Tables of Fourier Transforms and Fourier Transforms of Distributions

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Fritz Oberhettinger Professor emeritus. Oregon State University P. O. Box 84. Seal Rock. OR 97376/USA
Mathematics Subject Classification (1980): 42A38. 44A 10. 44A 15
ISBN-13: 978-3-540-50630-0 e-ISBN-13: 978-3-642-74349-8 001: 10.1007/978-3-642-74349-8
Library of Congress Cataloging-in-Publication Data Oberhettinger, Fritz. (Tabellen zur Fourier Transformation. English) Tables of Fourier transforms and Fourier transforms of distribution/Fritz Oberhettinger. p. cm. Rev. and enl. translation of: Tabellen zur Fourier Transformation. 1957.
1. Fourier transformations. 2. Mathematics-Tables. I. Title. QA404.0213 1990 515'.723-dc20 90-9507
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© Springer-Verlag Berlin Heidelberg 1990
Preface
These tables represent a new, revised and enlarged version of the
previously published book by this author, entitled "Tabellen zur Fourier
Transformation" (Springer Verlag 1957). Known errors have been correc
ted, apart from the addition of a considerable number of new results,
which involve almost exclusively higher functions. Again, the follow
ing tables contain a collection of integrals of the form
(Al
(B)
(C)
ge(y) = J f(x)eixYdx Exponential Fourier Transform -00
Clearly, (A) and (B) are special cases of (C) if f(x) is respec
tively an even or an odd function. The transform parameter y in (A)
and (B) is assumed to be positive, while in (C) negative values are
also included. A possible analytic continuation to complex parameters
y* should present no difficulties. In some cases the result function
g(y) is given over a partial range of y only. This means that g(y)
for the remaining part of y cannot be given in a reasonably simple
form. Under certain conditions the following inversion formulas for
(A), (B), (C) hold:
(B') f (x) 2 J gs(y)sin(xy)dy 11 0
(C' ) -1 00 -ix
f(x) = (211) J ge(y)e Ydy
In the following parts I, II, III tables for the transforms (A),
(B) and (C) are given. The parts I and II are subdivided into 23 sec
tions each involving the same class of functions. The first and the
second column (in parenthesis) refers to the location of the corre
spondent page number for the cosine- and sine transform respectively.
*The domain of analyticity is the strip in the direction of the real axis of the complex y plane.
VI Preface
Compared with the before-mentioned previous edition, a new part IV
titled "Fourier Transforms of Distributions" has been added. In this,
those functions f(x) occuring in the parts I-III have been singled
out which represent so-called probability density (or frequency dis
tribution) functions. The corresponding normalization factors are
likewise listed.
The author wishes to express his thanks for the expertise of
Mrs. Jolan Eross in the completion of this book.
Seal Rock. January 1990 Fritz Oberhettinger
Contents
1. 1 Algebraic Functions .......................................... 3
1.2 Arbitrary Powers ............................................. 8
1.7 Hyperbolic Functions ......................................... 33
1.9 Gamma- and Related Functions ................................. 47
1.10 The Error- and the Fresnel Integrals ......................... 48
1.11 The Exponential- and Related Integrals ....................... 53
1.12 Legendre Functions ........................................... 56
1.13 Bessel Functions of Arguments x, x' and l/x ................... 66 l:;
1.14 Bessel Functions of Argument (ax' + bx + c) ................... 77
1.15 Bessel Functions of Trigonometric and Hyperbolic Arguments .................................................... 83
1.16 Bessel Functions of Variable Order ........................... 86
1.17 Modified Bessel Functions of Arguments x, x' and l/x .......... 89 l:;
1.18 Modified Bessel Functions of Argument (ax' + bx + c) .......... 93
1.19 Modified Bessel Functions of Trigonometric and Hyperbolic Arguments .................................................... 97
1.20 Modified Bessel Functions of Variable Order .................. 100
1.21 Functions Related to Bessel Functions 102
1.22 Parabolic Cylinder- and Whittaker Functions .................. 106
1.23 Elliptic Integrals ........................................... 109
2.1 Algebraic Functions ........................................... 115
2.2 Arbitrary Powers .............................................. 120
2.7 Hyperbolic Functions .......................................... 145
2.8 Orthogonal Polynomials ........................................ 152
2.10 The Error- and the Fresnel Integrals .......................... 155
2.11 The Exponential- and Related Integrals ........................ 158
2.12 Legendre Functions .....................................•...... 160
2.13 Bessel Functions of Arguments x, x 2 and l/x .................... 164 !:i 2.14 Bessel Functions of Argument (ax 2 + bx + c) .................... 173
2.15 Bessel Functions of Trigonometric and Hyperbolic Arguments ..................................................... 178
2.16 Bessel Functions of Variable Order ....................................................... 179
2.17 Modified Bessel Functions of Arguments x, x 2 and l/x ...................... 180
2.18 Modified Bessel Functions of Argument (ax 2 + bx + !:i c) .... , ...... 183
2.19 Modified Bessel Functions of Trigonometric and Hyperbolic Arguments ..................................................... 187
2.20 Modified Bessel Functions of Variable Order ................... 188
2.21 Functions Related to Bessel Functions 1.89
2.22 Parabolic Cylinder- and Whittaker Functions ................... 192
2.23 Elliptic Int~grals ............................................ 194
Part III. Exponential Fourier Transforms (Tables III) .............. 197
Part IV. Fourier Transforms of Distributions (Tables IV and V) ...... 209
Appendix ........................................................... 249
1.1 Algebraic Functions 3
0 x>a
0 x>b + y-2(COS(bY)-COS(ay )]
1.3 x x<a 4y-2cos(ay)sin2(~ay) 2a-x a<x<2a
0 x>2a
-1 x>a x
0 x>a
x-~ x>a
1.8 -1 xb + sin(ay) (Si (ay+by)-si (ay) ]
1.9 0 xb
1.10 -1 xb + sin (ay) (si(aY)-Si(aY-bY) J a>b
loll 0 xb + sin(ay)si (ay+by)-Ci (by) 1 x (a+x) J
4 I. Fourier Cosine Transforms
f (x) gc(y)
n=2,3, ••• - sin(aY+~TIn)Ci(ay»)
n-l ( m-l)') -m n-m-l + I (n-l) i a sin (~TIn-~TIm) (-y)
m=l
(cos(aY+~TIn)si(ay+by) (n-l) !
(a+x) -n x>b -sin(aY+~TIn)Ci(ay+by»)
n=2,3, ••. n-l (m-l) ! -m n-m-l + I (n-l) ! (a+b) sin (~TIn-~TIm-by) (-y) m=l
1.14 x~(a+x)-l (2Y/TI)-~-TIa~cos(aY){1-C(ay)-s(ay)}
- TIa~sin(ay){c(ay)-s(ay) }
+ sin (ay) {C (ay) -S (ay) })
1.16 (a+x) -~ xb + sin(ay) {S(ay+hy)-S(ay)})
1.17 0 xb + cos(ay){~-C)ay+by)})
1.18 (a+x)-3/2 2a-~-(2TIy)~(cos(aY){1-2S(ay) }
- sin z{1-2C (ay) })
(x-a) -~ x>a
(~TI/y) {cos(ay)-sin(ay) }-TIa {l-C(ay)-S(ay)} 0 x<a
1.21 (a-x) -1 cos(ay)Ci(ay)+sin(aY){~TI+Si(ay)}
(Cauchy principal value)
-1< 1.22 0 x<a TIa 2{l-C (ay) -S (ay) }
-1 x (x-a) -~ x>a
0 x>a + sin(ay)S(ay)}
1. 24 0 xb + sin(ay) {C(ay+by)-S(ay+by)})
1. 25 (a2+x2)-1 ~TIa-le-ay
1.26 x(a2+x2 )-1 _~(e-ay Ei(ay)+eay Ei(-ay»)
1. 27 (a2 _x2 )-1 xb + sin(ay) {si(ay+by)-si(ay-by)} a>b
1. 28 (x2_a2) -1 xb - cos(ay) {Ci(by-ay)-Ci(by+ay)})
b:>a
1. 29 (a2 _x2 )-1 ~1Ta -1 sin (ay) (Cauchy principal value)
1. 30 2 2-1 x(a -x ) cos (ay)Ci (ay)+sin(ay)Si (ay)
1. 31 x-~(a2_x2)-1 -3/2 ~ -1 ~TIa sin(aY)+(~TIy) a S ~(ay)
0,
1. 32 (a2+x2)-~ Ko (ay)
1. 33 -~ 2 2-~ x (a +x ) (~1TY)~I_~(~aY)K~(~ay)
(a2+x2) (b2+x2») -1
~TI(a2_b2)-1(b-le-bY_a-le-ay) 1. 34
1. 35 (x+(a2+x2) ~)-l 2 -1 (a/y) -(ay) Kl (ay)
1. 36 2 )2 -1 {b +(a-x } -1 -by TIb . e cos (ay)
+{b2+(a+x)2}-1
f (x) gc(y)
1. 37 2 -1 (a+x) (b +a+x) -by .
TIe sin (ay) 2 -1 + (a-x) (b +a-x)
1. 38 r-1 (r+a)J., (J.,TI/y)J.,e-ay
1. 39 r-1 (r+a)-J., TI(2a)-J.,Erfc{(ay)J.,}
1. 40 x-J.,r-1 (r+x)-J., 2-J.,TIa-1e-J., aYI o (l~ay)
1.41 x-J.,r-1 (r+x)-3/2 2-J.,a- 2sinh(J.,ay) Kl (J.,ay)
1. 42 x-J.,{a+x+ (2ax) J.,}-1 TI(2a)-J.,e aYErfc{(ay)J.,}
1. 43 (a2_x2 )-J., x<a J.,TIJo (ay)
0 x>a
00
1.44 0 xa
0 x>a
1. 46 x<a
0 x>a
1.47 0 x<a -J.,TIYo (ay) (x2_a 2 )-J., x>a
1.48 0 x<a -1 J.,TIa (l-!:zTIay{Jo(ay) 8_1 (ay)
-1 2. 2 -!:z x (x -a ) x>a + Ho (ay)J1 (ay)})
1. 49 0 x<a -J.,TIa-n(sin(J.,nTI)J (ay) (x2_a2)-!:z
n +cos(!:znTI)Yn(ay))
2 2 J., -n I (k!(n+k-l)! (1.., )-k. ( !:zk)) • (x+(x -a )) x>a
n=l, 2,3, ••• - k=1 (2k)! (n-k)! 2ay Sln ay+ TI
x>a
1.52 (x4+2a 2x 2cos b+a 4 )-1 .,1Ta- 3exp(-ay cos.,b)sin(.,b+ay sin.,b cosec b
-1T<b<1T
1.53 x 2 -.,1Ta- l exp(-ay cos.,b) sin (.,b-ay sin.,b)cosecb
1.54
1.55
1.56
1.57
-1T<b<1T
X2m(a2n+x2n)-1
n,m=l,2,3, •••
2m 2n-l
m = 0,1,2, ••.
-1 2m-2n+l n "1Tn a I sin ((2m+l) ak+ay cos(ak ))
k=l
1.2 Arbitrary Powers
f(x) gc(y)
2.1 (a+x) v ~iy-V-1(exp(~i~V-iay)r(1+v,-iay) Rev <0 -exp(iay-~i~v)r(1+v,iay) )
2.2 (a+x)v xh -y(1+v,-iay) }
-exp(iay-~irrv) {y(1+v,iay+iby)
-y (1+v ,iay) })
2.3 v -1
x (a+x) ,-1 Rev<1 ~a vr (1+v) {eiaYr (-v,iay) +e -iaYr (-v,-iay) }
2.4 (a_x)v x<a ~iy-V-l(exp(~irrv+iaY)Y(I+v,iay) 0 x>a -exp(-iay-~i~v)y(l+v-iay) )
Rev>-1
2.5 xV (a-x)].l x<a ~B(v+l,].l+I)av+].l+1
0 x>a (IF l (l+v;2+v+].l;iaY)+IFl(l+v;2+v+].l;-iay)
Re(v,].l) >-1
2.6 0 x<a ~avr(l+v){r(-v,iay)+r(-v,-iay) } -1 v
x (x-a) x>a
0 x>a
2.9 2 2 -v-~
(x -a ) -~rr~(~v/a)vr(~-v)Y (av) x>a • v .. 0 x<a
-~<Rev<~
2.10 (2 2) v-~ x a -x , x<a v -a(a/y) sv_l,v+l(ay)
0 x>a
Rev>-~
2.11 0 -2V-l~ ( x<a ~rrsec(rrv)a 1-~rray J v (ay)Hv _ 1 (ay) x-l(x2_a2)-v-~,x>a -H (aY)Jv_ 1 (a ) J} -1 <Rev <~
1.2 Arbitrary Powers 9
f(x) gc(y)
2.12 0 x<2a -~1[ ~r (~-v) (2a/y)-v (x2-2ax)-v-~ x>2a .(J (ay)sin(ay)+Y (ay)cos(ay»)
v v -~<Rev<~
2.13 (x2+2ax)-V-~ -~1[~r(~-V) (2a/y)-V
-~<Rev<~ .(y (ay)cos(ay)-J (ay)sin(ay») v v
2.14 (2ax-x2) v-~ x<2a 1[~ r (~+v) (2a/y) v cos (ay) J v (ay)
0 x>2a
Rev>-~
2.15 (2ax_x2)V-~ x<a ~1T~r (~+v) (2a/y) v 0 x>a
• (Jv(ay)cos (ay)+ RV(ay)sin(ay») Rev>-~
2.16 x-~r-1(r+x)v (~1[y)~avI_~v_~(~aY)K~_~v(~ay) Rev<~
2.17 x-V-~r-1(r+a)V ~1[ ~sec (~1T+~1TV) 0v_~ (z) (0 -v-~ (z) +o_v_~ (-z) ) Rev<~ • (~1T/a) ~ Z= (2ay) ~
2.18 (u+x) -v -v -1 ( 1Tvcosec(1Tv)a y sin(~1[v)Iv(ay)
Rev>O + ~i Jv(iay)-~i Jv(-iay»)
2.19 -1 -v 1[COSec(1Tv)a-v(~ J (iay)+~ (-iay) u (u+x) v v
Rev>-1 -cos(~1Tv)I (ay») v
2n -2v-1 -1 2.20 x u (-1)n1T~(2a)-V(r(~+v») d 2n/dy2n) (yVKv(ay»)
O~n<~+Re v
2.21 x vu-211- 2 ~aV-211-1B(~+~v,~_~V+ll} -1 <Rev<2+2Rell 2 2 ~ 1+211-v
1F2(~+~v;~+~V-ll,~;~a y }+~1[ (~y)
+ r (~V-ll-~) 3 2 2 r (l+ll-~V) 1F2(1+11;1+11-~V'~Il-~v;~a y )
2 2 ~ 2 2 ~ 2 2 ~ u=(a +x ) ,v=(a -x ) ,w=(x -a )
10 I_ Fourier cosine Transforms
f(x) gc(y)
2.22 x-v-~v-l (~./a)~sec(.v) (O_V_~(z)+O_V_~(-z») _(a+v)v+(a_v)V) x<a -(0 ~(z)+o ~(-Z»)lZ=(2iay)~ v- v-
0 x>a -~<Rev<~
2.23 XVv 2jl x<a ~aV+2jl+lB(1+jl'~+~V)lF2(~+~Vl~,jl+~V 3 2 2
0 x>a + 21-~a y ) Re (v,jl) >-1
2.24 0 x<a -.aV(sin(~.v)Jv(ay)+cos(~.v)Yv(ay») w-l(x+w)v+(x_W)V)
-l<Rev<l x>a
2.25 0 x<a -~.av(~.Y)~(J~v_~(Z)Y_~v_~(Z) x-~-l (x+w) v + J_~v_~(z)Y~v_~(z»), z=~ay + (x-w) V) x>a
-~<Rev<~ w=(x2_a2)~
2.26 xV (a2_x2)-1 ~.aV-lsin(ay)+(./a)~ ~~:~~)r(~+~v)y~ -l<Rev<2 - S_~_v,~(ay)
cauchy principal
1.3 Exponential Functions 11
-1 -ax -bx b2+ 2 3.2 x (e -e ) ~log(~)
a +y
3.3 ~ -ax x e ~u~(a2+y2)-3/4cos(~arc tan (y/a) )
x-~e-ax ~
3.4 (~u)~(a2+y2)-~(a+(a2+y2)~)
3.5 n -ax n!an+1 (a2+y2)-n-1 I (_l)m(n;;) (y/a)2m ,x e
n=0,1,2, ••• m SUIlUllation over m such that 0~~+1
xn-~e -ax (_l)n(~U)~ dn -~ 3.6 (a2+y2)-~(a2+y2)~_a)
n=1,2,3, ••. dan
3.7 v-I -ax 2 2 -~v ( ) x e r (V) (a +y) cos varc tan (y/a)
ReV>O
3.8 0 xb 'cos(by+(v+1)arc tan)y/a») (x-b) e
Rev>-l
3.10 x -1 (e -1) -x -1 logy-~(~(iy)+~(-iy»)
3.11 -1( -1 X-I) x ~-x +(e -1) _~log(1_e-2uy)
3.12 x(eax_1)-1 -2 (u 2 ~y -~ acosech(uy/a»)
3.13 xV-1 (eax_1)-1 ~a -v r (v) (l; (v,l+iy/a) +l; (v,l-iy/a») Rev>-l
3.14 xV- 1 (eax+1)-1 r(v){y-VcOS(~uV)+~(2a)-V(l;(V,~+~iy/a) Rev>O +l;(v,~-~iy/a)-l;(v,~iy/a)-l;(v,-~iy/a»)J
3.15 x-2 (l_e-ax)2 l+4a2 arc cot(~(y/a)3+~y/a) a log (2 2 )-y
y +a
3.17 e _ax:l ~(u/a)~exp(-~y2/a)
f(x) gc (y)
1; -ax 2 3/2 -z ( 1 2
3.18 x e \rr(1;y/a) e I 3(Z)-I1 (Z)), Z=Sy /a -"4 "4
x-1;e-ax 2 !orr(1;y/a)1;e- zI 1 (z) z=~y2 /a
3.19 ,
8 -"4
2n _a2x 2= -1; n 1; -n-l -2n-l 2 2 2 3.20 x e (-1) rr 2 a exp(-1;y /a )He 2n (-a-Y)
n=I,2, 3, •..
