Tables of Mellin Transforms

Fritz Oberhettinger

Tables of Mellin Transforms

Springer-Verlag Berlin Heidelberg New York 1974

Fritz Oberhettinger Professor of Mathematics, Oregon State University, Corvallis, Oregon, U,S,A.

AMS Subject Classification (1970): 44-02, 44A10, 44A15

ISBN-13: 978-3-540-06942-3 e-ISBN-13: 978-3-642-65975-1 001: 10,1007/978-3-642-65975-1

Library of Congress Cataloging in Publication Data

Oberhettinger, Fritz, Tables of Mellin transforms, Bibliography: p, 1. Mellin transform-Tables, I. Title, QA432,024 515',723 74-16456

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks, Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1974,

Preface

This book contains tables of integrals of the Mellin

transform type

(a) 1> (z) J z-l q,(x)x dx o

Since the substitution x = e- t transforms (a) into

(b) 1> (z)

the Mellin transform is sometimes referred to as the two sided

Laplace transform. The use of the Mellin transform in various

problems in mathematical analysis is well established. Parti

cularly widespread and effective is its application to problems

arising in analytic number theory. This is partially due to

the fact that if ¢(z) corresponding to a given q,(x) by (a)

is known, then ¢(z) belonging to xaq,(x) or more general to

xaq,(xP ) (p real) is likewise known. (See particularly the

rules in sections 1.1 and 2.1 of this book.)

A list of major contributions conce~ning Mellin trans

forms is added at the end of the introduction. Latin letters

(unless otherwise stated) denote real positive numbers while

Greek letters denote complex parameters within the given range

of validity. The author is indebted to Mrs. Jolan Eross for

her tireless effort and patience while typing this manuscript.

Oregon State University

Corvallis, Oregon

May 1974

Fritz Oberhettinger

Contents

Part I. Mellin Transforms

Introduction. . . • . • • • . • . . . . . . . . . . . . • • • • . . . • . • . . • • • . • . 1

Some Applications of the Mellin Transform Analysis. ••. •••...•. •.... •• .• . ... ..••. .. .. •• 6

1.1 General Formulas.................................... 11

1.2 Algebraic Functions and Powers of Arbitrary Order ... 13

1.3 Exponential Functions............................... 25

1.4 Logarithmic Functions............................... 34

1.5 Trigonometric Functions ..••. .....•......•...•.•...•. 42

1.6 Hyperbolic Functions. . . • . . . . . • . . . . . . • . . . . . . . . . • . . . . . 61

1.7 The Gamma Function and Related Functions ....•.•...•• 68

1.8 Legendre Functions.................................. 69

1.9 Orthogonal Polynomials.............................. 83

1.10 Bessel Functions.................................... 93

1.11 Modified Bessel Function .....•...........•....••.... 115

1.12 Functions Related to Bessel Function ....•...•.•..... 133

1.13 Whittaker Functions and Special Cases •••.•.•••• ..... 138

1.14 Elliptic Integrals and Elliptic Functions .....•..... 155

1.15 Hypergeometric Functions ............................ 160

Part II. Inverse Mellin Transforms

2.1 General Formulas..................................... 163

2.2 Algebraic Functions and Powers of Arbitrary Order •... 164

2.3 Exponential and Logarithmic Functions ..•.........•... 173

2.4 Trigonometric and Hyperbolic Functions ...•........... 182

2.5 The Gamma Function and Related Functions .....•....... 191

2.6 Orthogonal Polynomials and Legendre Functions ...•.•.. 205

2.7 Bessel Functions and Related Functions ..........•.... 216

2.8 Whittaker Functions and Special Cases .••........•...• 244

Appendix ..•...•.....•.•.•.•.................•...•.• " 259

Part I. Mellin Transforms

Introduction

The integral

(1) M[cjJ(x),z] = (z) = J x z - l cjJ(x)dx

is called the Mellin transform of the function cjJ(x) with res

pect to the complex parameter

(2) z = a+i-r

The substitution

Laplace integral

(3) (z)

-t x = e transforms (1) into a two-sided

Dr into the sum of two one-sided Laplace integrals of parameter

z and -z

(3' ) (z)

Denote the abscissas of absolute and ordinary convergence by 6

and a respectively for the first integral in (3) and by 6'

and a' for the second integral. Then it is evident that the

domains of absolute and ordinary convergence of the integral (1)

consist of the respective strips.

