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7/26/2019 Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, Pi, and the Ladies Diary
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Gauss, Landen, Ramanujan, the Arithmetic-Geometric Mean, Ellipses, , and the Ladies DiaryAuthor(s): Gert Almkvist and Bruce BerndtReviewed work(s):Source: The American Mathematical Monthly, Vol. 95, No. 7 (Aug. - Sep., 1988), pp. 585-608Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2323302.
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2/25
Gauss,Landen,
Ramanujan,
heArithmetic-Geometric
ean,
Ellipses, rr,nd the
LadiesDiary
GERTALMVIST, UniversityfLund
BRUCE
BERNDT*,Universityf llinois
GERT
ALMKVISTeceived
is Ph.D. at theUniversityf Californian
1966
and has been t
Lund ince
967.His main nterestsre lgebraic
-theory,
invariant
heory,
nd
elliptic unctions.
BRUCEBERNDT
eceived
is
A.B. degree romAlbion
College,Albion,
Michigan
n
1961 nd his
Ph.D. fromhe
UniversityfWisconsin,adison,
in196.
Since
977,
he
hasdevotedll
ofhisresearchffortso
proving
he
hitherto
nproven
esultsn
Ramanujan's
otebooks.is
book,
Ramanujan's
Notebooks,
art (Springer-Verlag,985), s thefirstfeitherhree rfour
i
volumes to be
publshed
on
this
project.
Virtuendsense,
with emale-softness
oin'd
(All that ubdues nd
captivates ankind )
In
Britain's atchlessair
esplendent
hine;
They ule ove's mpire y rightivine:
Justly
heir harmshe stonished orld
dmires,
Whom
oyal
Charlotte's
rightxample
ires.
1. Introduction.he
rithmetic-geometric
eanwas first iscovered
y
Lagrange
and
rediscovered
y
Gauss a few
years
aterwhilehe was a
teenager.
owever,
Gauss's
major
contributions,ncluding
n
elegant ntegral epresentation,
ere
madeabout7-9 yearsater. he firsturposef this rticles, then,oexplain he
arithmetic-geometric
ean nd to describe ome
of ts
major
properties, any
f
which re due to
Gauss.
*Research
artially
upportedy
the
Vaughnoundation.
585
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3/25
586
GAUSS, ANDEN,
RAMANUJAN...
[August-September
Because f
tsrapid
onvergence,he
rithmetic-geometric
eanhas
been ignifi-
cantly
mployed
n
the
astdecaden
fast
machine
omputation.
second urpose
of
this rticle s
thus o
delineatets
role n
the omputationf
7T.
Weemphasizethat thearithmetic-geometriceanhas muchbroader
pplications,
.g., to the
calculation
f
elementary
unctions
uch s logx,
ex, sinx,
and
cosx. The
inter-
ested
reader
houldfurther
onsult he
several
eferences
itedhere,
specially
Brent's aper
14]
and the
Borweins'
ook 13].
The
determinationf
the
rithmetic-geometricean
s
intimately
elated o the
calculation
fthe
erimeter
f n
ellipse.
ince he ays
f
Kepler nd
Euler, everal
approximate
ormulas ave
been
devised o
calculate
he
perimeter.
he
primary
motivation
n
derivinguch
pproximations
as
evidently
hedesire o
accurately
calculate he
elliptical
rbits f
planets. third
urpose f
this
rticles
thus o
describehe onnectionsetweenhe rithmetic-geometricean nd theperimeter
of
an
ellipse, nd to
survey
any
f
the
pproximate
ormulashat
avebeen
given
in
the
iterature.he
most
ccurate f these
s due to
Ramanujan, ho
lso
found
some
extraordinarilynusual nd
exotic
pproximations
o
elliptical
erimeters.
The
atter
esultsrefoundn
hisnotebooks
ndhave
never
een
published,nd so
we shall
pay particularttention
o these
pproximations.