v -ax 2 1; -1;-1;v 1 2 3.21 x e 1; (1;rr) (2a) sec (!orrv) exp (-sy fa)
ReV>-1 • (D (z)+D (-z)) , z=(2a)-\ v v 2 2
-1 2 2 ( -by by ) 3.22 (b 2+x2) -Ie -a x \rrb exp{a b ) e Erfc(u)+e Erfc(v)
v = (ab±1;y/a) u
(b2+x2) -m-le -ax 2 1; -!om-!o 1;m -m-l -1 1 2 2
3.23 !orr 2 a b (m! ) exp (-sy / a+!oab ) m -lk m=I,2,3, •.. . I (m+k)!2 2 a-!okn-k[e!obYD (u)
k=O -v +e -!obYD (v) 1
-v u = v (2a) 1; (1±!oy/a), v = m+l-k
3.24 (b+ix)v+(b-ix)V) !o -!ov-!o 2 1 2 (1;rr) (2a) exp(!oab -BY fa) 2
• (e 1;bY D (u) +e -1;bYD (v)) ,~= (2a) 1;(b±!oy/a) -ax 'e v v
2 2 2 3.25 exp(-ax-bx ) \(rr/b)1;(e u Erfc(u)+ev Erfc(v)),
u = !o(a±iy)b-1; v
3.26 2 2 x exp(-ax ) 1; -3/2 2 2 \rr a (1-1;y /a)exp(-\y fa)
3.27 v-I 2 x exp (ax-bx ) -1;v (1 2 2 ) 1;r(v) (2b) exp a(a -y )/b
Rev>O • (exp(\iay/b)D (u)+exp(-\iay/b)D (v)) -v -v u = (2b) -1; (a±iy) v
3.28 x-1;e-a / x (!orr/y) 1;e -z (cos z-sin z) , z = (2ay) -!o
3.29 x- 3/ 2e-a / x (rr/a)!oe-zcos z, z = (2ay)!o
1. 3 Exponential Functions 13
f (x) gc(y}
3.30 x-v-le -a/x (y/a}~(exp(\iTIV}K (u}+exp(-\iTIv}K (v)) v v Rev>-l u = 2(±iay}~ v
3.31 x-~exp(-ax-b2/x} TI~(a2+y2}-~e-2bu(u cos(2bv}-v sin(2bu})
u ~ 2 2 ~ ~ = 2- (a +y ) ±a)
v
-3/2 2 TI~b-le-2bucos(2bv} , 3.32 x exp(-ax-b Ix} u,v as before
3.33 V-l 2 V( -~v -~v ) u (a±iy)~ x exp(-ax-b Ix} b u Kv (2bu}+v Kv (2bv} = v 0>
(_ay}n 3.34 -2 2 2 ~ -1 x exp(-a /x ) TIa I n~l'(~+~n}
0
3.35 -ax~ (~TI}~ay-3/2{cos Z (~-C(z) )+sin Z (~-S (z) ~, e
2 Z = \a /y
3.36 -~ -ax~ (2TI/Y) ~{cos Z (~-S(z) )-sin Z (~-C(Z})j, x e
Z = \a2/y
x e \ (a/y)~ J\(z}sin(z+i}-Yl/4(Z}COS(z+iD
1 2 Z = Sa /y
3.38 v-l -ax~
x e (2y) - vr (2 V) [e - T-ZD -2v (~ay -~ (l-i) )}
Rev>O + eT+ZD_2V(~aY-~(1+i}) , 1. 2/ ~. z=S1.a y, T= 1. TIV
-br -1 3.39 e abs Kl (as)
00
3.40 -2 -br f Ko{a(t2+y2)~}dt r e b
3.41 -1 -br Ko (as) r e
3.42 x-~r-le-br (~TIY}~I_\(~aS-~ab)K\(~as+~ab)
3.43 r- 3/ 2e-br (~b/TI}~Kk(~as-~aY}Kk(~as+~ay) 4 4
r = s =
f (x) gc(y)
·exp (-br)
3.47 xV-~(r+a)-Vr-1e-br ~(~)~cosec(~nv+i)D ~(U){D ~(v) a -v- v- Rev>-~
+Dv_~(-V) }
+ D_v_~(-iv)}
u = (2a) ~ (s±b) ~ v
3.48 -1 ~ 2 r (r+a) exp(-bx ) \exp(~a2b-~y2/b) (e~aYU~K\(~bU2)+e-~ayv~
.Kk(~bv2») j u = a±~y/b • v
3.49 (r+x)vr -1e -br a v cosec (nv) (ncos {varctan (y /b) }I (as) -v
n - f exp(ab cos t)cosh(ay sin t)cos(vt)dt)
0
• •
3.51 u- 3/ 2e bu x<a ~n(~nb)~(J,-\(~ay-~aV)J_\(~ay+~av) 0 x>a
+Jk(~ay-~aV)Jk(~ay+~av») • •
3.52 w- 3/ 2e-bw x<2a ~(~~b)~cos(ay) (J_\(~ay-~aV)J_\(~ay+~av) 0 x>2a
+ Jk(~ay-~aV)Jk(~ay+~av») • •
3.53 w- 3/ 2e bw x<2a ~ (~~b) ~cos (ay) (J_\ (~ay-~aV) J_\ (~ay+~aV) 0 x>2a
+ Jk(~ay-~aV)Jk(~ay+~av») • •
- 2 2,~ r - (a +x )
u = w = v = v
+(a+u)2Ve -bU)u -1
3.55 xVexp(-axc )
Rev>-l, c>l
. (D_2v_~(z2)+D_2v_~(-z2)) z = 1
(2a)~(b±v)~ 2
_v_l oo n -1 'IT -nc -y I(-a) (n!) r(l+v+nc)sin{"2(v+nc)}y
0
s = 2(~ y/a) 3/2, p q
2 2 ~ v= (b -y ) ,
= s eXP(±t'ITi)
1.4 Logarithmic Functions
0 x>a
4.3 2 2-1 (x -a) log (x/a) ~~a-l(sin(aY)Ci(ay)-cOS(ay)si(ay»)
4.4 (x2_a 2 )-110g(bx) ~~a -l~in (ay) (Ci (ay) -log(ab) )-cos (ay) si (ay~ Cauchy principal
value
4.5 (a2+x2)-110g(bx) -l( -ay ay -ay-) ~~a 2e log(ab)+e Ei(-ay)-e Ei(ay)
2 2 4.6 0 x<l ~(Ci(~y») -~ (si (~y) )
-1 x>l x 10g(2x-l),
-1 2 2 4.7 x log (l+x) ~ (Ci (~y») +~(si(~y»)
logl (a+x)/(b-x) 1 -1 { 4.8 y ~~cos(by)-~~cos(ay)+cos{by)Si(by)
+coS{aY)Si{aY)-Sin(aY)Ci(aY)-Sin(bY)Ci(bY~
4.9 log (a-x) x<a y-l{sin(ay ) (Ci(ay)-y-log y)-coS(ay)Si(ay } 0 x>a
4.10 log (a-x) xb +sin (ay) (Ci (ay) -Ci (ay-by»)
a>b -cos (ay) (Si (ay) -Si (ay-by) )}
4.11 log (a+x) xb +sin(ay) (Ci(aY+bY)-Ci(ay)~
4.12 xv-llog x r(v)y -v cos (~~v) ('I' (v) -~~tan (~~v) -log y)
O<Rev<l
+(a-ix)-llog(a-ix)
1.4 Logarithmic Functions 17
2 2 -l:i x co
-1 4.15 (a -x) log (il) , x<a -l:irrJ (ay)log(2)-l:;rr I n J 2n (ay) o _ 0 x>a n=o
4.16 X-llog(l+X+(l+X)~) 7 t-1Ko(t)dt Y
4.17 log(a2_x2 ) xb -si (ay-by) )+sin (ay) (Ci (ay+by) -Ci (aY-bY)))
a>b
4.18 2 2 2 2 log((a +x )!(b +x)) -1 -by -ay rry (e -e )
4.19 2 2 2 2 log! (a +x )/(b -x )! -1 ( -ay ) rry cos(by)-e
4.20 x -llog (1+x2/a2 ) Ei(-ay)Ei(ay)
4.21 x-llog!a+x! -rrsi (ay) a-x
4.22 (a2+x2)-l:ilog(a2+x2) -(y+log(2y/a))Ko (ay)
4.23 log(1+a2/x2 ) rry-l(l_e-ay )
4.25 log!1-a2/x2 ! -1 2 2rry sin (l:iay)
4.26 log(a2-x2 ) x<a -1 sin (ay) (Ci (2ay) -y-log (l:;y/a)) y
0 x<a -y-lcos(aY)Si(2ay)
4.27 (a2_x2)-~log(a2_x2) 2 \rr Yo(ay)-~rrJo(ay) {y+log(2y/a)} x<a
0 x>a
0 2 4.28 x<a -\rr Jo(ay+~rrYo(ay) {y+log(2y/a)} 2 2 -~ 2 2 (x -a) log(x -a )
x>a
4.29 2 -~
l:irrcos (ay) (rrY (ay) -2 {y+log (2y/a)}J (ay)) (2ax-x )
olog(2ax-x2 ) o 0
f(x) gc(y)
4.30 (x2+2ax)-loj loj1T {{Y+10g (2~) } (Y (ay) cos (ay)
'log (x2+2ax) a 0
+Jo (ay) sin (ay )))
4.31 log(loj+loj(1+a2/x2)loj) lojrry-l(l+ L (ay)-I (ay)) o 0
4.32 (a2+x2)-loj ~rr2{I (ay)- Lo (ay) } 2 2 loj 0
'log (l+a /x ) +a/x)
(1+x2 )-loj(1+x2 )loj-l) loj
-(2y/rr)-loj(eYEi(-2y)+lojrre-y ) 4.33 'log {x+ (1+x2) loj}
4.34 (a2+x2)-loj loj (8 1 (iay) +8 1 (-iay) +log a K (ay)) 2 2 loj - ,0 - ,0 0
'log{x+(a +x ) }
4.35 log (~u+~v) x>a ~y-l(~rrJ (ay)+si(ay)) 0
u=l±a/x, 0 x<a v
4.36 log(~+~(l+a/x)~) ~1Ty-l(l-cos(~ay)J (~ay)-sin(~ay)Y (~ay)) o 0
4.37 0 x<l -(rr/y)~(cos(Y-~1T)Ci(2Y)Sin(Y-~1T)si(2Y)) (l+x)-~
'log(x+(X2_l)~) x>l
x-llog(x+(x2_l)~) -lojrr f t-ly o(t)dt x>l y
4.39 -ax e log x 2 2 -1 ( -(a +y) ay+loja log(a2+i)
+ y arctan (y/a))
4.40 \1-1 -ax 2 2 -~v {( 2 2) x e log x (a +y) r(v) cos(vz) 'I'(v)-lojlog(a +y)
Rev>O -z Sin(VZ~; z=arctan(y/a)
-1 2 ~irr(K (v)-K (U))-log(2y~/a) (K (v)+K (u)) 4.41 x exp(-~a /x)log x o 0 0 0
u=a (±iy) loj v
1.4 Logarithmic Functions 19
f (x) gc(y)
-ax 2 2 2 -l{l 2 2 2 4.42 e (log x) (a +y) Grr a+(y+~log(a +y »)
( 2 2 2) • ya+~a log(a +y »)+2yz-az
z=arctan (y/a)
-1 cosech (rry fa)
4.44 -ax log (l-e ) ~ay-2_~rry -1 coth (rry/ a)
4.45 (a2+x2)-n-~ -n: n{ n 1 (2n): (2y/a) Kn(ay) (Y+log(2y/a)-2kI12k-l)
2 2 n-l k-n Kk(ay) } 'log(a +x )
n=l,2,3, •.. +~n\Io (~ay) (n-k)k:
-1 ( n n) n-l k (l:;s/-n 4.46 r (r+x) - (r-x) n!cos(nz) L a (n-k)k! Kk(as)
'e -brlog (~E.) k=o
a a -2an z sin(nz)Kn(as);z=arctan(y/b) ;
2 2 ~ n=l,2,3,;r=(a +x ) s =(b2+i)l:;
4.47 (a2_x2)n-~ -n (2n) : (~rrY (ay) ~rr (2y/a) n! n 2 2 'log(a -x ) x<a n 1
0 x>a +2Jn (ay) L 2k-l k=l n=1,2,3, •..
n-l (~ay)k-n ) -{y+log(2y/a)}J (ay)+~n! L (n-k)k! Jk(ay)
n k=o
4.48 (2ax_x2)n-~ 2 cos(ay)A(y) with A(y) = gc(y) 2 'log(2ax-x ) x<2a as given in the previous result 4.47
0 x>2a
n=l, 2,3, •..
1.5 Trigonometric Functions
l:;7T y=a
0 y>a
0 y>a1+a2+·· .an x ..... sin(anx) ... x
5.3 xV - 1sin(ax) k sec(l:;7Tv) 47T (I-v) ((y+a) -v -sgn (y-a) \ y-a I-V)
-l<Rev<l
5.4 2 2-1 x(b +x) sin (ax) l:;7Te -ab cosh (by) y<a
-l:;7Te-bYsinh(ab) y>a
5.5 -1 2 2-1 x (b +x) sin (ax) -2 l:;7Tb (l-e -ab cosh(by») y<a
l:;7Tb- 2e-bYsinh(ab) y>a
-bx 2 2)-1 ( 2 2)-1 5.6 e sin (ax) l:;(a+y) (b +(a+y) +l:;(a-y) b +(a-y)
5.7 -1 -bx sin (ax) ( 2ab ) x e l:;arctan 2 2 2 b -a +y
-1 2 5.B x sin (ax) Uog \1- (2a/y) \
x-1sin(ax)sin(bx) 2 2
0 y>2a
5.11 -3 3 x sin (ax) 122 ii7T (3a -y ) y<a
!:>7TY 2 y=a
0 y>3a
( -1 ) 2m (_1)m2-2mm7T ((m!)-2y 2m-1+ m
5.12 x sin (ax) L (-1)nA , y~2am n=l n
m=1,2,3, ••• 0 y~2am
A = (2an+y)2m-1+!2an_l!2m-1
1.5 Trigonometric Functions 21
m=l, 2,3, ••• m F(y) = L A y~
n=o n
k-1 m F(y) = L B + L A (2k-1)a~~(2k+1)a
n=o n n=k n
F (y) = 0 y~(2m+1) a
An 2m 2m =(_1)n((2n+1)a+z) ±((2n+1)a-~)
B (n+l+m) ! (m-n) ! n
k=1,2,3, ••• m
-ax. 2n (_l)n 2-2n-2 -1 -1 An n+l.;y±l.;ia .
5.14 e (s~n X) --(A -B ). = ( 2n+1 ) ~ 2n+1 n n 'B n=0,1,2, ••• n
5.15 e-ax(sin x)2n-1 (_1)nn-12-2n-2(A~1+B~1); A n-l.;+l.;y±l.;ia B n= (2n )
n=1,2,3, ••• n
5.16 (sin~x) v-I x<l I-v :L ( )-1 2 cos(l.;~)r(v) r(l.;+l.;v+l.;y/~)r(l.;+l.;v-l.;y/~)
0 x>l
-(y+a)log(y+a)-(y-a)10gly-a l)
5.18 -1 x (I-cos ax) 2 2 l.;log 11-a /y ) I
5.19 x2 (1-cos ax) l.;~(a-y) y<a
0 y>a
5.20 -1 -2 -1+l.;y(log(1+y)-logI1-yl+l.;logI1-y-2 1) x -x s~n x
5.21 v-I cos (ax) V (V) cos (l.;~V) (I y-a rV + (y+a) -v) x O<Rev<l
5.22 2 2-1 -1 -ab (b +x) cos (ax) l.;~b e cosh (by) y<a -1 -by
l.;~b e cosh (ab) y>a
5.23 -bx cos (ax) l.;b{(b2+ (a-y) 2) -1 +(b2+(a+y) 2)-~ e
5.24 _bx2
cos (ax) l.; ( 2 2 ) e l.;(~/b) exp -~(a +y )/b cosh(l.;ay/b)
f(x) gc(y)
(cos (~1TX) ) \1-1 -1
5.25 x<l 21 -\1r(\1) (r(~+~\1+Y/1T) r(~+~\1-Y/1T») 0 x>l
Re\1>O
-1 r n 2 2 -1 -na 5.26 (cosh a-cos x) ,X<1T -y sin(1TY)cosech a (-1) £n(n -y) e
0 ,X>1T n=·O
5.27 (cosh a-cos -\1 x) , X<1T
_\100 n 2 2-1 -y sin (1TY) (sinh a) !(-1) £n(\1)n(n -y )
n=O 0 ,X>1T
-n p (cotha) -\1
a) 5.28 (cos z-cos a) P_~+y(cos 0 ,x>a
5.29 (a2+x2)-1 ~1Ta-1(1_b2)-1(ea_b)-1(ea-aY+beay) y<l .(1-2b cos x+b2 )-1
Ibl<l
Ibl<l +(ea_b)-1(be-an-aA+bn+1eaA»)
y=n+A, O~A<l, n=1,2,3, •••
5.31 2 2-1
(a +x) (cos x-b) -1 a -1 ~1Ta (e -b) cos h(ay) y<l
2 -1 • (1-2b cos x+b )
I bl <1
z=\y /a
z=\y /a
5.34 -1 sin (ax2 ) ~1T(~_C2(z)_S2(z») 2 x 1 z=\y /a
5.35 -2
sin (ax2 ) ~1Ty(S(Z)-C(z»)+(1Ta)~sin(z+\1T) 2 x ; z =\y /a
5.36 _ax2 2
.sin(~arctan(b/a)-\by2(a2+b2)-1)
5.37 2 2
e-ax cos(bx ) ~ 2 2 -\ ( 2 2 2 -1) ~1T (a +b) exp -\ay (a +b )
'cos(~arctan(b/a)-\by2(a2+b2)-1)
5.38 xl:isin (ax2) (l:i~)l:iy/a)3/2(cOS(Z+~~)J 3(Z)-Sin(Z+~~)Jl(Z») -4" 4"
5.39 xl:icos (ax2) 3/2( 1 . 1 ~~(l:iy/a) cos(z+S~)Jl(z)+s~n(z+[~)J 3(z)
z=.!.y2/a 4" -4" 8
5.40 x -l:isin (ax2) -l:i~(l:iy/a)l:iSin(z-~~)J 3(z) , z=.!.y2/a 8 -4"
5.41 x-l:icos (ax2) l:i~(l:iy/a)l:iCOS(Z-~~)J l(z) 1 2 , z=sY /a -4"
5.42 v-I. 2 x s~n (ax) ~(2a)-l:iVr(V)i(eXp(~U-~i~V) (D_V(Ul:i)+D_V(-Ul:i»)
-2<Rev<2 D (vl:i) +D (_vl:i»), u=-v=l:iiy2/a -exp(-~u+~i~v) -v -v
5.43 v-I 2 ~(2a)-l:ivr(v) (exp(~u-~i~v) (D (ul:i) +0 (_ul:i) x cos (ax) -v -v 0<Rev<2
+exp(-l:iu+~i~v) (0 Vl:i)+D (_vl:i»)) u=-vl:iiy2/a -v -v
5.44 sin(a3x3) 6: (y/a)l:i(I 1(z)+I1 (Z)+J 1(z)-J1 (z) -3 3 -'3 3
-2J1 (z) (z)+2J 1 (z)+2iJ1 (iz)-2iJ 1 (iz») 3 -3 3 -3 3
z=2(3a/y) -"2
3 cos (a3x 3) ~(~)l:i3-l:i(3l:iK (u)+ J l(u»);
-"2 5.45 J 1 (u)+ u=2 (3a/y) 6a a . 1.