S < Re z < -6'; a < Re z < -a'

For the inversion of the integral (1)

2 I. Mellin Transforms

(4) -1

¢(x) = M [<!l(z) ;xl

exists the following theorem.

Let <!l(z) be a function of the complex variable z = a+iT,

regular in the strip S = {z:a < a < b} such that <!l(z) + 0 as

uniformly in the strip a+n ~ a < b-n for any arbitrary

small n > o.

Then if

for each a in the open interval (a,b) and if a function ¢(x)

is defined by

(5) ¢ (x) I c+ioo

2'Tfi J c-ico

-z x CI!(z)dz

for x > 0 and a fixed cs(a,b) then

<!liz) J ¢(x)x Z- 1 dx

Some relation between the Mellin transform and other integral

transform ~.

Consider the following integral transforms of a given function

¢ (x) •

(a) Fs[¢(t);xl = (2/rr)J, J ¢(t)sin(xt)dt

Fourier sine transform

(b) Fc[¢(t);xl = (2/rr);' f ¢(t)cos(xt)dt

Introduction

(c) L [<P (t) ;xl

(d) hv [¢ (t) ;xl

(e) kV[<P(t) ;xl

(f)

Fourier cosine transform

J <P(t)e-xt dt

Laplace transform

Hankel transform

K transform v

7 <P(t) (xt)~Y (xt)dt v

Y transform v

(g) hv[<P(t) ;xl J <P(t) (xt)~~(xt)dt

Generalized Stieltjes transform

(1' ) 1 ooJ v-I Wv[<P(t) ;xl = f(v) ¢(t) (t-x) dt x

Weyl's fractional integral.

Then the corresponding relations are valid.

(b')

(c' )

(d' )

I:; (2/'1l) f(z)sin(~'1lz)M[<P(x) ;l-z]

~ M{Fc[<P(t);x];z} = (2/'1l) r(.z)cos(I:;'1lz)M[<p(x);l-z]

M{L[<P(t) ;x] ;z} = f(z)M[¢(x) ;l-z]

M{hv [<P (t) ;x]; z} = 2 z-1:; f (J..,+l:;v+l:;z) M[<P (x) ; l-z] f (3/4 +l:;v-l:;z)

3

(e' ) z- 3,.....

M{kv [<jl(t);x];z}=2 2r('a+J,v+J,z)r(\,-J,v+J,z)M[<jl(x);I-z]

(f' ) M{YV[<jl(t);X];z}

2 z-"rr -lsin (J,rr (v-z- 3/2) ] r (HJ,z+J,v) r (\,+J,z- J,v)

M[<jl(x) ;l-z]

(g , ) M {hv [ <jl (t) ; x] ; z}

= 2 z-J,tan(J,rr(J,+z+v)] r(\,+J,v+J,z) M[<jl(x) ;l-z] r (3,i,+"V-J,z)

B(z,v-z)M(<jl(x) ;l-v+z]

r (Z) r (v+z) M [<jl (x) ; v+zJ

If, for instance <jl(x) is such that both, its Hankel transform

(d) and its Mellin transform (1) is known, then the relation

listed under (d') gives an additional result. For tables of

integral transforms of the types (a) - (i) see list of references

at the end of this introduction.

Laplace and finite Mellin transforms.