Also
contributing
o
this
circle f
ideas is
the
English
mathematician
ohn
Landen.
n
the
tudy fboth
he
rithmetic-geometric
ean
nd the
determination
of
elliptical
erimeters,
here rises
is most
mportant
athematical
ontribution,
whichs nowcalledLanden's ransformation.anyverymportantndseemingly
unrelated
uises f
Landen's
ransformationxist
n
the
iterature.hus,
fourth
purpose
f
this
rticles to
delineate
everal
ormulationsf
Landen's
ransforma-
tion
s well as
to
provide short
iography
f
this
ndeservedly,ather
bscure,
mathematician.
For
several
ears, anden
ublished
lmost
xclusively
n
the
Ladies
Diary. his
is,
historically,hefirst
egularly
ublished
eriodicalo
contain
section
evoted
to the
posing
f
mathematical
roblemsnd their
olutions.
ecause n
important
feature f
the
MONTHLY
has
its roots
n
the
Ladies
Diary,
it
seems
hen
dually
appropriatenthis aper oprovide brief escriptionf theLadiesDiary.
2.
Gauss and the
arithmetic-geometric
ean.As we
previously
lluded,
he
arithmetic-geometric
ean
was first et forth
n
a
memoir f
Lagrange30]
pub-
lished
n
1784-85.
However,
n
a
letter,
ated
April
6,
1816,
o a
friend,
. C.
Schumacher,
auss
confidedhathe
independently
iscovered
he
rithmetic-geo-
metricmean
n
1791
t the
ge
of
14. At about he
ge
of
22
or
23,
Gausswrote
longpaper
23]
describing
is
many
iscoveriesn
the
rithmetic-geometric
ean.
However,
his
work,
ike
many
thers
y
Gauss,
was
not
published
ntil fter
is
death.Gauss's
fundamental
aper
hus
id
not
ppear
ntil 866
when .
Schering,
theeditor f Gauss'scompleteworks, ublishedhepaperas partof Gauss's
Nachlass.
Gauss
obviously
ttached
onsiderable
mportance
o his
findings
n the
arithmetic-geometric
ean,
or everal fthe
ntries
n
his
diary,
n
particular,
rom
the
years
799 to
1800,pertain
o the
arithmetic-geometric
ean.Some
of these
entries
re
quite
vague,
ndwe
may
till
otknow
verything
hat
Gauss
discovered
aboutthe
rithmetic-geometric
ean.
For
an
English
ranslationf
Gauss's
diary
together
ith
ommentary,
ee a
paper y
J.J.
Gray 24].)
By
now,
hereaders anxious o earn bout
he
rithmetic-geometric
ean
nd
what
he
young
Gaussdiscovered.
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1988]
GERT ALMKVIST
AND BRUCE BERNDT
587
Let
a and b denotepositive
umbers
ith
a
>
b. Construct
sequence
f
arithmetic
eans nd
a
sequence
f
geometric
eans s
follows:
1
a,l=
-(a
+b),
b=
4,b
1
a2
=
-(a,
+
bl)9 b2= Ha1,1,
1
a
=
+
-(an
+
bn),
bnl
=
b
Gauss
[23]
gives
our
umerical
xamples,
f
whichwe
reproduce
ne. Let
a
=
1
and
b
=
0.8. Then
al
=
0.9,
b,
=
0.894427190999915878564,
a2
=
0.8972135954999579392829 b2
=
0.8972092687327349
a3
=
0.897211432116346,
b3
= 0.897211432113738,
a4
=
0.897211432115042, b4
=
0.897211432115042.
(Obviously,
auss
did
not hirk
rom umerical
alculations.)
t
appears
rom his
example
hat
an)
and {
bn)
converge
o the
ame
imit,
nd that urthermorehis
convergence
s
very apid.
his
we nowdemonstrate.
Observe hat
b
1,
we see that
11)
reduces
o
a
Wenow et
n
tend o
x.
Since
n
tends o
M(a,
b)
and
xn
tends o
0,
we
conclude
that
a
a7T
K(x)-=M(a,
b)
K(O)
2M(a,
b)
Landen's
ransformation
10)
was
ntroduced
y
him n
a
paper
31]
published
n
1771
and
n
more
eveloped
orm
n
his
most amous
aper 32]
published
n
1775.