3 3 -3
5.48 x-l:isin(a/x) l:i(l:i~/y)l:i(sin(z)+cos(z)-e-Z)
5.49 x -3/2sin (a/x) l:i(l:i~/a)l:i(sin(z)+cos(z)+e-Z)
5.50 -1 cos (a/x) l:i~(2~-lK (z)-y (z») x o 0
5.51 x-l:icos(a/x) l:i(l:i~/y)l:i(cOs(z)-sin(z)-e-Z)
f (x) gc(y)
5.52 x- 3/ 2cos(a/x) ~ ( . -Z) ~(~~/a) cos(z)-sin(z)+e
5.53 v-1 sin (a/x) !..t~ (y/a) -~v sec (~~v) (Jv (z) +J_v (z) +Iv (z) -I_v (z) ) x
-1<Rev<2
5.54 v-1
cos (a/x) !..t~ (a/y) ~v cosec (~~v) (J_ v (z) -Jv (z) +I_v (z) -Iv (z)) x
-l<Rev<l a ~v -1 =~~(-) (cos(~~v){2~ K (z)-Y (z)} y v v
-sin(~~v)J (z») v
5.55 -1 x log(bx) sin (a/f) ~~(Jo(z)log b+~Jo(z)log(a/Y)-Ko(Z)}
5.56 -1 I log(b(a/y)~) (K (z)-~~Y (z») z=2 (ay) ~ x log(bx) cos (a/x) ; o 0
5.57 -1 ~ x sin (ax ) ~(s(!..ta2/y)+c(!..ta2/y)}
5.58 -~ ~ (2~/y~(C(z)sin z-S(z)cos z) z=!..ta2/y x sin (ax ) ;
5.59 x-!..tsin(ax~) 3/2 (. 1 1 -~~(~a/y) s~n(z-8~)J 1(z)+cOS(z-8~)J3(zl}
-4 4 1 2
x-3/4sin(ax~) ~(~a/y)~coS(z+~~)J1(Z) 1 2 5.60 ; z=ga /y
4
5.62 -1 -ax~ sin(ax~) ~~Erf(z) z=a(2Y)-~ x e ;
5.63 xV-1sin(ax~-~~v) ~ -v 2 -~
-ax~ - (~~) (2y) exp (-!..ta /y) 02v-1 (ay )
·e ; Rev>O
• {exp (!..tiz2_~hv) (02V-1 (u) -02v-1 (-u))
+exp(-!..tiz2+~i~V) 02v-1 (V)-02V_1(-V»)}
u=a(2y)-~(-i)~, v=a(2Y)-~(i)~
e
5.66 x-~cos(ax~) (n/y)~sin(~n+~a2/y)
5.67 x-~cos(ax~) -~n(~a/y)3/2(sin(z+jn)J 3(Z)+cos(z+jn)J 1 (z») -4" -4"
1 2 z=1ia /y
5.68 x-3/4cos(ax~) n(~ay)~cos(z-jn)J l(Z)l 1 2
z=sa /y -4"
5.69 x-~e-bxcos(ax~) n~(b2+y2)-~exp(_~a2b(b2+y2)-1) ( 2 2 2 -1) ·cos ~arctan(y/b)-~a y(b +y )
-~ -ax~ 2 5.70 x e (~n/y) ~e -~a /y
·cos (ax~) -sin(ax~)
• (exp (l:!iz2-~inv) (02v-1 (u)+02v_1 (-u»)
+exp(-l:!iz2+~inv) (02V_1(V)+02V_1(-V)~ u=a(2Y)-~(-i)~, v=a(2y)-~(i)~
5.72 2 2-1 x(b +x) tan (ax) ncosh(by) (l+e2ab)-1,
5.73 2 2-1
5.74 2 2-1 (a +x) sec (bx) -1
~na cosh (ay)sech(ab),
(Cauchy principal value), y<b
5.76 x -2/3sin (ax·l / 3) ~n(a/Y)~(I_l/3(Z)+Il/3(Z)-J_l/3(Z)+Jl/3(Z) 3/2 -~ z=2 (a/3) y +2iJ l / 3 (iz) -2iJ_l / 3 (iz) +2J;L/3 (z) -2J_ l / 3 (.z»)
26
£ (x)
5.78 xVsin(axc ) ,c<l
-c-l<Rev<c-l
5.81 u- 3/ 2sin(bu)
5.85 x-l..;r- 1cos(br)
gc(y)
-l..; ( l..; ) l..; (3y/a) 3 K1/ 3 (z)+1fJ1/ 3 (z)+1fJ_1/ 3 (z) ,z,same as before
-v-c-l I n ( ) -y (-1) sin l..;1f{v+(2n+l)c} r{1+v+(2n+l)c} n=O
'y-2nca 2n+l/(2n+l)!, lal<<<>, y>O -1 -(l+v)/c «> n
c a L(-l sin{l..;1f(1+v+2nVc}r{(1+v+2n)/c} n=O
'a-2n/ cy2n/(2n)!, a>O , I yl <<<>
-v-l 00
-2nc 2n/(2 )' .y an., lal<oo, y>O
-1 -(l+V)/cI n . c a (-1) cos{l..;1f(l+v+2n)/c)r{(l+v+2n)/c} n=O
'a-2n/ Cy2n/(2n)! a>O Iyl <00
cos (ay)Ko (as)
l..;1f (l..;1fb) l..;J _~ (l..;ay-l..;aS) (sin (ay) J~ (l..;ay+l..;i:lS)
-cos(ay)Y~(!..;ay+!..;as)
l..;(1fb)!..;I_~(l..;ay-!..;aS)K~ay+l..;as)
yb
y>h
y>b
yb
yb
y>b
Ko(aS) y>b
5.89 r-5/ 2sin(br) b(21Tb)~(13(~ay-~aS)K3(~ay+~as) '4 '4
-I 1 (~ay-~aS)K1 (~ay+~aS)) y>b -'4 '4
5.90 r -5/4cos (br) b(21Tb)~(1 3(~ay-~aS)K3(~ay+~as) -'4 '4
-11(~ay=~aS)K1(~ay+~as)) y>b
'4 '4
5.91 -1 -~ xr (r-a) sin(br) ~ -1 ~ (~1T) s (b+s) cos(as-~1T) yb
5.92 xr-1(r-a)-~cos(br) !:i -1 ~ -(~1T) s (b+s) sin(as-~1T) yb
5.93 (r+x)\}+(r-x)\}) ~1Ta\}({(b+Y)/(b-Y)}~\}+{(b-Y)/(b+Y)}~\}) -1 sin(br) .(COS(~1T\})J (as)-sin(~1T\})Y (as)) yb
5.94 (r+x)\}+(r-x)\}) -~1Ta\}({(b+Y)/(b-Y)}~\}+{(b-Y)/(b+Y)}~\}) -1 cos (br) '(sin(~1T\})J (as)+cos(~1T\})Y (as)) yb
5.95 2 2 -1 -1 (c +x) r sin(br) ~1Tc-1(a2_c2)-~e-cYsin{b(a2_c2)~} y>b
5.96 -2 sin (br) -1 -ay ay r ~a (e Ei(ay)-e Ei(-ay))
b Y {a(t2_y2) }dt -~1T f ya
0 x>a
x-~u-lcoS(bu)x<a (~1T)3/2Y~J_~(~aV-~ab)J_~(~av+~ab) 0 x>a
u- 3/ 2sin(bu)x<a (~1T)3/2b~Jk(~av-~aY)Jk(~av+~ay) 4 4
0 x>a
u -3/2cos (bu) x<a (~1T)3/2b~J_~(~av-~aY)J_~(~av+~ay) 0 x>a
-1 cos (bw) x<2a 1Tcos(ay)Jo(av) w
0 x>2a
w- 3/ 2cOS(bW) x<2a 1T(~1Tb)~COS(aY)J_~(~av-~aY)J_~(~av+~ay) 0 x>2a
w- 3/ 2sin(bw) x<2a 1T(~1Tb)~cOS(aY)Jk(~av-~aY)Jk(~av+~ay) 4 4
0 x>2a
sin (bw) x<2a 1Tab -1 cos(ay)v J 1 (av)
0 x>2a
-1 V cos (bV) x>a -~1TYO (as) y>b
0 x<a -(~1T)3/2b~J_~(~ay-~as)Y~(~ay+~as) y>b v- 3/ 2cos (bV) x>a
u=(a2_x2)~; w=(2ax-x-x2)~
f{x) gc{y)
5.109 0 x<a _(~~)3/2b~J~{~ay-~aS)Y~{~ay+~aS) y>b u-3/ 2sin{bU) x>a
5.110 0 x<a ~(~Y)~I_~{~ab-~aS)K~{~ay+~as) ya
5.111 0 x<a (~~)~{a2+k2)-~exp{-b{a2+k2)~)cosh{kY) ya
5.113 u -1 -bu e x<a 0 ya ~~Jo{as) y>b
5.114 x-~u-1sin{bu) x<a (~~)3/2Y~J_~{~aW-~ab)Y_~{~aw+~ab) -.x-~u-1e -bU x>a
00
5.115 exp{a cos x)x<~ d{a)+sin{~y) f exp{-a cosh t-yt)dt 0 x>~
Yo
5.116 (sin x)-~e-2a sin x, ~(~a)~cos{~~y) (I_~_~y) (a)I_~+~y{a) x<~ -I~_~y{a)I~+~y{a)
0 x>~
5.117 (sin -~ 2a sin x, ~(~a)~cos{~~y) (I_~_~{a)I_~_~{a) x) e x<~
0 x>~ +I%+~y{a)I~_~y{a»)
5.118 (cos -I:; -2a cos x, 1:;~{~a)I:;(I_~_~y{a}I_~+~y{a) x} e x<l:;~
0 x>l:;~ -I~+~y{a}I~_~y{a})
5.119 {cos -~ 2a cos x, x} e ~~(~a}I:;(I_~_~y{a)I_%+~(a) x<~~
0 x>~~ +I~+~y(a)I~_~y{a»)
2 2 l:i 2 2 l:i 22 l:i s={b -y ) :S={y -b ) :w={b +y l
f(x) gc(y)
5.120 (cos x)-l:>e-a sec X, ~l:>D l:>(Z)D l:>(z) y- -y- ; z=(2a)l:>
x<l:>~
0 x>l:>~
5.121 (cos x)-l:> l:> l:>~(~a) J~+l:> (a)J~ l:> (a) • y .- y
·sin(2a cos x) x<l:>~
0 x>l:>~
5.122 (cos x)-l:> l:> l:>~(~a) J ~+l:> (alJ ~ l:> (al -. y -.- y
cos(2a cos xl x<l:>~
0 x'>l:>~
5.123 (sin x)-l:> ~(a~)l:>cos(l:>~YlJ~+l:> (a)J~ l:> (al • y .- y ·sin(2a sin x) x<~
0 x>~
5.124 (sin x)-l:> ~(a~)l:>cos(l:>~y)J ~+l:> (alJ ~ l:> (al -. y -.- y
cos(2a sin x) x<~
0 x>~
5.125 sin(a sin xl x<~ l:>~cot (l:>~yl {J (a) -J (al }=-l:>~{E (al+E (al} y -y y-y 0 x>~
5.126 cos (a sin xl x<~ l:>~{J (a)+J (al }=l:>~cot(l:>~Yl{E (al-E y(a l } y -y y- 0 x>~
5.127 sin(a cos x) x<l:>~ (l:>~yl {J (al -J (al =-hsec (l:>~yl y -y 0 x<l:> • {E (al +E (al} y -y
5.128 costa cos x) x<l:>~ \~sec(l:>~y) {J (al+J (al }=\~cos ec(l:>~yl y -y 0 x>l:>~ ·{E (al -E (al} y -y
5.129 (sin xl-3/ 2 x<~ 2 (~al 3/2cos (l:>~yl (J -\-l:>y (a) J -\+l:>y (al ·sin (2a sin x)
+J 3/ 4_l:>y(a)J 3/ 4+l:>y(al) 0 x>~
(cos x)-3/2 3/2 5.130 x<~ 2(~a) {J ~ (alJ \+ (a) -.-y - y ·sin(2a cos (l:>x)
+J3/4+y(a)J3/4_y(a)} 0 x>~
f (x) gc(y)
5.131 log{sin( 1Tx) } x<l -1 sin Y(Y+IOg2+~~(1+~Y/1T)+~~(1-~Y/1T) ] -y
0 x>l
5.132 log{cos(~1Tx)} x<l -1 sin Y(Y+IOg2+~~(1+Y/1T)+~~(1-Y/1T) ] -y
0 x>l
5.133 {sin(1Tx)}V-l x<l 21-Vr(V)cos(~y) (r(~+~v+~Y/1T)r(~+~v-~Y/1T)]~ 'log{sin(1Tx)}
'(~(V)-~~(~+~v+~Y/1T)-~~(~+~v-~Y/1T)-lOg 2] 0 x>l
Re v>O
5.134 {cos (~1TX) }v-l x<l I-v ( tl 2 r(v) r(~+~V+Y/1T)r(~+~V-Y/1T)
'log{cos(~'ITx)} '(~(V)-lOg 2-~~(~+~V+Y/1T)-~~(~+ v-Y/'IT)]
0 x<l
Re v>O
5.135 (cos x -cos a)-~ 2-~'IT(P_~+y(COS a) {log(sin a)-y-log 4
log (cos x-cos a) , x<a -~(~-y)}-Q ~ (cos all 0 ,x>a - -y J
5.136 (a2+x2)-1 -l( -ay 'ITa {~u+~log(~r)}e
.log(r cosh u±r cos (bx) -u-ab + cosh(ay)log(l±e )
m _ n -1 -au ] + l (+1) n e sinh (ay-abn) ; n=l
m<~y/b<m+l; u>O
5.137 (a2+x2)-1 ~'ITa cosh (ay)log(l+e )+log(~r)e -l( -2ab -ay
.logl r cos (bx) I m n -1 ] ;m<~y/b<m+l + l (-1) n sinh (ay-2abn) n=l
5.138 x-2 (a2+x2 )-1 ~'ITa-3(ay+e-aY)lOg 2-cOSh(ay)log(1+e- 2ab )
m (-1)nn-1 {sinh(ay-2abn)+sinh(2abn) loglcos(bx) I -ab- l
n=l
1.6 Inverse Trigonometric Functions
0 x>l
0 x>l
6.3 (l_x2)-~ -v sin(~1Tv)s 1 (y) - ,v 'cos(varccos x) x<l
=~1Tsec(~1Tv){Jv(y)+J_v(Y) } 0 x>l
6.4 x-~(l-x2)-~ l:i ~1T(~1T) J~v_~(~Y)J_~v_~(~Y)
'cos(varccos x) x<l
'cos(varctan(x/a)
Rev>O
6.7 xV(l+x2)~v ~1T~r(l+V) (I_~_v(~Y)Sinh(~Y) 'sin(varccot x)
-I~+v(~y)coSh(~y) JY-v-~ -l<Rev<O
6.8 xV(l+x2)~v -1T-~r(l+V)y-V-~Sin(1Tv)cOSh(~Y)K~+v(~Y) cos (varccot x)
-l<Rev<O
6.9 arctan (a/x) -1( -ay- ay ] ~y e Ei(ay)-e Ei(-ay)
6.10 arctan (a/x2) 1Ty-le - yz sin(yz) , z=(~a)~
arctan{(a/x)n} 1 n
,sin(ay cos{(m-~)1T/n}]
6.12 arctan(ax-~),z=a2y ~1Ty-l cos z C(z)-S(z) -sin z l-C(z)-S(z)
6.13 -1 . x<l l:i1T(Ci(Y) + j t-1J (t)dt) x arCSl.n x
0 x>l Y 0
6.14 (a2_x2)-~ ~1T2 (Ho (ay) -Yo (ay) )
'10g(~{a+(a~-x2)~}1
(x2_a2) -~ x<a
2 -2 7.2 (sech (ax) J ~1Ta y cosech(~1Ty/a)
(sech (ax) ) 2n 2n-3 -2 n-l ( ) 7.3 2(2n-rrf COSeCh(l:;1Ty/a)m~l (l:;y/a)2+m2
n=2,3, •••
J 2n+l 2n-l -1 n ( 2 2) 7.4 (seCh(aX) 2 1Ta
(2n) ! sech(l:;1Ty/a) IT (l:;y/a) + (m-l:;) n=1,2,3 ••• m=l
7.S (seCh(aX»)V Rev>O
v-2 -1( )-1 2 a r(v) r(l:;v+il:;y/a)r(~v-il:;y/a)
7.6 (coseCh(aX) 1v v -1 -1 2 1Ta sin(l:;1Tv)r(l-v)cOSh(l:;1Ty/a)A O<Rev<l
A = (COSh (1Ty/a)
-cos (1TV) 1r (l-l:iv+l:;iy/a) r(l-l:;v-l:;iy/a) J
( -1 -1 7.7 cosh(ax)+cos b) 1Ta cosec b cosech(1Ty/a)sinh(by/a) b<1T
7.8 (coSh(aX)+cosh bJ-l -1 a cosech b cosech(1Ty/a)sin(by/a)
f (x) gC(y)
7.10 (cOSh(aX)+cos b)-~ 2-~TIa-lsech(TIy/a)p ~+' / (cos - l.y a b)