Tables of Laplace and inverse Laplace transforms (see list of re-

ferences at the end of this introduction) can be used to obtain

additions to the transform tables presented here. Let

'" (6) <!> (z) f f(t)e-tzdt

o

Then the substitution t - log x transfol":as (6) into a finite

Mellin transform

Introduction

(7) 1> (z) 1 z-l f x ¢(x)dx with ¢(x) o

For instance, the Laplace transform pair

fit) J v (a sinht)

leads to the Mellin transform pair

-1 ¢(x) = Jv[~a(x -x)], x < 1

Ox> 1

f (log .:h.) • x

1> (z)

5

whi'ch is listed in Part I, under 10.77. Vice versa, the pair of

the inverse Laplace transform type

1> (z) I (bz~) K (az~), v v fIt)

yields the inverse Mellin transform type pair

1> (z) I (bz~)K (az~), v v ¢(x)

x < 1

o x > 1

with y 1

log (x) . This result is listed in Part II under 7.85.

6

Some Applications of the

Mellin Transform Analysis.

I. Mellin Transforms

Only a few examples will be singled out.

(A) Application to certain integral equations (Sneddon, p. 277,

Titchmarsh, p. 303). Solutions of the following integral

equations can be given in the form of the inverse Mellin

Transform.

(a)

(b)

(c)

g(y) J f(x)K(xy)dx, y > 0, with

M[f (t) ; z] M[g(t) ;z] M[K(t) ;z]

g(y) + J f(x)K(xy)dx = f(y), y > 0, with

M[f(t);z] M[g(t) ;z]+M[K(t) ;z]M[g(t) ;l-z] I-M[K(t) ;z]M(K(t) ;l-z]

g(y) J K(y/x)dx, y > 0, with

M[f(t) ;z] M[g(t) ;z-l] M[K(t) ;z-l]

where in (a) and (b) the kernel function K depends on the

product xy and in (c) on the quotient x/yo

(B) A summation formula. An infinite series of the form

I f(n+a) can be transformed into an integral expression n=O

Introduction

of the inverse Mellin type (Sneddon, p. 283).

I n=O

-1 f (n+a) = (2Tfi) o+ioo f z;(z,a)~(z)dz

a-ioo

[z;(z,a) is the Hurwitz zeta function] with max(l,o)<o<y

where ° and yare the abscissas of absolute convergence

of the Mellin transform of the function f(x) involved in

the above infinite sum. Hence

z-l ~(z) = f f(x)x dx

(e) An asymptotic expansion theorem (Doetsch, Vol.2, p. 115).

This theorem is widely used in problems of analytic number

theory.

c+ioo Theorem 1 Let CP(x) = f -z x ~(z)dz such that

c-ioo

7

(a) ~(z) is analytic in a left half plane Re z ~ c except

for singular points of one-calued character (poles or

essential singularities) >.. , >.. , >.. , ••• ; o 1 2

c > Re >.. > Re >.. > ••• + o 1

The principle part of the Laurent expansion of ~(z) at

(b) In every strip of finite width Co ~ 0 < c, CP(O+iT) + 0

as ITI + 00 uniformly in o.

8 I. Mellin Tranforms

(c) Between two singularities AV and AV+l there exists a

f\ (real) with Re(AV+l ) < S < Re(A), (v=O,1,2,"')

such that the integral

converges uniformly for O<x<X. = v [This is the case,

for example, if w(z) = w(a+iT) = O(ITl a ) for fixed a

wi th Re a < O. J

Then

(2d) -1 c+ioo J -z x w(z)dz ¢(x)

c-ioo

converges for 0 < x ~ Xo and

n ¢(x) L

v=O

b(V) [b(v) + 2 (-log x) + •••

1 11

r -1 -A (- log x) v + .•• J x v

Sn+ioo

J + (27Ti)-1 -z x w(z)dz Sn-ioo

as x + 0 (through positive values) , where the last -S

term (the integral) is o(x n).