There xist everal
ersionsf Landen's
ransformation.
ften
anden's ransfor-
mation
s
expressed
s an
equality
etweenwodifferentials
n
the
heory
f
elliptic
functions17], 37].The importancef Landen's ransformations conveyedy
Mittag-Leffler
ho,
nhis
very erceptive
urvey
37, .
291]
n the
heory
f
lliptic
functions,
emarks,
Euler's ddition heoremnd the
transformationheoremf
Landen and
Lagrange
were
the two fundamentaldeas of which
he
theory
f
elliptic
unctions as n
possession
hen
his
heory
as
brought
p
for enewed
consideration
y Legendre
n
1786."
In
Section
,
we
shall
rove
he
ollowing
heorem,
hich
s often alled anden's
transformationor
omplete
lliptic
ntegrals
f
the
first ind.
THEOREM
2.
If
0
2
VJb
He [28, p.
368]
furthermore
emarks
hat 1/2)(a +
b) >
V,
and
so
concludes
that
L
T
h-(a
+
b).
2
Kepler ppears o be
using he dubious
rinciple
hat uantities
arger
han he
same
number
must
e
about
qual.
Approximationsf
several
ypes,
epending
pontherelativeizesof a and
b,
exist
n
the
iterature.n
this
ection,
e
concentraten
estimateshat rebest or
close
to
b.
Thus,
we
shall write ll of
our
approximations
n
terms f
X
(a - b)/(a + b) andcomparehemwithhe xpansion
25).
Forexample, epler's
second
pproximation
an be
written
n
theform
L
7T(a
+
b)(1
-
A
)
We now how
how
heformula
/
1
00
C\
L(a, b)
=
4J(a,
b)
=
M( bIa2-
2E
2ncn),
(27)
arising romTheorems ' and 4, can be used to findapproximationso the
perimeter
f an
ellipse.
Replacing
M(a, b) by a2
and
neglecting
he termswith
n
>
2,
we
find
hat
L(a,
b)
-
(a2
-
- 12 = T
This
formula
was first btained
y
Ekwall
19]
n
1973 as a
consequence
f a
formula
y Sipos
from 792
54].
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600
GAUSS, LANDEN,
RAMANUJAN...
[August-September
If
we
replaceM(a, b)
by a3
in
27)
andneglect ll terms
ithn
>
3, we
find,
after ome
alculation,hat
2(a
+
b)2 (
a _
b)4
L(a,b) 27Tr
2
(Va
+
V)
+
2V4/a
b
Vab>
This formula
s complicated
nough
o dissuade s from
alculating urther
p-
proximations
y thismethod.
We nowprovide
table
f approximations
or (a, b) that ave
beengiven
n
the
iterature.t the
eft,
e ist he
discovereror source)
ndyear fdiscovery
if
known).
The approximation
(X) forL(a,
b)/7T(a
b) is given n the
second
column
n
two
forms.
n
the
astcolumn,he
first onzero erm
n thepower
eries
for
AA-7T(a
+
b)
=A )-
-2 '2;1
)
is offered
o that
he ccuracy f the
pproximating
ormulaan
be discerned.or
convenience,
e
note
hat
F(
-
-
'
-
2
;
1;
A?)
=
1
+
-
A?
+
43 A
+
4
+
47 A
+
48
A1+
Kepler 28], 1609
-
+=
(1
-
X2)1/2
3
Euler
21],
1773
2 1 +
b2)1/
2
+
a
+b
4
Sipos [54],
1792
2(a
+
b)
2
7
Ekwall
[19],
1973 (f
+
)2
1
+
1
64
Peano
[42],
1889 --
=---(1
-
K')1/2
34
2 a +b
2 2
6
2
(a3/2_+ b3/2
2/3
Muir
38],
1883
a+ b
2
1
1
1
64
=
/3
{(1
+
X)3/2
+
(1
-X)3/22/3
(
1
(a-b\22
Lindner
35,
p.