7.11 (coSh(aX)+cosh b)-~ -~ -1 2 TIa sech(TIy/a)p_~+iy/a(cosh b)
7.12 (COSh (ax-cos b)-~ -~ -1 ( b) ) 2 a Q ~+' / (cos b)+Q ~ . / (cos - l.y a - -l.Y a O<b<TI
(a+cosh x)~ -k -1 ( 7.13 ~i2 'TIY sech(TIy) P~+iy(b)-P~_iy(b)
-(b+cosh x)~la,b>-l -p~+iy(a)+p~_iy(a) )
7.14 -1 (~TI)~(l+coSh t)-l t=y (2TI/3) ~ (1+2cosh s) 1
s=x(2TI/3)~
(cosh a+cosh -v (~TI)~(r(v) )-l(Sinh a)~-vr(v+iy)r(v-iy) 7.15 x)
Rev>O ~-v a) • p ~+' (cosh - l.y
7.16 -v (~TI)~(r(v) )-l(Sin a)~-vr(v+iy)r(v-iy) (cos a+cosh x)
O<a<TI, Rev>O . ~ v .p ~+. (cos a) - l.y
7.17 (cosh -v (~TI) ~r (I-v) sinh ~-v -~+v a-cosh x) a) p ~+' (cosh a) - l.y
,x<a
7.18 0 x<a (2TI) -~r (~+v) (sinh a)ve iTIv
(cosh x-cosh a)v-~ ( -v -v • q ~ . (cosh a)+q ~+' (cosh - -l.Y - l.y a) ) ,x>a
-~<Rev<~
7.19 cosh (ax) a<TIb b-1 cos(~a/b)cosh(~/b) cosh (TIbx) , cos (a/b)+cosh(y/b)
1.7 Hyperbolic Functions 35
f(x) gC(y)
7.20 sinh(ax) ,a<1fb ~b-1 sin (a/b) s~nh 11fbx) cos la7b) +coSh{y7EI
7.21 x cosech(1fax) ~a-2(seCh(~y/a) )2
7.22 -1 x (cosech x_x-1 ) -log (l+e-1fY )
7.23 x-1 (1-sech x) 10g(r(i+~iy)r(! ~iy) J
-10g(r(~+~iy)r(~-~iY) J-10g(~y)
7.24 sinh (ax) -1 ( cosh(bx) a<b ~1fb sin (1fa/b) cos (1fa/b) +cosh (1fy/b)
+~b-1(~(i-~E+~i~)+~(~ ~E-~i~)
-~(4+~~ - ~i~)-~(~~~~i~) J 4 b b 4 b b
7.25 -1 sinh(ax) ,a~1fb ~10g( COSh(~~/b)+sin(~a/b)J x cosn {1fExl cosh(~Y7b)-sinl~a7b)
-1 . 2 ) 1 ( l+cosh(~Lb) J
7.26 s~nh lax ,a~~nb x sinh (1fbx) ~ og cos(2a7b)+cosh(y7b)
7.27 v-1 -1 2 -v r (v) ( I; (V, ~+~iy) +1; (v, ~-~iy) J x (cosech x-x ) -1<Rev<2
-sin(~1fv)r(v-1)y 1-v
7.28 cosh (~ax) -1 coshb+cosh (ax) ~1fa cos(by/a)sech(~b)sech(1fy/a)
7.29 cosh(~ax) -1 cos b+cosh(ax) ~1fa cosh(by/a)sec(~b)sech(1fy/a)
O<b<1f
7.30 cosh (ax)-cosh(bx) ~10 (cos (b/c) +cosh (y/c) ) x sinh (1fcx) g cos (a/c) +cosh (y/c)
a,b<1fc
7.31 2 2 (a +x )sech(~1fx/a) 2a 3 (sech (ay) )3
2 2 3 4( r 7.32 x(a +x )cosech(1fx/a) sa sech(~ay)
f (x) gc(Y)
7.33 2 -1 (l+x) sech (1TX) 2coSh(~y)-eYarctan(e-~Y)-e-Yarctan(e~y)
7.34 (1+x2)-lsech(~1Tx) ye-Y+cosh -2y y log(l+e )
7.35 -1 -ax sinh (bx) 101 (y2+(a+b)2) x e og 2 2 a~b
y + (a-b)
7.38 x (1+x2) -ltanh (~1TX) -ye-y-cosh y log (1_e-2y )
7.39 x (l+x2 ) -l tanh (!o1TX) cosh y log coth(~Y)-~1Te-y
-2sin y arctan(e-Y)
7.40 (l+e1Tbx)-lsinh(ax) _~a(a2+y2)-1+b-1 sin(a/b) cos (l/b) cosh (2y/b)-cos (2a/b)
a<b1T
a<b1T
7.42 -2 cosech(bu) ~1TC -1 e -cyv cosec (bv) ur
_loon 2 2 2 -1 -1 -yvn -1Tb L (-1) c (v -c) v e ·c =n1T/b
n=l n n n ' n
7.43 -2 sech (bu) -1 -cy -1 ~ n 2 2-1 r ~1TC e sec (bV)+1Tb (-1) c (v -c ) n=o n n
-1 -yvn ;cn=(n+~)1T/b ·V e n
-2 sinh~aUt -1 -cy -1 00
7.44 r ; a~b ~1TC e sin (av) cosec (bV)-1Tb L (-l)nc sinh bu n=l n
. 2 2 -1'-1 -yvn cn=n1T/b ·s~n (acn ) (v -cn) vn e ;
cOSh (au) ~wc-1e-cYcos(av)sec(bv)+wb-1 00
-2 L n 7.45 r cosh (bu) ; a~b (-1) c n n=o
.cos(ac ) (v2 _c 2 )-lv-1 -yv
e n;cn=(n+~)w/b n n n .
-2 sinh (ax) ~wc-1e-CYsin(ac)cosec(bc)- 00
7.46 r s~nh (bx) ; a~b b-1 L (_l)n
n=l
-2 cosh (ax) ~wc-1e-cYcos(ac)sec(bc)+Wb-1 00
7.47 r cosh (bx) ; a~b L (-1) n n=o
2 2-1 -ycn cn=(n+~)w/b .cos(acn ) (c -cn) e ;
_loon -1 -yvn 7.48 sinh (au) cosech(bu) -wb L (-1) c sin(ac)v e
a<b n=l n n n
cn=nw/b
00 -yv 7.49 cosh (au) sech(bu) -1 L (_l)n C cos(ac )v-1e n wb n n n
a~b n=o
cn=(n+~)w/b
7.50 u-1sinn(au)sech(bu) wb -1 00 -1 -yvn L (-l)n s in(acn )vn e
n=o a~b c =(n+~)w/b
n
1 cosh (au)cosech(bu) ~wb-1 00
-1 -yvn 7.51 L (-1) n EVe cos (acn ) u n n
a~b n=o
00
-2 sinh (au)cosech(bu) ~wa(bk)-le-yk+wb-1 L n -1 -7.52 u (-1) c sin(ac)v
a~b n=l n n n
cn=nw/b oe-yvn
n=o n n n c n= (n+1) w/b
n
f (x) gc(y)
7.55 sech2(ax~) as before
cos(~arctan(Y/b)-~a2Y(b2+y2)-1)
7.57 -1 cosh (bu) x<a ~TIJo(av) ya ~TIJo(av) y>b
7.58 sinh(bu) x<a -1 ~TIabv 11 (av) ya -1 ~TIabV J 1 (aV) y>b
7.59 u3/ 2sinh(bu) x<a ~TI(~TIb)~J~(~ay+~aV)J~(~ay-~av) 0 x>a
7.60 -3/2 u cosh (bu) x<a ~TI(~TIb)~J_~(~ay+~aV)J_~(~ay-~av)
0 x>a
7.61 -1 sinh (bu) x<a ~TIlo(av) ya 0 y>b
7.62 -~ -1 ~TI(~TIY)~I_~(~ab-~aV) x u sinh (bu) x<a
x-~u-1sin(bU) x>a (I_~(~ab-~aV)-2~TI-1K~!~ab+~aV) ] ya
7.64 0 x>2a TI(~TIb)~cOS(aY)J_~(~ay+~aV)J_~(~ay-~av) (2ax_x2)-3/4
• cosh{b(2ax-x2)~} x<2a
f (x) gC(y)
7.65 (2ax_x2)-3/4 TI(~TIb)~COS(aY)J~(~ay+~aZ)J~(~ay-~aZ) sinh{b(2ax-x2)~}
z= (i-b2 ) ~ x<2a
-1 sin (bv) v x>a 0 y>b
u=(a2_x2)~
v=(x2_b2)~
7.67 -ax( r 2-v-2b-lr(1+ ) ( r{~(a-vb+iy)/b} e sinh (bx) v r{l+~(a+vb+iY)lb} Rev>-l,b<Rev<a
+ r{~(a-vb-i~)/b} r{l+~(a+vb-~Y)lb}
2 ~ -1 2 2 7.68 e-bX cosh(ax) ~TI b exp{~(a -y )/b}cos(~ay/b)
7.69 -2 2 x sinh (x) ~ _x2/ 8 -3/2
(~TI) e -~TIyErfc(y2 )
7.72 sin(x2/TI)sech x 2 -~ ~TIsech(~TIy){cos(~TIY )-2 }
7.73 2 cos(x /TI)sech(x) ~TIsech(~TIy){sin(~TIy2)+2-~}
7.74 sin (ax2 ) Erdelyi, A. et. al. : Tables of Integral
7.75 2 secn(ox
cos (ax) Transforms Vol.l,p.10, New York, 1953
7.76 exp (-a cosh x) Kiy (a)
7.77 exp (-a sinh x) S . (a)=~TIi cosech(TIY) (J. (a)-J . (a) o,~y ~y -~y
+ J . (a)-J. (a)l -~y ~y
f(x) gc(Y)
7.78 exp(-a cosh x) cos h(Y arctan (b/a) JKiy(a2+b2)~J ·cos(b sinh x)
7.79 (cosh x)-~ ~ (a/~) K~+~iy(a)K~_~iy(a)
·exp (-2a cosh x)
7.80 (sinh x)-~ ~~(a~)~(J~_~iy(a)Y_~_~iy(a) ·exp(-2a sinh x)
-J_~_~iy(a)Y~_~iy(a)+J~+~iy(a)Y_~+~iy(a)
-J ~+~iy(a)Y~+~iy(a) J
7.81 (sinh x)-~ 2-~(r(~+iY)D ~ . (U)D ~ . (v) - -1Y - -l.Y ·exp(-a cosech x)
~r(~-iY)D_~+iy(U)D_~+iy(v) J; ~ (±2ia)~
7.82 -3/2 2(~a)3/2coS(~~Y) (I_~_~y(a)I_~+~(a) (sin x) ·sinh(2a sin x) -I3/4+~y(a)I3/4_~y(a) J
, x<~
0 , x>~
7.83 (cos x)-3/2 3/2 ( (~a) I~_~y(a)I_~+~y(a) ·sinh(2a cos x)
-I3/4_~(a)I3/4+~y(a) J ' x<~~ 0 x>~~
7.84 (sin x)-~ ~(a~)~coS(~~Y)I~_~(a)I~+~y(a) ·sinh(2a sin x) ,x<~
0 , x>~
7.85 (sin x)-~ ~(a~)~coS(~~Y)I_~+~y(a)I_~_~y(a) ·cosh(2a sin x),x<1r
0 , x>~
7.87 (cos x)-~ ~ ~TI(TIa) I ~+~ (a)I k ~ (a) - y -.- y
• cosh (2a TI
cos x) ,x<2'
0 TI , x>I
7.88 sinh(a sin x) X<TI -~TI{E (ia) +E (ia) } y -y 0 X>TI
7.89 cosh(a sin x) X<TI ~1T{J (ia) +J (ia)} y -y 0 X>TI
7.90 sinh(a cos x) x<~TI ~iTIsec(~TIy){E (ia)+E (ia)} y -y 0 x>~
7.91 cosh(a cos x) x<~TI ~TIsec(~TIy){J (ia)+J (ia)} y -y 0 x>~TI
7.92 sin(a cosh x ~TIsech (~TIY) {J. (a) +J . (a)} l.y -l.Y
7.93 cos (a cosh x) ~iTIcosech(~TIY){Jiy(a)-J_iy(a)}
7.94 costa sinh x) cosh(~TIy)K. (a) l.y
7.95 sin(a sinh x) ~TIsech (~TIY) (I. (a) +I . (a) l.y -l.Y
-2sech(~TIy){E. (ia)+E. (ia)}) l.y -l.Y
7.96 cos (a cosh x) cosh (~TIY) cos (~y log (~+a) ) K. {(b2_a2)~} -a l.y
• cos (b sinh x) ~iTICOSeCh(~TIY)Cos(~y 10g(:~~) J
, ab
7.97 sin(a cosh x) sin h(~TIY)Sin(~y 10g(~+a) JK. {(b2_a2)~},ab l.y -l.Y
7.98 (coSh x)-~ ~ -~TI(aTI) {J~+~. (a)Yk~' (a) l.y .- l.y
'sin(2a cosh x) +J~_~iy(a)Y~+~iy(a) }
f(x) gc(y)
·cos(2a cosh x) +J_~_~iy(a)Y_~+~iy(a)}
7.100 (sinh x)-~ ~ ~(a~) {I~_~iy(a)K~+~iy(a)
·sin(2a sinh x) +I~+~iy(a)K~_~iy(a)}
7.101 (sinh x)-~ ~ ~(a~) {I_~_~iy(a)K~_~iy(a) .cos(2a sinh x)
+I_~+~iy(a)K~+~iy(a)}
7.1·02 cosh x log (1_e-2x) 2 -~ ~ 2 2-2 -~~y(l+y) tanh(2Y)+(y -1) (y +1)
7.103 (cosh x+a)-~ ~2-~sech(~y) (P_~+iy(a){-Y+~10g(1-a2) ·log(cosh x+a),-l<a<l
-10g4-~~(~+iy)-~~(~-iy)}+~Q ~+' (a) - ~y
+~Q ~ . (a) 1 - -~y
7.104 (cosh x+z)-~ ~2-~Sech(~y) (P_~+iy(Z){-Y+~109(Z2_1) ·log(cosh x+z), x>l
-10g4-~~(~+iy)-~~(~-iy)}+~ ~+' (z) ) - ~y
+~q ~ . (z) J - -~y
7.105 -~ ~2-~(p_~+iy(Z){-Y-1094+~109(Z2_1) (z-cosh. x)
·log (z-coSh x) -~~(~+iY)-~~(~-iy)}-~q_~+iY(z)
,cosh x<z -~q ~ . (z) J 0 ,cosh x>z - -~y
7.106 0 ,cosh x<z 2-~({q_~+iy(Z)+q_~_iy(Z)}{-Y+~109(Z2_1) -~ -1094}-~(~+iy)q_~+iy(Z)-~(~-iy)q_~_iY(z) J (c.:lsh x-z)
·log (cosh x-z)
7.107 1 ( l+cosh(ax) og cos b+cosh(~x) J
2~y-1cosech(~Y/a)Sinh2(~bY/a) , b<~
7.108 1 (coSh(aX)+sin b) og cosh(axJ-sin b ~y-1sinh(bY/a)sech(~~/b) , b<~
f(x) gc (y)
7.109 log ( cosh(ax)+cos ~) ~y-lcosech(~y/a) (COSh (cy/a)-cosh(by/a) J cosh(ax}+cos
,b,c.:;;~
7.110 log (1+a 2sech2x) -1 2 2 ~ 2~y cosech(~~y)sin [~y log{a+(l+a ) }]
7.111 log (1-a2sech2x) ,a<l -2~y -1 cosech(~~y)sinh2(~y arcsin a)
7.112 log{tanh(ax)} -1 -~~y tanh (l,oy/a)
7.113 B~llog{(A+B)/(A-B)} ,-l<a<l -~ 2 2 2 ~ sech (~y)P ~+' (a)
x+l)~, - J.y
7.114 (cosh ax) -v
'log (cosh ax) 2v-2{ar(v) }-lr(~v+~iy/a)r(~v-~iy/a) Rev>O '{~(v)-log 2-~~(~v+~iy/a)-~~(~v-~iy/a)}
7.115 (1+x2)-lsech(l,o~x) 2-~(~e-y+2Sinh y arctan(2-~cosech y)
-cosh y log{(cosh y+2-~) (cosh y_2-~)-1})
7.116 2 -1 x(l+x) cosech(~x) ~ye-y-~+cosh -2y y log (l+e )
7.117 (l+x2)-lSinh(ax)
• cosech (~x) ~e-Y(y sin a-a cos a)+~sin a cosh y