Similarly

Theorem 2

(a) The function w(z) be analytic ~n a right half plane

Re z ~ a except for singular points of one valued

character A, A , with a<Re A <Re A < ••• + + 00.

o 1 o 1

Introduction 9

(b)

The principal part of the Laurent expansion of ¢(z)

at Z = A be v

-r b (v) (Z-A ) -1 + b (v) (Z-A ) -2 + •.• + b (v) (Z-A) v+

1 V 2 V rv v

In every strip of finite width

¢(o+iT) + 0 as ITI + 00 uniformly in o.

a , o

(c) Between two singularities Av and Av+l there exists

a f\ (real) with Re Av<Sv<Re Av+l ' (v=0,1,2,"') such

that the integral

f

converges uniformly for

a+ioo

x > X > O. v

Then (211i) -1 f -Z x ¢(z)dz converges for a-ioo

b (v)

¢(x) - ¥ [b(v) - -IT- log x + ..• v=O 1

as

r -1 + (-1) v

+ (211i)-1

(rv -1) !

r -1 (log x) v

S +ioo n f

Sn-ioo x-Z¢(z)dz

x + + 00 where the last term is

-A + •. "jx V

-Sn a (x ) •

x > X

These theorems can be applied as follows: If the

and o

asymptotic behavior for x + 0 or x + +00 of a given

function ¢(x) is to be investigated one forms its


Mellin transform 4(Z) by (1) and represents ¢(x) by

the Mellin inversion formula (5) in the form required

in the above theorems 1 and 2.

References

Churchill, R. V., 1958: Operational Mathematics, McGraw-Hill, New York.

Doetsch, G., 1950-1956: Handbuch der Laplace Transformation, 3 vols. Birkhauser Verlag, Basel.

Erdelyi, A. et.al., 1954: Tables of Integral Transforms, 2 vols. McGraw-Hill, New York.

Oberhettinger, F., 1957: Tabellen zur Fourier Transformation, Springer Verlag, Berlin.

Oberhettinger, F., 1972: Tables of Bessel Transforms, Springer Verlag, Berlin.

Oberhettinger, F. and Badii, L., 1973: Tables of Laplace Transforms, Springer Verlag, Berlin.

Van der Pol, B. and Bremmer, H., 1950: Operational Calculus based on the Two-Sided Laplace Integral, Cambridge University Press, London.

Sneddon, I. N., 1972: The Use of Integral Transforms, McGraw-Hill, New York.

Titchmarsh, E. C., 1948: Theory of Fourier Integrals, Oxford University Press.

Widder, D. V., 1971: An Instroduction to Transform Theory, Academic Press, New York.

1.1 General Formulas 11

1.1 General Formulas

00

z-l q, (x) (z) = f q,(x)x dx 0

(21Ti)-1 c+ico

1.1 f -z (z) (z)x dz c-ioo

1.2 q,(ax) a > 0 -z a (z)

1.3 xaq,(x) (z+a)

1.4 q,(xP ) p > 0 -1 p (z/p)

1.5 q,(x-p ) P > 0 -1 p (-z/p)

1.6 x v q, (axP ) a,p > 0 p-la-(z+V)/P<l>[(z+v)/pl

1.7 xVq,(ax-p ) a,p > 0 p-la(z+V)/P<l>[_(z+v)/pl

1.8 q, (x) (log x) n (n) (z)

1.9 q,(n) (x) (-1) n f{z) (z-n) r(z-n) provided that

r (n+l-z) = r (l-z) (z-n) lim xz-k-lq, (k) (x) = 0 x-'-O

k = 0,1, ••• n-l

00

q, (x) <jl{z) J z-l = q,{x)x dx

0

d n

1.10 (x dx) q,{x) (-z) n<jl{z)

(~ x) n

1.11 q, (x) (l-z) n<jl (z) dx

I-a d n (-a)nr{z/a)

1.12 (x dx) q,{x) r (-n+z/a)

<jl (z-na)

a t 0 n r (n+l-z/a) <jl (z-na) = a r (l-z/a)

