5 1
4
4 41
k k
k k k
10 .99557428756427633462... 7
2
21 .99557428756427633462... 7
211
22
2
1
k
k
kk
k
.995633856312967456342... 1
786432
3
8
5
8
1
8
7
84 4 4 4 ( ) ( ) ( ) ( )
= 11
4 1
1
4 15 51
( )
( )
( )
( )
k k
k k k
=
05
2/)1(
)12(
)1(
k
k
k Prud. 5.1.4.4
.99585422505141623107... 2
2 2
( )kk
k
.99591638044188511306... arctan k
kk 21
.99592331507778367120...
sinhsin ( ) sin ( )/ /
81 1 1
13
1 4 3 48
2 kk
= 2cos2cosh16
sinh3
7 .99601165442664514565... ( )kk
2
8
1 .99601964118442342116... 142 3844
3
0
ek
k kk
!( )
.99608116021237162307... 5207
49601160
8
8
J312
15 .996101835651158622733... 32
3
238
81 3 2
4
1
kk
kk
.99610656865... g8 J310
.99615782807708806401... 5
15365
1
2 1
5
50
( )( )
( )
k
k k AS 23.2.21, J345
.99623300185264789923... 127
12096008
127 8
128
18 1
81
( )( ) ( )k
k k J306
.99624431360434498902.. 219
4096 2
5
= 11
4 1
1
4 15 51
( )
( )
( )
( )
k k
k k k
.996272076220749944265... tanhtan
i
i
e e
e e
.99633113536305818711...
2/1
0 22
2
1)(
)(
4/34
3logdk
kk
kkK GR 6.153
8 .99646829575509185269... 2 2 9 31
33
0
log ( )sin
cos
x xdx
x
.997494864721368727032... ( )( )
12
2
2
kk
k
k k
119 .99782676761602472146...
1
5
6
234
0,5,1
)1(
)1266626(
kke
k
ee
eeee
.99784841149774479093... 3
8
11
22
7 3
4
16 6 1
2 2 1
2 2
32
log( )
( )
k k
k kk
=k k
kk
2
2
1
2
( ( ) )
8 .99802004725272736007... ( )kk
2
9
.998043589097895... 9 J312
.9980501956570772372... 3236
55801305 3
9
9
g J310
.9980715998379286873... 23
1296 31
1
6 1
1
6 1
4
4 41
( )
( )
( )
( )
k k
k k k
.99809429754160533077...
( )( ) ( )
8255 9
256
1 1
91
k
k k AS 23.2.19
2 .998095932602046572547...
1 )12(k
kk
kk
HH
.99813603811037497213...
kk e
e
e
22
2
)1(12
tanh1 J944
.9983343660981139742...
1
333
)38(
1
)58(
1
64
1
2128
3
k kk
17 .99851436155475931539... 1016
81
512
27
4
3
1 2 3
41
log( )( )( )k k k Hk
kk
.99851515454428730477... 3 2 11
220
logx
dx
.99857397195353054767... 19
2 3 5 31
1
4 1
1
4 1
2 4
14 6 61
( )
( )
( )
( )
k k
k k k J327
2 .9986013142694680689... 9 2
.99861111319878665371... HypPFQk
k
k
, , , , , ,( )
( )!
1
6
1
3
1
2
2
3
5
6
1
46656
1
60
.99868522221843813544... ( )( )
( )( ) ( )6
1
491520
1
4
3
4
1
2 15 5
60
k
k k AS 23.2.21
.998777285931944... h4 J314
33 .99884377071514942382... 5 1
8
4
2
4
0
e k
k k
k
k
!( )
.999022588776486298... 48983
4591650240
10
10
.99903950759827156564... 73
684288010
511 10
512
110 1
101
( )( ) ( )k
k k J306
1 .99906754284631204591... 6 2
.99915501340010950975...
4
4 4 42
1 1
11
4sin sinh
kk
1 .99916475865341931390... e e e
e
k
k
e e
k
2 1 2
1
1
2 1
/ sinh
( )!
1 .99919532501168660629... 7
11
1 .99935982878411178897... 3 2
120
7 4 3
216 1
log x dx
x
.99953377142315602297... 3 2
.9995545078905399095... 61
1843207
1
2 1
7
70
( )( )
( )
k
k k AS 23.2.21
.99955652539434499642... 2307 2
13107201
1
4 1
1
4 1
7
7 71
( )
( )
( )
( )
k k
k k k
.999735607648751739... 11
1944 3
5
5
h J314
.9997427569925535489... 2305
933121
1
6 1
1
6 1
5
5 51
( )
( )
( )
( )
k k
k k k
.9997539139218932560...
sinhcosh sin ( ) sin ( )/ /
12
3
21 15
2 1 6 5 6
= 3coscosh2
3cosh
24
sinh 25
= 1112
2
kk
.99980158731305804717... ( )
( )!
1
70
k
k k
.99984521547922560046... 24611
165150720 21
1
4 1
1
4 1
8
8 81
( )
( )
( )
( )
k k
k k k J327
.99984999024682965634... ( ) ( ) ( )81
330301440
1
4
3
47 7
=( )
( )
1
2 1 80
k
k k
.99992821782510442180... 1681
933120 31
1
6 1
1
6 1
6
6 61
( )
( )
( )
( )
k k
k k k J327
.99994968418722008982... 6385
412876809
1
2 1
9
90
( )( )
( )
k
k k AS 23.2.21
.99997519841274620747... ( )
( )!
1
80
k
k k
.9999883774094057879... 301
524880 3
7
7
h J314
.99998844843872824784...
1
77
7
)16(
)1(
)16(
)1(1
100776960
333672
k
kk
kk
.9999972442680777576... ( )
( )!
1
90
k
k k
.99999727223893094715...
188
8
)16(
)1(
)16(
)1(1
31410877440
25743
k
kk
kk
J329
.99999972442680776055... ( )
( )!
1
100
k
k k