a.:;;~ olog(1+2e-Ycos a+e-2y )
-cos a sinh y arctan{sin a (eY+cos a)-I}
7.118 2 -1 (l+x) cosh (ax) ye-Ycos a+ae-Ysin a+~cos a cosh y
sech (~~x) 'log{1+2cos(2a)e-2Y+e- 4y)}
a.:;;~~ -2y
+sin a sinh y t {e sin(2a)} arc an 2 l+e- y cos (2a)
2 -1 ~~e-Ysin a+sin a sinh y cos a 7.119 (l+x) sinh.(ax) arctan (-. -h--) sJ.n y • cosech (~~x)
log (cosh y+s~n a) a.:;;~~
-~cos a cosh Y cosh y-sJ.n a
7.120 log (l+cos a sech x),a~~ -1 cosech(~y){coSh(~~y)-cosh(ay)} ~y
f (x) gc(y)
7.121 log{coth(~x)}log(sinh x) rry-1tanh(~rrY){-Y-1og 2-\O/(~+~iy) -\o/(~-~iy)-\O/(l+~iy)-\O/(l-~iy)}
7.122 arctan{sinh a sech(bx)} ~rry -1
sin(ay/b)sech(~rry/b)
7.123 (1+z2)-~vcos(v arctan z) 2 (c/rr) -~{r (v) }-lcosh (~rry) (1_c2 ) \-~v Rev>O , z=c sin hx .r(v-iy)r(v+iY)P~~~, (l/c) - ly
1.8 Orthogonal Polynomials
8.1 P2n (x) x<l (-1)n(~rr/Y)~J2n+~(Y) 0 x>l
8.2 P2n+1 (x) x<l n+1 I I -2 -~ (-1) (2n+1).(n.) y s-~,2n+3/2(Y)
0 x>l
8.3 P (a-bx2 ) n -~
rr(2b) In+~(u)Yn+~(v)
a-1<bx 2<a+1 v = ~yb-~{(a+1)~±(a-1)~} 0 otherwise u
a~l
8.4 xV- 1p (x) x<l rr~2-vr (v) (r (1+~v+~n) r (~+~v-hn) ]-1 n
0 x>l 2 2F3(~v,~+~v;~,~+~v-~n,1+~v+~n;-\y ) Re(v+n) >0
8.5 (a2+x2)-n-~ n 2n{ -1 2n (-1) y (2n) !} y Ko (ay) 2 2-~ 'P2n{x(a +x) }
8.6 (sechx) 2n+1 (_1)n rr-122n-1{(2n) !}-2cos h (~rry) 'P 2n (tanh x)
.(r(~+n+~iy)r(~+n-~iY) ]2
8.7 u-n-~P2n(u-~a sinh x) n -1 -2n-1 -2 (-1) \1r (~a) {(2n)!} cos h (~rry
u=b 2+a2sinh2x .(r(~+n+~iy)r(~+n~iY) ]2
2F1(~+n+~iy,~+n-~iY;2n+1;1-b2/a2)
1.8 Orthogonal Polynomials 45
, x<a (-1)n~'lTJ2n(ay)
0 , x>a
2 2 -~ x ~'IT(~'lTY)~J~n_~(~aY)J_~n_~(~ay) 8.9 x-~(a -x) Tn(i),x<a
0 ,x>a
8.10 -1 cos (bU)T 2n (x/a) ,x<a u (-1)n~'lTT2n(y/z)J2n(az) ; z = (b2+y2)~
0 ,x>a
u = (a2_x2)~
-1 cos (bu) T2n (u/a) ,x<a n (b 2+/) ~ 8.11 u (-1) ~'lTT2n(b/z)J2n(az); z =
0 ,x>a
u = (a2_x2)~
-1 sin(bu)T2n+l (u/a) ,x<a n z=(b2+y2)~ 8.12 u (-1) ~'lTT2n+l (b/z)J2n+l (az);
0 ,x>a
u =(a2_x2)~
8.13 -1 cos (bu)T2n (u/a) , x<2a n 2 2 ~ u (-1) 'lTcos(ay)T2n(b/z)J2n(az) ;z=(b +y )
0 , x>2a
8.14 -1 u
u sin(bu)T2n+l (i)' x<2a n (-1) 'lTCOS (ay)T2n+l (b/z)J2n+l (az)
0 , x>2a (b 2+y) ~ z =
u=(2ax-x2)~
~'IT{(n_l)!}-lyn-le-ay
(b2+x2)~ • (Hen(Z)+Hen(Z) ] z = (2a) ~ (b±~y/a) u = Z
8.17 sin (bu) U2n (x/a) n -1 2 2 , x<a (-1) ~'lTabz U2n (y/z) J 2n+l (bz) ;z=(b +y )
0 , x>a
f (x) gc(y)
8.18 2 exp(-bx )He 2n (ax) ~~~b-n~~(-z?eXp(-~y2/b)He2n{~aY(bZ)-~}
z=~a2-b
2 m ~ 2m 2 2m 2 8.19 e-~x Hen (x)Hen+2m (x) (-1) n! (~~) y exp(-~y )Ln (y )
~+~n 2 ~ ~n 8.20 ~ D~ ~+ (2a)D~ ~ (2a) u exp (-2a u) n- y n- -y
·He {2a(1+u)~}x<~~ n
0 x>~~
_~x2 (x2 )
~ -1 2 2 8.21 e L (~~) (n!) exp(-~y ){He (y)} n . n
8.22 2 2 exp(-~x )Ln(~x ) ~ 2 2
(~~) exp(-~y )Ln(~y )He2n(~Y)
·He2n(~x)
8.23 -ax v-2n v-2n e x L2n_1 (ax) ~i(-l)n+lr(v) {(2n-l) !}-ly2n-l
Rev>2n-l ( -v -v] • (a-iy) -(a+iy)
8.24 -ax v-2n-l (_1)n~r(v){(2n)!}-ly2n{(a+iy)-v+(a_iy)-v} e x
v-2n-l .L2n (ax)
2 m ~ -1 2 8.25 x2me -~x L 2m (x2 )
n (-1) (~~) (n!) exp(-J.;;y)Hen (y)Hen+2m (y)
2 2 8.26 x2ne-J.;;x Ln-J.;;(J.;;X2 ) (J.;;~)J.;;e-J.;;y y2nLn+J.;;(J.;;y2
n n
2 2 8.27 e-~x {L-~(J.;;x2)}2 (J.;;~)J.;;e-J.;;y {L-~(J.;;y2)}2
n n
0 ,x>a • J v+2n (ay)
Rev>-~
f(x) gc(Y)
8.30 (a2_x2) vp (v ,v)(~) 2n a ,x<a ~(-1)n(2a/y)v+~~~{(2n) :}-lr(v+2n-l)
0 ,x>a ·Jv+2n+~(ay)
8.31 (l-X)V(l+X»)J (-1)n22n+V+)J{(2n)!}-lB(2n+V+l,2n+)J+l)
+(l+X)V(l-X»)J) .y2n(eiY1Fl(2n+V+l;4n+V+)J+2;-2iY)
p(V,)J) (x) ,x<l +e- iY1Fl (2n+V+l;4n+V+)J+2;;2iy) ) 2n
0 ,x>l
8.32 (l-X)V(l+X»)J (_1)n+122n+v+)J+l{(2n+l)!}-1
_ (l+x) v (l-x) )J) .B(2n+v+2,2n+)J+2)y2n+l
p(V,)J) (x) ;x<l • (ieiY1Fl(2n+V+2;4n+V+)J+4;-2iY) 2n+l
0 ;x>l -ie-iY1Fl (2n+V+2;4n+V+)J+4;2iy) )
1 9 Ga~a- and Related Functions .
9.1 Ir(a+ix)12 ~2-2ar(2a){sech(~y) }2a
2
r(t+ibx) b sech(~y/b)K{sech(l:ty/b)}
9.3 1 r (l:t+ibx) 14 2~-2b-lsech(l:tY/b)K{tanh(l:ty/b)}
9.4 1 3 -2 h (~x) r4""'"ix) 1 } -1 -~ ~ ~ (~~cosh y) log{(l+cosh y) +cosh y) ~}
9.5 {r(a+bx)r(a-bx)}-l 22a-3 {r (2a-l) } -lb -1 {cos (~y /b) }2a-2 ,y<~b a>~ 0 ,y>~b
9.6 {r(a+bx)r(c-bx)}-l b-l{r(a+c-l)}-1{2cos(~y/b)}a+c-2
+{r(a-bx)r(c+bx)}-l ·cos{~~y/b (c-a) } ,y<~b
a+c>l:l 0 ,y>~b
9.7 r(a+ibx)r(a-ibx) 3 -1 ~ b cosec(2~a)P2a_l{cosh(l:ly/b)}
·r (l:l-a+ibx)r(l:l-a-ibx)
O<a<l:l
r(b+iax)r(b-iax) 2~-b-cTI3/2a-1r(2C)r(2b)r(b+c)
·r(c+iax)r(c-iax) (t-b- C J • sinh(~y/a) P~:~:~(COSh(~y/a) a,b,c>O
f(a+icx)f(a-icx) TI~2b-a-~c-1e-iTI(a-b+~){r(b_a)}-1
• (f(b+iCX)f(b-iCX»)-l [ t-a-~ ( 1 sinh(~y/c) q:~~~i/2 Cosh(~y/c») b>a>O
. 'I' (l+x) -log x ~'I'(l+~Y/TI)-~log(~Y/TI)
-1 -1 2x +TI 'I'(~x/TI) -1 y +'I'(y)-log y
_TI-11og(~x/TI)
'I'(~+ix)+'I'(~-ix)-log 4 -1 TI{y ~cosech(~y)}
f (v+ix) r (v-ix) 2TIf(2v) {2cosh(~y)}-2v • {'I' (v+ix)+'I'(v-ix)} {'I'(2v)-log(2cosh ~y)}
Rev>O
·{'I'(l+~x/TI)+'I'(l-~x/TI)} 0 y>l
1;(~+ix) 2TI2 I 9
4 -ty (2TIn e -3n
5 2 -2Y 2 -2y
e ) exp (-n TIe ) n=o
(1+4x2) -11; (~+ix) ~TI(COSh(~y)+~e3(o,ie-2Y) J
1.10 The Error- and the Fresnel Integrals
Erfc(x)
f(x) gc(y)
-Erfc(bx)}
2 _~ \ 2 2 10.6 eX Erfc (x) -~~ e ~ Ei(-\y )
2 _~ -k 2_ 2 10.7 ie -x Erf (ix) ~~ e ~y Ei(\y )
10.8 -1 _x 2
ix e Erf (ix) -~~Erfc(~y)
10.9 a 2x2 e Erfc(ax+b)
-~ -1 k 2 a 2 2 2 2 -~~ a e 4Y / Ei(-b -\y /a )
10.10 Erfc (x~) 2-~(1+y2)-~((1+y2)~+1)-~
10.11 x-1Erf(x~) log y-~10g(1+(1+y2)~)-10g({1+(1+y2)~}~-2~)
10.12 X-~Erf (x~) (2~)-~ arctan((2y)~(y-a)-1) -10g((1+y2)-~{y+1+(2Y)~})
10.13 x-1(1-ErfC(X~) ) log y-~10g{1+(1+y2)~}-10g({1+(1+y2)~}~-2~)
10.14 eXErfc(x~) (2Y)-~{1+(2Y)~}-1
10.15 x-1Erfc(ax-~) -Ei(-a(2iY)~)-Ei(-a(-2iY)~)
10.16 -1 -a/x (a ~] x e Erfc (x) ~~(H (u)+H (v)-Y (u)-Y (V») o 000 u"'2 (±iay) ~ v
10.17 Erfc (a/x) _~ co 2n
-2a~ ~ (ay) {n! (2n+l) !} n=o
'(~(2n+2)+~~(n+1)-10g(ay) )
10.18 Erfc(a{b+(b2+x2)~}~) -~ 2 4 2 -~( 4 2 \ 2)-~ 2 a exp (-ba ) (a +y) (a +y ) +a
4 2 ~ 'exp{-b(a +y ) }
Erfc{a(l+sec x)~} 2
D 1(2~a)D . 1(2~a) 10.19 -a e y- -y-
x<~~
x>~~
fix) gc (y)
10.20 Erf{b sech(ax)} 1T >'a -lb SeCh(>'1Ty/a)2F2(>'+i~,>'-i~1i,11-b2)
10.21 Erfc(a cosh x) -1 2 2 >,a W >, >," (a )exp(->,a ) - , ~y
10.22 2 exp{>,a cosh(2x)} >, 2 sech(>'1Ty )K.,iy(>,a )
'Erfc (a cosh x)
10.23 exp{-.,a2cosh(2x)} \1Ti sech (>,1TY) (IiY (.,a2) +I_ iy (.,a2) J 'Erf (ia cosh x)
2 1T>'sech(1Ty)D .,+" (a2>')D>, " (a2") 10.24 (sech x)"ea sech x
- ~y - -~y
->,log(a>'+y") J
10.26 .,-S (ax) >, -1 2 2 -., 2 2.,., .,(.,a) y (a -y) {a-(a -y ) } y<a
., -1 2 2 -., 2 2 >, >, .,(>,a) y (y -a) {y-(y -a ) } y>a
10.27 >,-C(x) >, -1 2 2 ->, 2 2.,., -.,(>,a) y (a -y) {a-(a -y ) } y<a
>,(.,a)>'y-l(y2_a 2)->'{y+(y2_a 2).,}>, y>a
10.28 x->'c(ax) >'(>'1T/y)" y<a
0 y>a
10.29 >,-s(ax2) -1( 2 2 2" 2 1 y C(\y /a)cos(\y /a)+S(\y /a)s~n(\y fa)
10.30 .,-c(ax2) -1( 2 "2 2 2 J y C(\y /a)s~n(\y /a)-S(\y /a)cos(\y fa)
10.31 x-3/ 2{sin(bx)C(bx) 0 y>b
-cos(bx)S(bx)}
10.32 .,-{C(ax)}2_{s(ax)}2 -1 2 Y sin (\y fa)
10.33 l-C(ax)-S(axl 0 y<a .,a"y -1 (y-a)-" y>a
10.34 -1 2 x S (ax ) \{si(\y2/al-Ci(\y2/a)}
1.10 The Error- and the Fresnel Integrals 51
f (x) gc(y)
10.35 -1 2 x c(ax) _~{Ci(~y2/a)+si(~y2/a)}
cos(ax2 )S(ax2 ) -l>:; ( 2 2 10.36 l>:;(2TIa) cos(~y /a)Ci(~y fa)
-sin(ax2 )C(ax2 ) +sin(~y2/a){TI Si(~y2/a)}]
sin(ax2 )s(ax2 ) -l>:;( 2 2 10.37 ~(2TIa) cos(~y /a){TI+si(~y /a)}
+cos(ax2 )c(ax2 ) -Sin(~y2/a)Ci(~y2/a)]
10.38 {~-S(ax2)}cos(ax2) -l>:;(2TIa)-~(sin(~y2a)Si(~y2/a) -{~-C (ax2 )}sin (ax2 ) +cos(~y2/a)Ci(~y2/a) )
10.39 {~-c(ax2)}cos(ax2) ~(2TIa)-l>:;(sin(~y2/a)Ci(~y2/a) +{l>:;-s(ax2 )}sin(ax2 ) -cos(~y2/a)si(~y2/a) ]
10.40 x- 1 {c(ax2 )cos(ax2 ) ~TI(~-s(~y2/a) 1 +s(ax2 )sin(ax2 )}
10.41 x-1{c(ax2 )sin(ax2 ) l>:;TI (l>:;-C (~i fa) )
-s(ax2 )cos(ax2 )}
-1 z+e -z}; z=2 (ay) l>:; 10.42 S(a/x) ~y {sin z-cos
10.43 C (a/x) ~y-1{sin z+cos z-e -z}; z=2(ay)~
10.44 l>:;-S (ax~) ~ -3/2 -~TI y a COS(Z-~TI)J~(Z) ; z=~2/y
~-c(ax~) l>:; -3/2 5 z=.!.a2/y 10.45 -~TI Y a cos(z-BTI)J_~(z)
8
10.46 x-1 {sin UC(U) ~TI{Jo(Z)+Io(Z)+Ho(Z)-Lo(Z)} ; z=2ay~ -cos US(u)};u=a2/x
10.47 x-1 {sin uS(u) ~TI(H (z)+L (z)-J (z)-I (Z») ; z=2(ay)~ +cos UC(u)};u=a2/x
o 0 0 0
-cos(bu)S(bu) }
f (x) gc(y)
10.49 (z/a)-~{sin xS (z) -~iTr('TJa)~cOSeCh(~TrY) (I k~' (~a)Ik~' (~a) -.- ~y .- ~y
+cos zC(z)} -I~+~iy(~a)I_~+~iy(~a) ] ;z=a sinh x
10.50 (z/a)-~{sin zS(z) ~iTr(Tra)~cOSeCh(~TrY) (J~+~iy(~a)J_~+~iy(~a) +cos zC (z) }
-J~_~iy(~a)J_~_~iy(~a) J ; z=a cosh x
10.51 (z/a)-~ sin zC(z) 1 ~ ( 8Tr (Tra) sech (~TrY) Jk+~' (~a)J ~+k' (~a) 4 ~y - 2~y