439],
1?-j
I
1
1904-1920 ( 8(a+b)
j
-1x6
Nyvoll 41],
1978
( lX2)2
28
Selmer
49,
975 1 +
4(a
-
b)
Selmer
49], 1975
1 +
(5a
+ 3b)(3a
+ 5b)
3
-1
+
x2
1-
-X2
-
4
1
X2
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18/25
1988]
GERT
ALMKVIST
AND
BRUCE
BERNDT 601
Ramanujan
[44],
[45],
1914
-
a3)(ab
1
a
+
b
-A
Fergestad 49], 19513- 4- 2
Almkvist
1],
1978
2
2(+
-(V-
'i)
(a +b){(
V)+ 2V
a
+
bv%b
15
-
X8
(1+
+1
X22?X i-x2-
21
-2
~~~~~~~~~~2
(1
+
1 +2)114 2
Bronshtein nd
1
64(a
+
b
4-
3(a
-
b)
Semendyayev 16
(a
+ b
2
(3a +
b)(a
+
3b)
219
[15],
1964
64
-
3X4f
Selmer
49],
1975=64-1X
6{
2
166a+b)
(
a- b\2
2V2(a
+6+b2
Selmer
49],
1975
a
2
+
b
1
3-?- 2--1-X
2 8
2 2
Jacobsen nd 256
-
48X2 21X
33x1
Waadeland
[26],
1985
256
-
112X2
3X
218
Ramanujan
1+
3X2
3
[44],
[45],
1914
10+4-3f27
The
two
approximationsy
India's
great
mathematician,
.
Ramanujan,
ere
first tated
y
him
n
his notebooks
46,
p.
217],
nd then ater t theend of
his
paper
[44],
45, p. 39],wherehe saysthattheywere discoveredmpirically.
Ramanujan
44],45]
also
provides
rror
pproximations,
ut
they
re
in
a
form
different
rom
hat
iven
ere.
ince
a -b 1
-
11-
e2
e2
a +b
1?
1j-
e2
T4
we
find
hat,
or hefirst
pproximation,
(e2/4)
6
__2
__
~~(a?b)j~~a(1?
1-e2)
~29
2. Proceeding y nduction, e deduce that
an+1
2
af
for n > 2, and theproof f 34) is complete.
From
32)
and
(34),
it follows
hat
3
-
r4
-?2
7T/6.
Second,
we calculate
a when
=
1.
Thus,
X
=
1
and
0
=
a.
Therefore,
rom
25)
and
(32),
1 /11\
4
1 + 4sin2-a
=F--,--;1;1I
=
-. (35)
2 \2 2J
This evaluation
follows
from
general
theorem
f Gauss
on the evaluation
of
hypergeometric
eries t
the
argument [4, p.
2].
Moreover,
his
particular
eries s
found
n
Gauss's
diary
under
he date
June,
798
[24].
Thus,
1 1 1
sin2
-a
=-
-
-
=
0.0683098861.
2
l
4
It follows
hat
a
=
30'18'6"~.
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604
GAUSS, ANDEN,
RAMANUJAN...
[August-September
Third,
we calculate
when
=
0. From 30) and
32),
sin2o
4
sin2-0
lim in2a lim
lim
2
00
21
=
M
im
2
adin
F=
ai~
A ?
n 1
4
Thus,
tends
o
ir/6
s e tends o 0.
Ramanujan
46, p. 224] offers nother
heorem,
hichwe do not state,
ike
Theorem
but which ppears o be
motivatedy
his second pproximation
or
L(a, b).
Ramanujan46,p. 224] states wo dditional ormulasachof which ombines
two
pproximations,
ne for near and the ther
or close o1. Again,we
give
just
one of
the
pair.
A completeroof
fTheorem belowwould
e too engthyor
this
aper,
nd
so
we
shall
ust
ketch
he
main deas
ofthe
proof.
ompleteetails
may
be found
n
[7].
THEOREM . Set
tan
L(a, b)-= r(a
+ b)
,
0
Recommended