-cos zS (z) +Jk~' (~a)J ~~. (~a)] ;z=a cosh x .- ~y - - ~y
10.52 cos zc(z)+sin zS (z) ~Tr~SeCh(~TrY) {H~l) (a)+H(~) (a) }-Tra sech(TrY) ~Y -~y
z=a cosh x .(r(~-~iy)r(~+~iY) ]-ls_~,iy(a)Jr
10.53 sin zC(z)-cos zS(z) ~Tr~SeCh(~TrY) {H~l) (a)+H(~) (a) }+~Tra sech(7fY) ~y -~y
z=a cosh x .(r(t -~iy)r(t+~iY) J-ls~,iy(a)~
10.54 cos z{~-S(z)} ~Tr-lr(! ~iy)r(t+~iY)S~,iy(a) -sin z{~-C (z) }
z=a cosh x
10.55 cos z{~-C(z)} ~Tr -lr~ ~iy) r (t+~iY) S ~ . (a) - ,~y
+sin z{~-S(z)}
z=a cosh x
10.56 sin zC(z)-cos zS(z) ,.,-',e", "'y) (,'n "'+"-"'r' J"y"" z=a cosh2x +sin (~7f+~a+~i Try) J ~. (~a)
- ~Y
10.57 cos zc(z)+sin zS(z) ,.,-',ech"·y) (CO""+"-"'r)J"y"') z=a cos +cos(~Tr+~a+~iTrY)J~. (~a) - ~y
10.58 cos uC (v) +sin uS (v) ~Trsech (TrY) (COSh (~Try) {J. (a) +J . (a)} ~y l-~y
u=a cosh .x +i sin h(~Try) {J. (a)-J . (a)} v=2a cosh2(~x)
~Y -~y ~
10.59 sin uC(v)-cos uS(v) ~TrSeCh(TrY)(COSh(~TrY){J. (a)+J. (a)} ~y -~y
u=a cosh x -i sinh(~Try){J. (a)-J . (a)}]
v=2a COSh2(~x) ~y -~y
1.11 The Exponential and Related Integrals 53
1 11 The Exponential and Related Integrals
11.1 Ei (-ax) -1 arctan (y/a) -y
11. 2 Ei(-bx) x<a y-l(sin(aY)Ei(-ab)-arctan(Y/b)
0 x>a -~iEi(-ab-iay)+~iEi(-ab+iay) }
11.3 eaxEi(-ax) 2 2 -l[ ) (a +y) a log(y/a)-~rry
e-axEi(-bx),a~-b 2 2 -l( 2 2 11. 4 -(a +y) ~a log{(a+b) +y }
-a log b+y arctan{y/(a+b)}}
11.5 e -axEi (ax) _(a2+y2)-1{a log(y/a)+~rry}
11.6 e-a~-r(bx) a>b _(a2+y2)-1[~a 10g{(a-b)2+y 2} , =
-a log b+y arctan{y/a-b)})
11. 7 -1[ -ax=- rrarctan(a/y x e El-(ax)
-eaxEi(_ax) J
11.8 Ei (-ax) Ei (-ax) -1 2 2 ~rry 10g(1+y /a )
11.9 Ei (_ax2 ) -1 -~ -rry Erf(~ya )
2 -~rr(rr/a)~exp(\y2/a)Erfc(~ya-~) 11.10 e ax Ei (_ax2 )
11.11 2 - 2 exp(-ax )Ei(ax ) k 2 -~ ~irr(rr/a) 2exp (_\y /a)Erf(~iya
exp (a cosh x) 2 2 ( 11.12 ~rr cosech (rry) I. (a)+I . (a) l-y -l-Y
Ei(-a cosh x) _e~rrYJ. (ia)-e-~rrYJ . (ia) J
l-y -l-Y
exp (-a cosh x) 2 2 ( 11.13 >,71 cosech (rry) cosh (rrY){I. (a)+I. (a)}
-Ei(a cosh x) l-y -l-Y
_e~rrYJ. (-ia)-e-~rrYJ . (-ia) r ~y -l-Y
2 ->,rr(rr/a)~exp(\y2/a)Erfc(b~+~ya->') 11.14 e ax Ei{-(b+ax2 )}
11.15 eaxEi(-u)+e-axEi(_v) _2a(a2+y2)-lK {b(a2+y2)"} 0
u 22., I v=a{ (b +x ) ±x}
f(x) gc(Y)
eauEii-a(x+u) } 2 2 -~ 2 2 ~ 11.16 4ab(a +Y) S 1 {b(a +y ) } 0,
+e-auEi{-a(x-u)}
u=(x2_b2)~
11.17 si(ax) _~y-llog{(y+a)/(y-a)} Yfa
-bx l 2 2 11.18 e si (ax) _~(b2+y2)-1 ~y 10g (b2+(y+a)2] b + (y-a)
+rrb-b arctan {2ab (b 2+y2_a2 ) -10
11.19 Si (bx) x<a ~y-l(2 sin(ay)Si(ab)+Ci(ay+ab)-Ci(lay-abl)
0 x>a +logl (y-b)/ (y+b) I] ,Yfb
~b-l{2sin(ab)Si(ab)-Ci(2ab)-Y-log(2ab)},y=b
0 y>a
2 2-1 ( -by =--11.21 x(b +x) Si (ax) \rre {El.(by)-Ei(-ab)}
-ebY{Ei(-bY)-Ei(-ab)}] y<a
\rr(e-bY{Ei(-ab)-EI'(ab) }] y>a
-~rry -1 y>a
11.23 Ci (bx) x<a ~y-l(2sin(aY)Ci(ab)-Si(ay+ab)-Si(ay-ab) ) 0 x>a
11.24 (b 2+x2) -lCi (ax) ~rrb-lcosh(bY)Ei(-ab) y~a
\rrb- 1 (e-bY{EI' (ab) +Ei (-ab)-EI'(by) }
+ebYEi(-bY») y~a
e-axCi(bx) 2 2 -1( 2 2 2 2 2 2 11.25 -\(a +y) a 10g{(b +a -Y ) +4a y )}
222 ] -4alog b+2y arctan{2ay/(a +b -y )}
11.26 sin(ax)Ci(ax) 2 2-1 a(y -a) log (y/a)
-cos (ax)si(ax)
f(x) gc(y)
+cos(ax)Ci(ax)
11.28 x-l{cos (ax) Si (ax) -\1Ilogl (y+a)/(y-a) I -sin(ax)Ci(ax)}
11.29 {si(ax)}2 -1 ~1IY log (l+y/a) y<2a
~1Iy-llogl (y+a)/(y-a) I y>2a
11. 30 {Ci(ax)}2 -1 2 2 ~1IY log(y /a -1) y>2a
~11 -llog (l+y/a) y<2a
11. 31 si(ax2 ) 1Iy-l{S(\y2/a )_c(\y2/a )}
11. 32 Ci (ax2 ) -1 2 2 -1IY {C(\y /a)+S(\y fa)}
11.33 cos(ax2 )si(ax2 ) 11 (~1I/a) ~ (sin z{S(z)-~}+cos z{C(z)-~}) -sin(ax2 )Ci(ax2 ) z=\i fa)
11.34 cos (ax2 )Ci (ax2 ) 11 (~1I/a) ~ (cos z{S(z)-~}-sin Z{C(z)-~}) +sin(ax2 )si(ax2 ) 2
z=\y /a
-1 co
t-1J c=2(ay)~ 11. 35 x si (a/x) -11[ o(t)dt c
11.36 x-l{sin (a/x)Ci (a/x) 'JrKo{2(ay)~} -cos(a/x)si(a/x)}
11. 37 sin zCi(z) ~1Isp.ch (~1Iy) S . (a) o,~y
-cos zsi(z)
+sin zsi(z)
1.12 Legendre Functions
f (x) gc(y)
12.1 Pv (x) x<l -~~~V(V+l){r(~~V)r(l-~V)}-l
0 x>l -~ Y s_~,~+v (y)
12.2 lJ-l x Pv (x) x<l ~~2-lJr(lJ){r(1+~lJ+~V)r(~+~lJ-~V)}-1 0 x>l 2
RelJ> 0 '2F3(~lJ,~+~lJ1~+~lJ-~V,~V+~lJ'~1-~y
12.3 (1_x2) -~lJplJ (x) v x<l ~~2lJ-l{r(~ ~lJ+~V)r(l-~lJ-~V)}-l
0 x>l • (lJ+v) (lJ-V-l)y-~+lJS_lJ_~'V+~(Y) RelJ < 1
12.4 xA-l(1_x2)-~lJplJ(x) ~~2lJ-A(r(1+~A-~lJ+~V)r(~+~A_~lJ_~V»)-1 v ,x<l .r(A)2F3(~A,~+~A1~+~A-~lJ-~V,~,1+~A-~lJ
0 x>l 2 +~V1-~y )
ReA>O, RelJ< 1
12.5 0 x<l (~~/Y)~{Sin(~~V)YV+~(Y) Pv (x) x>l
-cos(~~V)JV+~(Y)} -l<Rev<O
12.6 (x2_1)~lJplJ(x) 2lJ+l~~{r(_~V_~lJ)r(~+~v_~lJ)}-ly-~-lJ v
• SlJ_~, V+~ (y) Re(lJ+v»01 Re (lJ-v)<1
12.7 0 x<l -(~~)~ylJ-~(COS(~~lJ-~~V)JV+~(Y) (x2_1) -~lJplJ (x) x>l +sin(~~lJ-~~V)YV+~(Y) ] v -~<RelJ<l,RelJ>Rev>-l
-RelJ
12.8 {X(X+l)}-~lJp~(1+2X) -~~~ylJ-~(COS(~Y+~~lJ-~~V)JV+~(~Y) RelJ<l,-l-RelJ<Rev<RelJ +sin(~Y+~~lJ-~~V)YV+~(~Y) J
12.9 Q2n+l(x/a) x<a n+l 3/2 ~ (-1) (~~) (a/y) J 2n+3/ 2 (ay)
q2n+l (x/a) x>a
n=O, 1,2, ...
f(x) gc(y)
2 -1 ~ -~ 2 12.10 PV(l+x ) -, 2 sin (1TV ){KV+~ (y2 )}
-l<Rev<O
Rev>-l
-1 2 2 -1 12.12 x qv(1+2a fx ) ~1Tr(l+v) (ay) W ~ (ay) cosec (1TV)
-v- ,0 ReV>-l
• (cos (1TV)Mv+~,o (ay) -W_v_~,o (ay) }
12.13 P v {(x2+a2+b 2 )f(2ab) } -1 ~ -21T (ab) sin(1TV)Kv+~(aY)Kv+~(by)
-l<Rev<O
12.14 Qv{(x2+a 2+b 2 )f(2ab) } ~ 1T(ab) Iv+~(bY)Kv+~(ay) a>b
Rev>-l
12.15 2 2
P v (2x fa -1) x<a ~1TaJ~+v(~aY)J_~_v(~ay)
0 x>a
-l<Rev<O
Pv(x ) x<l _1T-12-~Sin(1TV)Kv+~{Y(~i)~}Kv+~{Y(-~i)~}
2 pv(x ) x>l
-l<Rev<l
12.18 2 2
Pv(~x fa -1) x<2a -~1Ta sin (1TV) ({Jv+~ (ay) }2+{yv+~ (ay) }2}
2 2 Pv (~x fa -1) x>2a
-l<Rev<O
12.19 Qv (~x2 fa 2-1) x<2a 2 -~1T aJv+~(aY)Y_v_~(ay)
2 2 Qv (~x fa -1) x>2a
Rev>-l
f(x) gc (y)
12.20 222 PV{(a +b -x )/(2ab)} ~n(ab)~(JV+~(bY)YV+~(ay)
x<a-b
0 x>a-b -JV+~(aY)YV+~(by) ) a>b
12.21 0 x<a+b ~rr(ab)~(YV+~(aY)Y_V_~bY) 222 -JV+~(aY)J_V_~(by) ) Pv{(x -a -b )/(2ab)}
x>a+b
-1<Rev<O
12.22 -1 2 rr sin ( rrv ) n(ab)~Jv+~(bY)J_v_~(ay) a>b 222 qv{(a +b -x )/(2ab)}
x<a-b 2 2 2 Pv{(x -a -b )/(2ab)}
a-b<x<a+b
0 x>a+b
12.23 2n+l Z Q2n+l(zx) ,n=O,I, •• ~n(-I)n+l{(2n+l)!}-ly2n+lKo(ay)
2 2-~ Z= (a +x) , Re (v±ll) <0
12.24 z-v{p~(zx)+P~(-zx)} 2sin(~nll-~nv) {r(-v-Il) }-ly - V-lK (ay) 11
z=(a2+x2)-~,Re(v±Il)<0
12.25 z-V{pll(zx)_p~(-zx)} -n~21l{r(-~v-~Il)}-ly-V-l(2Vr(~v+~Il+~)
2 2-~ ·{I (ay)+I (ay)+2n-1sin(rrv)K (ay)} z=(a +x) ,Re(v±Il)<O 11 -11 11
-4ie {r (~+~v-~Il)} s (iay) -~inv -1 J V,1l
12.26 Pv(-cos x) x<rr 2 n {r(~-~v-~y)r(~-~v+~y)r(I+~v+~y)
0 x>n .r (l+~_~y}-1
12.27 Pv(COSh x) -~rr-2sin(rrv)r(-~v+~iy)r(-~v-~iy) -1<Rev<O .r(~+~v+~iy)r(~+~v-~iy)}-1
12.28 pv(a cosh x) v-3/2 -~ -1 -2 (na) sin (nv){cosh (ny) -cos (nv) }
-1<Rev<O,a~1 • r (~+~+~iy) r (~+~v-~iy)
( -v-~ -v-~ ] -2 ~ • P_~+iy(z)+P_~+iy(-z) ;z=(I-a )
1.12 Legendre Functions 59
f (x) gc(y)
12.29 qv(a cosh x) 2V-1(~TI/a)~r(~+~V+~iy)r(~+~V-~iy) a;;:'l,Rev>-l -~-v
'P_~+iy(z) : z=(1_a-2)~
12.30 (sinh x)~P~(cOSh x) ~TI-12-~r(~+~v-~v+~iy)r(~+~v-~~-~iY) Re (l+v-~) O,Re(v+~)<O ·r(-~v-~~+~iy)r(-~v-~~-~iy)
.(r(-V-~)r(l+v-V)r(~-~) )-1
12.31 -v-1 ~ sinh .x) qv(cotanh x) ~eiTI~2v{r(1+v)r(1+v_~)}-1
-l<Re(v+~)<O ·r(~+~v-~~+~iy)r(~+~v-~~-~iy)
·r(~+~v+~~+~iy)r(~+~v+~~-~iy)
12.32 (a2cosh2 x-1~v ~TI-~a~-V-12-~r(~v-~~+~+~iy)r(-~v-~~-~iY)
p~(a cosh x) ·r(~+~v-~~-~iy)r(-~v-~~+~iy) v
r r1 Re(v+~)<O:
'2F1 ~+~v-~~+~iy,~+~v-~~~~iY:~-~:1-a-2) a>l
12.33 P (1-2a2cos2x) x<~TI ~TIP~Y(z)P-~Y(z) : z=(1-a2)~ v v v 0 x>~TI
a;;:'l
12.34 2 2 x<~TI ~TIP~Y(z)p-~Y(z) z=(l+a2)~ pv(1+2a cos x) v v :
0 x>~TI
12.35 Pv(1+2 a 2 sinh2 x) ~sech(~TIY) (p-~iY(S) {Q~iY(S)+Q~iY1(S)} v v -v-
-l<Rev<O,a<l +p~iy(s) {Q-~iY(S)+Q-~iY(S)}J v v . -v-1
s=(1-a2)~
-~-v ~+v ~+v ) -P_~+~iy(S){Q_~+~iy(S)+Q_~_~iY(s) }
12.37 qv(1+2a 2 sinh2 x) -~iTIcosech(~TIY)
a<l : Rev>-l .(p~iY(S)Q-~iY(S)_p-~iY(S)Q~iY(S) ) v v v v
s=(1-a2)~
f(x) gc(Y)
12.38 qV(I+2a2Sinh2 x) -"pTa -1 cos ec (1TV)
a>1 ; -1<Rev<O ( -~-v ~+v ~+v J • P_~+~iy(S){Q_~+~iy(S)+Q_~_~iY(S)}
s=(I-a-2)~
-1<Rev<O ( -~-v 2
s=(l+a-2)~ • r(I+V+~iy)r(I+V-~iy){p_~+~iY(S)}
-r(-V+~iy)r(-V-~iY){P~~~~iy(S)}2J
12.40 q (1+2a2cosh2 v
x) -1 -~-v 2 !..o1Ta r(l+v+~iy)r(I+V-~iy){p_~+~iY(S)}
Rev>-1 s=(l+a-2)~
12.41 PV(2a2COSh 2 x-I) -~i sin(1Tv)cosech(1Tv)
-1<Rev<O , a<1 (p-~iY(S){Q~iY(S)+Q~iY (s) v v -v-l
_p~iY(S){Q~~iY(S)+Q:~~i(s)}] ,s=(I-a2 )
12.42 pv(2a2COSh2 x-I) !..oa-lsech(~'1TY)
( -~-v v+~ v+~ -1<Rev<O, a>1 • p ~+~. (5) {Q ~+~. (s)+Q ~~. (s)} - ~Y - ~Y - - ~Y
s=(I-a-2)~ v+~ -v-~ -v-~ J +p ~+~. (5) {Q ~+~. (s)+Q ~~. (s)} - ~Y - ~Y - - ~Y
12.43 q (2a2cosh2 x-I) !..o1Ta-lr(l+v+~iy)r(l+v-~iy) v
Rev>-I,a>1 -v-~ 2 {P_~+~iy(S)} ; s=(I-a-2)~
12.44 (2 2. 2 1 -1 v+~ -v-~ -2 ~ Pv a s~nh x-I) ,sinh x <-a ~1Ta P ~+~. (s)p ~~. (5) ;s=(I+a ) - ~Y - - ~Y
0 ,sinh x>..!.. a
12.45 0 ,sinh x<! -1( -v-~ v+~ a (21Ta) q-~+~iy (5) +q-~+~iy (5)
( 2. h 2 x-I) ,sinh x>! -v-~ v+~ J s=(I+a-2)~ Pv 2a s~n +q ~~. (s)q ~~. (s) a - - ~y - - ~y
-1<Rev<O
12.46 (sech 2n+2 x) Q2n+l (tanh .x (_I)n+122n-l{(2n+l)!}-2COSh(~1TY)
(r(n+l+~iy)r(n+l-~iY)J2
f (x) gc(y)
12.47 (sinh x)-~q~(cosh x) e i 1T~1T ~2 ~-2 r (~_~) 'J
Re ('J+~+1) >0, Re~<~ ·r(~+~'J+~~+~iy)r(~+~'J+~~-~iy)
.(r(l+~'J-~~+~iy)r(l+~'J-~~-~iY) J-1
12.48 2 2
(a cosh x-1) ~~ ~1T~e-i1T~2-~a~-'J-1{r(~'J) }-1
.q~~ (a cosh .x) a~l ·r(~+~'J-~~+~iy)r(~+~'J-~~-~iy)
Re (~-'J-1) <0 ·2F1(~+~'J-~~+~iy,~+~'J-~~-~iy;3/2+'J;a -2
12.49 -~-1 -~ . x) ~(_1)n+122na-2n-2{(2n+1) :}-2cos h(~lTY) z Q2n+1(az s1nh
n=0,1,2, •.. .(r(n+1+~iy)r(n+1-~iY) )2
z=b2+a2sin x .2F1 (1+~+~iy,1+~-~iy;2n+2;1-b2/a2)
12.50 log (sinh x)q'J (cosh x) ~1Tr(~+~'J+~iy)r(~+~'J-~iy)
Re'J>-l • (r(l+~'J+~iy)r(l+~'J-~iY) )-1(_r_10g 4
+~('J+1)-~~(~+~'J+~iY)-~~(~+~'J-~iy)
-~~(l+~'J+~iy)-~~(l+~'J-~iy) )
12.51 (b 2+a2sinb2 x) ~'J ~1T-12-'Jsin(~1T~-~1T'J)a'J-~{r(-'J)r(-'J-~) }-1
• (p~ (z) +P~ (-zl ) ·cosh(~1Ty)r(~~-~'J+~iy)r(~~-~'J-~iy)
·r(-~~-~'J+~iy)r(-~~-~'J-~iy)
z=a sinh x(b2+a2sinh2 X)-~2F1(~~-~'J+~iY'~~-~'J-~iy;-v;i-b2/a2) Re (~+'J<O
12.52 z-~'J-~P~(bz-~cosh 'J
x) 2'J-1a-~r(~+~'J-~~+~iy)r(~+~'J_~~_~iy)
Re ('J-~+l) <0 { -1 ~-x-1 • r(l-~)r(l+'J-~)} y 2F1(~+~'J-~~+~iy,
2 2 2 ,z=a +b cosh. x ~+~'J-~~-~iY;1-~;-a2/b2)
12.53 P~~+x(COS a) a<1T (~1T)~{r(l-~)}-l(sin a)~(cos y-cos a)-~-
y<a Re~<~ 0 y>a
12.54 COS(1TX)q_~+x(z)q_~_x(z) 2 2 -~ ( 2 2 -~J ~1T (z -sin ~y) K cos (~y) (z -sin ~y)
y<1T
f(x) gc(y)
12.55 (r(~+~+X)r(~+~-X) ]-1 ~1Tei21T~(z2_1)-~
·q~~+X(Z)q~~-X(Z) ( 2 -1 2 ] 'P~_~ 1+2(z -1) cos (~y) y<1T
0 y>'1T z>l
12.56 p~(a)p~x(a) , a<l ~PV[l-2(1-a2)COS2(~y) ] y<1T
0 y>1T
12.57 x -x z>l ~PV(1-2(1-Z2)COS2(~y) J Pv (z)PV (z) y<1T
z>l 0 Y>1T
12.58 sech(1Tx)P_~+ix(a) 2-~(a+cosh y-~ -1 <a<l
12.59 tanh(x1T)cosech(bx) -1 r n b (-1) EnCOs(n1Ty/b)Q~n1T_~(a) .p n=O -~+ix(a)
-b~~b
12.60 SeCh(1Tx)P~~+. (a) 2-~-~(1+a)~~(a+cosh y)-~~-~ - ~x
-l<a<l .p~{ (l+coSh y)~(a+cosh y)-~} ~
12.61 2 sech (1Tx)P_~+ix(a) 2-~1T-1(cosh y-a)-~
-l<a<l (10g{ (l+cosh y)~+(cosh y-a)~}
-log{ (l+cosh y)~-(co5h y+a)~}]
12.62 -1 sech (1TX) x
,{p~+ix(a)-p~_ix(a)} i2 3/ 2{(a+cosh Y)~-(l+cOSh y)~} -l<a<l
12.63 r(~+ix)r(~-ix)P_~+ix(a) 2(21T)~{(1-a2)~+coSh y}-~
a<l 'K(2~{1+(1-a2)-~COSh y}-~)
12.64 {~(~+ix)+~(~-ix)} (2a)-~(l+cOSh Y)-~(-2Y+lOg(1+a)-4 log 2
'sech(1Tx)P_~+ix(a) 2 log{ (l+cosh y)~+(a+cosh y)~} -l<a<l -2 log(a+cosh Y) )
f (x) gc(y)
12.65 r(~-~~+~ix)r(~-~~-~ix) ~ 1+~ 2 -~ ( 2-~ 1f 2 (l-a) q_~_~ (l-a) cosh y)
,p\+. (a) O<a<l - l.X
12.66 r(~+iX)r(~-ix)P~~~. (a) - ·l.X (~1f) ~r (ll) (1_a2 ) ~~-~ (a+cosh y)-~
-l<a<l ,Re~>O
12.67 r(~+ix)r(~-ix) 21-~(1_a2)~~-~r(211)ei1f~-~i1f
.sech(1fX)P~~~. (a) - l.X • (cosh y-a)-~~-~q~-~(z) ~-~
O<a<l, Re~>O z=2~cosh (~y) (coSh y-a)-~
12.68 r(~-~+ix)r(~-~-ix) (~1f) ~ (1_a2) -~~r (~-~) (a2+sinh2 y)~~-~
.cOSh(~1fX)P~~+. (a) - l.X .cos( (~-~)arctan(a-1sinh y) J O<a<l ,Re~<~
12.69 r(~-~~+~ix)r(~-~~-~ix) 1+~ 3/2 2 -k 2 1f sec (1f~) (l-a ) •
r1
.{p\+. (a)+p\+. (-a)} - l.X - l.X -l<a<l ,-~<Re~<~
12.70 sech(1fx) (P_~+iX(a»)2 -1 2 ~ 1f X{(l-a) sech(~y)}sech(~y)
O<a~l
'P_~+ix(a)p_~+ix(-a) 2 ~ 2 2 -~ z=(l-a ) {cosh (~y)-a }
-l<a<l
12.72 {sech(1fx)}2 1f sech(~y)K {l-(l-a )sech (~y)} -1 ( 2 2 ~)
'P_~+ix(a)p_~+ix(-a)
-l<a<l
f (x) gc(y)
12.73 rC~-p+ix)r(~-p-ix) 2 -~ ( 2 -1 2 ] (l-a) q_P_~ 2(1-a) cosh (~y)-l
2 .(pp . (a)]
-l<a<l
z>l 0 cosh y>z
12.76 sech(7Ix)P_~+ix(z),z<l 2-"Cz+cosh y)-"
12.77 2 {sech(7Ix)} P_,,+ix(a) -1 ~ -" (a-cos]"
71 2 (a-cosh y) arctan l+COS~ ~ a>l cosh y<a
7I-12-"(cosh y-a)-~(-10g(1+a)
+2 log{(l+cosh y)"(cOSh y-a)"}] , cosh y>a
12.7B r(,,-p+ix)r(,,-p-ix) 2 -" ( 2 -1 '2 J (a -1) q_~_P 1+2(a -1) cosh (~y)
.(P~~+iX(a) ]2
a<l Rep<~
12.79 (~(~+iX)+~(~-iX) ] (~cosh a-~cosh y)-~{-Y-log 4+1og(sinh a)
·P_~+ix(cosh a) -log(cosh a-cosh y)} y<a 2-~(cosh y-cosh a)-~ y>a
12.BO sech (7Ix) i7l(2cosh a+2cosh Y)-~{-Y-log 4
{~(~+ix)q_~+ix(cosh a) +log (sinh a) -log (cosha+cosh y) }
-~(~-ix)q_~_ix(cosh a)}
2 7I-l(a2_1)-~K{(a2_1)-~(a2_cosh2 ~y)~} 12.Bl {P_~+ix(a)} , a>l
cosh ~y<a
f(x) gc(y)
12.82 sech(~x){P_~+ix(a)} 2 ~-lz-~K{(a2_1)~z-~}, z=a2+sinh2 (~y)
,a>1
12.83 sech(~x)P_~+ix(a) z -~ K( z -~coSh ~y) , z=a2+sinh2 (~y)
{q_~+ix(a)+q_~_ix(a)}
~cosec(~Y)K{I-(a -I)cosech (~y)}
2 a>1 sinh (~y»(a2_1)~ +{q-~-ix (a)} ,
0 sinh (~y)«a2_1)~
(~~)~{r(~-ll) }-I(a2-1)~Il(a-cosh y) -ll-~
cosh y<a
0 cosh y>a
12.86 Sech(~x)pll~+. (a) 2-11-~(a+l)~Il(a+cosh y)-~-~ll - l.X
P~((I+COSh y)~(a+cosh y)-~) , a>1
12.87 r(ll+ix)r(ll-ix) (~~)~r(ll) (a2-1)~Il-~(a+cosh y)-ll
~-ll 'P-~+ix(a)
a>l, Rell>O
12.88 sech(~x)r(ll+ix)r(ll-ix) ~-ll ~ 2 _k ~ll 2 r (211) ~ (a -1) 4 (a-I) (a-cosh y)-~ll
,p~~~. (a) ,a>l:,Rell>O p =~ ( (~+~a) -~coSh (~y») cosh y<a - l.X
1-11 ~ll-~ -~ll-k 2 r (211) (a-I) (cosh y-a) •
i~ll-~i~ ~-ll((~ h 'e qll-~ cos y-~a)-~COSh(~Y») cosh y>a
12.89 11 11 q_~+ix(z)+q_~_ix(z) 2-~~3/2{r(~_Il) }-lei~llsinhll a
Rell<~ z=cosh a • (cosh y-cosh a) -ll-~ , y>a , 0 , y<a
f (x) g (y) c
12.90 sech (1Ix) (~11) ~ei 1I)lr ()l) (sinh a) )l-~
ll-~ )l-~ ·{q_~+ix(z)-q_~_ix(Z)} • (cosh y+cosh a)-ll
z=cosh a, Rell>O
12.91 11 -11 ~(a2_1)-~P ~{2(a2-1)-lsinh2 (~y) -l} P_~+ix(a)p_~_ix(a) ,a>l 11-
sinh(~y)«a2-1)~ "0 sinh(~y»(a2-1)~
1.13 Bessel Functions of Arguments 2 x,x and l/x
13.1 Jo(ax) (2 2)-~ a -y ,y<a
0 , y>a
13.2 x -~J (ax) (1Ia)-~(K{(~-~/a)~}+K{(~+~y/a~}] , y<a 0
(~1I)-~(a+Y)-~K{(~+~y/a)-~} , y>a
13.3 x-~log xJo(ax) -(1Ia)-~(K{(~-~y/a)~}+K{(~+~y/a)~}J .{y+log 4+~log(a2_y2)} , y<a
-~ 2 2 -(211) {1I+2y+4 log 2+1og(y -a )}
.(a+Y)-~K{(~+~y/a)-~} , y>a
13.4 log (bx) J 0 (ax) 2 2 -~ 2 2 -(a -y) {y-log(~ab)+log(a -y )} , y<a
2 2-~ -~1I(Y -a ) , y>a
13.5 J 2n (ax) n 2 2-~ (-1) (a -y) T2n (y/a) , y<a
n=0,1,2, ••. 0 , y>a
13.6 -~ (-1)n(~1I/a)~P2n(y/a) x J 2n+1 (ax) , y<a
n=0,1,2, •.. 0 , y>a
13.7 J v (ax) 2 2-~ (a -y) cos{varcsin(y/a)} , y<a
ReV>-l v 2 2 -~ 2 2 ~ -v -a sin(~1Iv) (y -a) {y+(y -a ) } , y>a
1.13 Bessel Functions of Arguments x,x2 and l/x 67
fIx) gc(y)
13.8 -1 x J v (ax) v-1cos{varcsin(y/a)} y<a
Rev>O -1 v 2 2 ~ -v v a cos(~7fv){y+(y _a ) } y>a
13.9 -2 x J v (ax) ~{V(V-l)}-la cos{(v-l)arcsin(y/a)}
Rev >1 +~{v(v+l) }-la cos{(v+l)arcsin(y/a)} ,y<a
-1 v 2 2 ~ l-v ~{v (v-l)} a sin (~7fv) {y+ (y -a ) }
-1 v+2 2 2 ~ -v-l -~{v(v+l)} a sin(~7fv) {y+(y -a )} ,y>a
13.10 XV J v (ax) 7f-~r(~+v) (2a)V(a2_y2)-V-~ ,y<a
-~<Rev<~ -7f-~r(~+v) (2a)vsin (7fv) (y2_a2)-V-~ ,y>a
13.11 -v x J v (ax) 7f~(2a)-v{r(~+v) }-1(a2_y2)v-~ ,y<a
Rev>-~ 0 ,y>a
13.12 -v x J v +2n (ax) (-1)n~(2n)! (~a)-vr~){r(2v+2n)}-1
Rev>~ ;n=0,1,2,. .(a2_y2)V-~C~n(y/a) ,y<a
0 ,y>a
-l<Rev<-~ (2a)v27f~{r(_~_v)}-ly(y2_a2)-V-3/2 ,y>a
13.14 l-v x J v (ax) l-v v-2 -1 2 2 2 a {r(v)} 2Fl (l,l-v;~;y /a ) y<a
Rev>~ v -1 -2 3 2 2 -(~a) {r(l+v)} y 2Fl(1'2;1+v;a /y ) y>a
13.15 X-~J v (ax) (~7f/a)~sec(7fv)cos(~7fv+~7f) Rev>-~ '{Pv_~(y/a)+pv_~(-l/a) y<a
(~7fa)-~sin(~7fV-~7f)q ~(y/a) v- y>a
13.16 (b 2+x2) -lJo (ax) ~7fb-le-bYI (ab) y>a 0
13.17 2 2-1 x(b +x) Jo(ax) cosh (by) Ko (ab) y<a
13.18 x-v (b2+x2)-lJ (ax) ~7fb-V-le-bYI (ab) y>a Rev>-3/2 v I v
68 I Fourier Cosine Transforms
f(x) gc(y)
13.19 xv+1(b 2+x2)-lJ (ax) bVcOSh(bY)Kv(ab) y<a v
-1<Rev<3/2
x (b +x ) v
xv+2n+1(b 2+x2)-1 n v+2n y<a
13.21 (-1) b cosh(by)Kv(ab)
13.22 (b 2+x2)-1 -1 -2 sin(~TIv)cosh(bY)Kv(ab) y<a
{Jv(ax)+J_v(ax)}
-l<Rev<l
13.23 x)JJ v (ax) (2TIa)-~cos(~TIv-~TI)J)r(1+v+)J)r(1+)J-v) -l-Rev<Re)J<~ .(a2_y2)-~)J-~{P-)J-~(y/a)+p-)J-~(_y/a) }
v-~ v-~ y<a
-~ 2 2 -~)J-k (~TIa) sin (~TI)J+~TIv) (y -a ) 4
-iTI(~+)J) )J+~( / ) ·e qv-~ y a y>a
13.24 x-~log xJv (ax) P2TI/a)~sec(TIv) ({Pv_~(Z)+Pv_~(-z)} Rev>-~ '{~TIsin(~TIv+~TI)-~cos(~TIV+~TI)log(a2-y2)
+o/(~+v)cos(~TIv+~TI) }-cos(~TIv+~TI)
• {Q-v-~ (z) +Q-v-~ (-z) } J 1 z=y/a , y<a
-~ 2 2 (~TIa) q ~(z) {~sin(~TIv-~TI)log(y -a ) V-
-~TICos(~TIv+~TI)-o/(~+v)sin(~TIv-~TI) }1z=l, a
y>a
XJv(x) TI~2v-1{r(~_v)}-1(y2+2Y)-v-~ x sin y<2
-l<Rev<~ TI~2v-1{r(~_v)}-1{(y2+2y)-v-~_(y2_2Y)-v-
y>2
x,x and l/x 69
f(x) gC(y)
13.26 -v x cos x J V (x) (~7T)~2-V{r(~+V) }-1(2y_y 2) v-~ , y<2
Rev>-~ 0 , y>2
13.27 -v .
x s~n x J v+1 (x) ~7T~2-v{r(~+v) }-l(l_y) (2y_y2)v-~ , y<2
Rev>-~ 0 , y>2
13.28 -1 x {si(ax)+~7TJo(ax) -dog (~y -~ { (y+a) ~+ (y-a) ~} ] , y>a
0 , y<a
-
13.30 x-\J (ax)}2 0 (~7T/Y) ~ (p -3/2 {(1-4a2 /l) ~} r , y>2a
13.31 J o (ax) J o (bx) 27T-lz-~X{2(ab)~z-~} y<la-bl
7T-l(ab)-~K{~(ab)-~z~} la-b I <y<a+b
0 y>a+b
2 2 z=(a+b) -y
13.32 x-~{J (ax)}2 (~7T/Y) ~r(l-o+v) {r(l-o-v) }-1 v
• (P-V{(1-4a2/y2)~}r Rev>-l-o y>2a -l-o
x- 2V {J (ax)}2 21 - v {r (~-v) }-2 ~7T
13.33 f 2 2 v-~ 2v dt ,y<2a (4a -y) cos t v z Rev>O
0 ,y>2a
z=arcsin (~y/a)
X-~J (ax) J (ax) k -~ V -v 2 ~ ,y>2a 13.34 v -v (~7T) 2y P-l-o (z)P_l-o (z) 1z=(1-4a /y)
13.35 Jv(ax)J_v(ax) ~a-lp 2 2 v-~(~y /a -1) ,y<2a
0 ,y>2a
0
f (x) gC(y)
13.37 JV(ax)JV(bx) TI-1(ab)-~qv_~(Z) y<!a-b!
Rev>-~ TI -1 (ab) -~Q ~ (z) v- !a-b!<y<a+b
-1 -~ -sin(TIV)TI (ab) qv_~(-z) y>a+b
-1 2 2 2 z= (2ab) (a +b -y )
13.38 Jv(bx)J_v(ax) -1 -~ y<a-b TI cos (TIv) (ab) q ~ (z) v- a>b ~ (ab) -~P ~ (-z) a-b<y<a+b v-
a y>a+b -1 2 2 2
z=(2ab) (a +b -y )
13.39 xV-~+1J~(aX)Jv(bX) 0 y<b-a
-1<Rev<Re~ ,b>a
13.40 XV-~-1J~(aX)Jv(bX) 2v-~-1b-va~r(v) {r(1+v)}-1 y<b-a
a<Rev<2+Re~ , b>a
13.41 J (ax)J (bx) ~ v Watson,G.N.J.Lond.Math.Soc.,Vo1.9,1936,p.21
13.42 XAJ (ax) J (bx) ~ v Bai1ey,W.N.Proc.Lond.Math.Soc.,Vo1.40,1936
13.43 {xvJ (ax)}2 v
3v -1 -~ -v 2 2-v 2 {r(~-v)} (2TIay) (y/a) (4a -y )
-~<Rev<~ ( v -iTIV . V ) ,y<2a • TIPv_~(z)-2e s~n(TIv)qv_~(z)
3v -1 -~-v -2 {r(~-v)} sin(TIv) (~TIay) (y/a)
• (y2_4a2)-ve- iTIVqV (z) v-~
13.44 X-v-~Jv(ax)J~(ax) TI-~aV+~-1{r(~+v+~)}-1
z -v-~ 2 2 2 v+~-\t Re(v+~»-1 /(cos t) cos{(~-v)t}(cos t-~y la ) t
0 y<2a
a y>2a
z=arccos (~y/a
13.45 Yo (ax) 0 y<a 2 2-~ - (y -a ) y>a
1.13 Bessel Functions of Arguments 2 x,x and l/x 71
f(x) gc(y)
13.46 Yo (ax) log (bx) ~'IT(a2_y2)-~ y<a 2 2 -~ 2 2 (y -a) {y+log(y -a )-log(~ab)} y>a
13.47 (b2+x2)-1{~'lTY (ax) 0
-~'lTb-le-bY{I (ab)log 0
b+Ko(ab)} y>a
0 -b log y<a
+Yo(ax)log x}
-2 ('lTY) K(2a/y) y>2a
13.50 J (bx)Y (ax) 0 y<a-b o 0
_'IT-l(ab)-~K{~(ab/z)-~} a~b
a-b<y<a+b
z=y2_(a_b)2
a~b -'IT-1(ab)-~K{~(ab/u)-~} a-b<y<a+b
-2'IT-lu-~K{2(ab/u)~} y>a+b
2 2 2 2 u=y -(a-b) , v=(a+b) -y
13.52 {Yo (ax)}2 -1 2 2 ~ ('ITa) K{ (l-l;oy /a ) } y<2a
-1 2 2 ~ 4 ('lTY) K{ (1-4a /y ) } y>2a
13.53 2 2
{Jo(ax)} +{Yo(ax)} -1 2 2 ~ 2 ('ITa) K{ (l-l;oy /a ) } y<2a
-1 2 2 ~ 4 ('lTY) K{ (1-4a /y ) } y>2a
13.54 Yo(ax)Yo(bx) 2'IT-1u-~K{2(ab/u)~} y<!a-b!
'IT-1(ab)-~K{~(ab/u)-~} ! a-b! <y>a+b
4'IT-1v-~K{(-U/V)~} 2 2 2 2 y>a+b u=(a+b) -y ; v=y -(a-b)
13.55 Yv(ax) 2 2-~
-tan(~'lTv) (a -y) cos{varcsin(y/a)}, y<a
-l<Rev<l 2 2 -~ ( -v 2 2 ~ v -sin(~'lTv) (y -a) a {y-(y -a ) } cot ('lTv)
v 2 2 ~ -v J +a {y-(y -a ) } cosec('lTv)} , y>a
f(x) gc(y)
13.56 XVyv (ax) 0 y<a
-~<Rev<~ _(2a)vu~{r(~_v)}-1(y2_a2)-V-~ y>a
13.57 -v x Yv (ax) -v ~ . 2 2 v-~ - (2a) u r (~-v) un (uv) (a -y ) y<a
-~<Rev<~ -v -~ 2 2 v-~ - (2a) u r (~-v) (y -a ) y>a
13.58 Yv(ax)cos(~uv) 0 y<a
+Jv(ax)sin(~uv) _~a-V(y2_a2)-~({y+(y2_a2)~}V+{y_(y2_a2)~}V) -1<Rev<1 y>a
13.59 xV{Yv (ax) cos (ax) _u~(2a)v{r(~_v)}-1(y2+2ay)-v-~
-J)ax) sin (ax) }
13.60 xV{J (ax)sin(ax) v 0 y<2a
+Yv(ax) cos (ax) } ~ v -1 2 -v-~ y>2a -u (2a) {r(~-v)} (y-2ay) , -~<Rev<~
13.61 sin(ax-~uv)Jv(ax) -v 2 -~ ( 2 ~ v ~a (y +2ay) {y+a+(y +2ay) }
-cos(ax-~uv)Yv(ax) +{y+a-(y2+2ay)~}vJ -1<Rev<1
13.62 x]Jyv (ax) (2au)-~sin(~u]J-~UV)r(1+]J+V)r(1+]J-V)
Re]J<~, Re(]J±v»-1 .(a2-y2)-~]J-~(p~~~~(Z)+P~~~~(-Z»)7z=y/a,y<a -~ 2 2 -~]J-~( -~iu-iu]J -(~ua) (y -a ) cos(~u]J+~uv)e
.q]J+~(Z)-Sin(~u]J-~UV)r(1+]J+V)r(1+]J-V) v-
'P~~~~ (z) J ' z=y/a , y>a
13.63 v 2 2-1
x (b +x) Yv(ax) v-1 -b cosh (by) Kv (ab) y~a
-~<Rev<5/2
{Yv (ax)+Y_v (ax) }
f (x) gc(y)
13.65 Jv (ax)Y_v (ax) -1 2 2 Rev>-J., -(~a) Qv-J.,(J.,y /a -1) y<2a
-1 2 2 -(~a) qv-J.,(J.,y /a -1) y>2a
13.66 Jv(ax)Yv(bx) -(ab)-J.,p ~(z) v- y<a-b
a.2:,b , Rev>-J., -(J.,ab)-J.,p J.,(z) v- a-b<y<a+b
-1 -J., -~ (ab) cos(~v)qv_J.,(-z) y>a+b
z=(a2+b 2_y2)/(2ab)
13.67 Jv(bx)Yv(ax) 0 y<a-b
a.2:,b, Rev>-J., - J.;, (ab) - J.,p v- J.;, (z) a-b<y<a+b
-~-l(ab)-J.,cos(~V)q J.;,(-z) v- y>a+b
222 z=(a +b -y )/(2ab)
13.68 J v (bx)Y_v (ax) ~-lsin(~v) (ab)-J.,q J.,(z) v- y<a-b
a>b , Rev>-l::; _~-l(ab)-l::;Q l::;(-z) a-b<y<a+b \1-
-~-l(ab)-l::;q\l_l::;(-Z) y>a+b
222 z=(a +b -y )/(2ab)
13.69 Jv (ax)Y_\I (bx) (ab)-J.;,(~-lsin(~\I)q\l_l::;(Z)-COS(~\I)Pv_J.,(Z)) a>b , Re\l>-J.;, y<a-b
-1 -J.;, -~ (ab) Qv-l::; (-z) a-b<y<a+b
-1 -J.;, y>a+b -~ (ab) q l::;(-z) \1-
222 z=(a +b -y )/(2ab)
13.70 2 2 -1 2 2 y<2a Jv(ax)+Y\I(ax) a sec(~\I)P l::;(l::;y /a -1) -1 \1- 2 2
-l::;<Rev<l::; a sec(~\I)p\l_l::;(l::;y /a -1) y>2a
13.71 Yv(ax)Yv(bx) (ab)-l::;(~-lq\l_l::;(z)+tan(~\I)P\l_l::;(z)),y<la-b -l::;<Rev<J., -l::;( -1 ) I (ab) ~ Qv_l::;(z)+tan(~v)P\l_l::;(z) ,Ia-b <y<a+
(ab)-l::;(~-lsin(~\I)q\l_J.;,(-Z)1 +sec(~\I)p l::;(-z) , y>a+b
2 2 2 v- z=(a +b -y )/(2ab)
b
f (x) gc (Y)
J (ax2 ) 1 -1 { 2 2 1 2 13.72 8~a Y J_k(z)-Jk(z)} ; Z = sY la 0
x1:;Jk (ax2 ) -1 1:; 2 2 z=loy 2/a 13.73 -loa (~y) {J_lo(Z)+H_lo(Z)} ; •
13.74 1:; 2 2
x J_lo(ax ) -1 1:; 2 1:;a (1:;~y) J_ lo (1:;y la)
13.75 x3/2Jk(ax2) -2 1:; 2 • ~a y(1:;~y) J 3(loy la)
-4'
13.76 X 3/2 J 3(ax2 ) -2 1:; 2 loa y(1:;~y) Jk(loy fa) -4' •
2 2 1:; 2 2 -k -az 2 2 2 13.77 x1:;e-ax J_lo(bx ) 1:; (1:;~y) (a +b ) 2e J_lo (bz) ; z=loy I (a +b
13.78 1:; 2 2 x cos (ax )J_lo(ax ) -1:; 1 1 2
1:; (ay) cos(g~-sY la)
13.79 x1:;sin (ax2)J_ lo (ax2 ) -1:; 1 1 2 1:; (ay) sin(s~-gy la)
13.80 x 1/3sin X2J 2 1 (x ) 1 1:; 2 1/6. ~ ~
s~ (1:;y) s~n(~16) -}'
13.81 x 1/3cos X2J 2 l(x)
1 1:; 2 1/6 ~ y2 g~ (1:;y) cos(~16)
-}'
13.82 2 2 x Jlo(ax ) -1 1 2 1 2
-loa {Jo(gy la)+Yo(gy la)}
13.83 2 2
x J_lo (ax) -1 1 2 1 2 loa {Jo(sY la)-Yo(sY la)}
13.84 x1:;J 21 (ax2 ) _loa-l(1:;~y)1:;J 1 (z)Y1 (z) ; z=y2 / (16a) -g -s S
13.85 1:; 2 2 x J l(ax )J1 (ax) -1 1:; -loa (1:;~) J 1 (z) -s s -g
.(sin(~/8)J_!(Z)+cOS(~/8)Y_!(Z) ]~z=y2/(16a) 8 8
f(x) gc(Y)
-v-a -1 -~ ( -i 1T/8
' 8 ' 8 2
; v-8" 8 , 8 ' 8
X2VJ (ax2) -v-~( 3 -1 3 2 13.87 J.o(~a) (r'4)} r (J.o+v) lF2 (J.o+v;~''4;-~) ! v 64a
-J.o<Rev<l 5 -1 3 2 -1 3 3 5 ~] -J.o{r('4)} r(~v)y a lF2(~v;2''4;- 2)
64a
13.88 X- 2vJ (ax2 ) -v-l v-~( 3 -1 v 2 a cos (J.o1T+1TV) {r('4)} r(J.o-v)
Rev>-l 3 2 5 -1 ·lF2(J.o-v;~''4;-~)-J.osin(J.o1T+1Tv) {r('4)}
64a
3 2 -1 3 3 5 2) .r(~V)Y a lF2(~v;2''4;-~) 64a
2 1 -1 2 2 1 2 13.89 Yo(ax ) -8"1Tya {JJ.o (z)+J_J.o (z) }; z=aY /a
13.90 x~Y_J.o(ax2) -1 ~ 2 -~a: (~1TY) H_J.o (J.oy fa)
13.91 3/2y ( 2) x J.o ax -2 ~ 2 -~a Y(~1TY) H 3(J.oy fa)
-'4
2 2 -1 z=!i/a 13.92 xJJ.o(ax )YJ.oax ) -J.oa {Jo(Z)+Ho(Z)} ; 8
13.93 ~ 2 2 . -1 ~ 2 2 x J l(ax)Y l(ax ) -~a (~1TY) J l(J.oy fa) -8" -a -a
2 2 J.oa- 1 {J (z)+Y (z)-2H (z)}; 1 2 13.94 xYJ.o(ax )
000 z=aY /a
2 2 -J.oa-1 {J (z)+Y (z)+2H (z)}; 1 2 13.95 xY_J.o(ax ) z=aY /a 000
13.96 -1 (a/x) 2{Jv (U)Kv (u)+Jv (V)Kv (V)} u (±2iay)~ x J ; = vRev>O v
13.97 x-1e-a/ xJ (b/x) J (ui~)K (vi~)+J {u(-i)~}K {v(-i)~} v v v v v Rev>-l v (2y)~{(a2+b2)~±a} = u
f(x) gc (y)
) }
Rev>-2 +sin(~~V)Iv(cY~)Kv(dY~)
c = (a+b)~±(a-b)~ , a,;:,b d
13.99 -1 cos (a/x) J v (b/x) -~~J (cy~){J (dy~)sin(~~v)+Y (dy~)cos(~~V) x v v v
Rev>-l +coS(~~V)Iv(cY~)Kv(dY~) a,;:,b ,
c = (a+b)~±(a-b)~ , a,;:,b d
13.100 -~ a a x cos(x)J2n_~(x) (-1)n~(~/Y)~J4n_l{2(2ay)~}
n=1,2,3, •••
n=0,1,2, •••
13.102 x-le-a/xYv(b/X) Y (ui~)K (vi~)+Y {u(-i)~}K {v(-i)}~ v v v v
;
13 .103 -1 sin (a/x) Yv (b/x) ~~Y (cy~){Y (dy~)cos(~~v)-Y (dy~)sin(~~v)} x v v v
-2<Rev<2 +K (dy~) {I (cy~)cos(~~v)+~ K (cy~)sin(~~v) v v ~ v
c = (a+b)~±(a-b)~ a>b d
,
13 .104 -1 cos (a/x)Yv(b/x) -~~Y (cy~) {J (dy~)sin(~~v)+Y (dy~)cos(~~v) x v v v
-l<Rev<l -K (dy~) {I (cy~)sin(~~v)+~K (cy~)cos(~~v)} v v ~ v
c = (a+b)~±(a-b)~ a,;:,b d ,
13 .105 X211J 2V (a/x) ~~411-2vr(~+Il_V){r(1+2v)r(V_ll)}-ly2V-211-1
-\<Rell <Rev-~ 'oF3(1+2V'~+V-ll'V-ll; (\ay )2}
+\4-11al+211r(V_ll_~)
-1 ( 2J '{[(V+ll+3/2)} oF3 ~,Il-v+3/2,Il+V+3/2; (\ay)
1.14 Bessel Functions of Argument (ax2+bx+c)~ 77
1 14 Bessel Functions of Argument (ax2+bx+c)l:i . f (x) gc(y)
14.1 J (ax~) -1 sin(\a2/y) 0
y
14.2 -!:i ~ x J 1 (ax) -1 2 1 2 4a sin (sa /y)
J (ax~)log(bx) -1( z{log(!:iab!:;/y)-l:iCi(z)}+l:i cos z Si(Z») 14.3 2y sin 0
z=\a2/ y
14.4 J (ax~) l:i -3/2(. 1 2 1 2 V
-\TI ay sln(sa /y-\TIv)Jl:iv_l:i(ga /y)
1 2 1 2 J Rev>-2 +cos(\TIV-sa /y)Jl:iv+~(sa /y)
14.5 x-l:iJ (ax!:i),Rev>-l l:i 1 2 1 2 v (TI/Y) cos (\TI+\TIV-sa /y)Jl:iv(ga /y)
14.6 xl:ivJv (axl:i) -1 v 2 y (l:ia/y) sin(\a /y-l:iVTI)
-l<Rev<!:;
Rev>-l 2 2-1 s=(a +y )
14.8 J (ax)J2 (bxl:i) ~ 2 2 2 2-1 s cos(\b ys)J (\ab s1; s=(a -y ) ; y<a v v v Rev>-l:i l:i. 2 2 2 2-1 y>a S sln(\b yS-TIv)J (\ab S);S=(y -a) , v
14.9 J (axl:i) J (bxl:i) v v
-1 2 2 Y Jv(!:;ab/y)sin{\(a +b )/y-~TIv)}
Rev>-l
14.10 Yo (ax~) (TIy)-l(Ci(Z)Sin z-{TI+si(z)}cos z),z=\a2/ y
14.11 x-l:iy (ax!:i) 0
~( . TI TI J 1 2 l:i(TI/y Jo(z)sln(z-4)+Yo(z)cos(z-4) ;z=aa /y
J (ax!:i)y (axl:i) ~y-l(sin ZYo(z)-cos ZJo(Z») 2
14.12 ;z=l:ia /y o 0
14.13 J (bxl:i)y (ax) l:i 2 2 2 2-1 o 0
s cos(\b ys)Yo(\ab s) ; s=(a-y) ; y<a
+2Jo (ax)Yo (bx!:;) sl:i sin (\b2yS)Y (\ab2S) 0
; s=(y2_a2)-1 ; y>a
+J (bxl:!)y (axl:!) o 0
14.15 J (axl:!) Y (bxl:!) v -v
+J (bxl:!)y (axl:!) -v v
-l<Rev<l
14.17 log (a2+x2)
.J {b(a2+x2)1:!} 0
14 .19 2 2 log(a +x )
.y {b(a2+x2)1:!} 0
14.20 J 2n (br)
14.22 rVJ v (br)
Rev>-I:!
22] -cos{\(a +b )/y}Jo(l:!ab/y)
-1 ( 2 2 Y sin{l:!nv+\(a +b )/y}Yv(l:!ab/y)
2 2 J -cos{l:!nv+\(a +b )/y}Jv(l:!ab/y)
2 2 -I:! 2 2 I:! (b -y) cos{a(b -y ) } yb
-2s-1 (sin(as1si(as)-cos(as) {log(ab/s)
-Ci(2as)} yb
2S-1 (cOs(

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Tables of Fourier Transforms and Fourier Transforms of Distributions