(21Ti) -1 c+ico

1.13 q, (x) q, (x) J <jl (s) <jl {z-s)ds 1 2 c-ioo 1 2

co tBq, 1.14

a J (xt) q, {t)dt <jl (z+a) <jl (l-z-a+B) x 1 2 1 2

0

co

tBq, {x/t)q, (t)dt 1.15 a J <jl (z+a) <jl (z+a+S+l) x

1 2 1 2 0

x -1 1.16 J q,{t)dt -z <jl (z+l)

0

00

1.17 J q,{t)dt -1

z <jl (z+l) x

1.2 Algebraic Functions and Powers of Arbitrary Order 13

1.2 Algebraic Functions and Powers of Arbitrary Order

00

z-l <P(x) 1> (z) = f <P(x)x dx

0

2.1 v

x x < a (v+z) -la v+z

0 x > a Re z > - Re v

2.2 0 x < a _(v+z)-lav+ l

v x x > a Re z < -Re v

2.3 x x < a (Z+l)-l(l+az+l_bz+l)+bz-l(bz_az)

b-x a<x -1

0 x > b = l+a-b+b log (bfa) for z = 0

2.4 -1 z-l CSC(1TZ) (a + x) 1Ta O<Re z < 1

2.5 (a + x) -n ( -1 ) n +l1T [ (n -1) ! 1 -1 ( z -1) (z - 2) ••• (z -n + 1 )

n = 2,3,4,'·· z-n °a esc (1TZ)

O<Re z < n

2.6 (a + x)-J, -J, z (1Ta) a f(z)f(J,-z)

O<Re z < "

q, (x) <jJ (x) = 7 z-l q,(x)x dx 0

(a + -1 z-l 2.7 x) x < a >,a [\)I(>,+>,z)-\)I(':;z)

0 x > a Re z > 0

2.8 (a + -1 b Re z > 0

2.9 0 -1 z -1 a ZY(-a/c,l,z) x < a c b Y(-b/c,l,z)-c

(c + -1 a<x

0 x > b

2.10 [(c+bx) (d+ax) J -1 -1 l-z rr(ac-bd) (ab) csc(rrz)

• [(bd) z-l_ (ac) z-l J , O<Re z < 2

2.11 (x + a) [(x +b) (x +c)J-1 rrcsc(rrz) [(b-a)b z- 1+(c-a)cz- 1 ] b-c c-b

0 < Re z < 1

2.12 -1 z-l cot(rrz) (a-x) rra

Principal value 0 < Re z < 1

2.13 [(a+ x) (b-x) 1 -1 -1 z-l z-l rr(a+b) [a csc(rrz)+b cot(rrz)]

00

<p(x) <l>(Z) f z-l = <p(x) x dx 0

2.14 [ (a-x) (b-x) ] -1 TIcot(TIz) (az-l_bz-l)/(b_a)

Principal value O<Re z<2

2.15 [(x+a) (x2+b2) ]-1 ~TI(a2+b2)-1[2az-lcsc(TIz)

-bz-lsec(~TIz)+abz-2csc(~TIz)]

O<Re z<3

2.16 (x2+a 2+b 2) "TICSC (~TIz)

• ([x2+(a+b) 2] [x 2+(a_b)2]}-1 I Iz - 2 • [ b-a + (b+a)z-2]

O<Re z<2

2.17 (x 2_b 2+a 2) "TIa -1

csc(~TIz)

'{[x 2+(a+b)2] [x 2 +(a_b)2]}-1 z-l I Iz- l • [ (a+b) +sgn (a-b) a-b ]

O<Re z<2

2.18 (l-xo.) (l_xn o.)-l -1 TI (no.) sin (TI/n)

n = 2,3,'" ·csc[TIz/(no.)]csc(TIz+TIo.) no.

O<Re z«n-l)o.

2 .19 (b+ax) -v z -v

(b/a) b B(z,v-z)

O<Re z<\)

16

2.20

2.21

2.22

2.23

2.24

q,(x)

(a_x)v

o

Re v > -1

0

(x-a) v

Re v

0

(c+ax) -v

-v (c+ax)

o

> -1

x < a

x > a

x < a

x > a

x b

x b

[(c+bx) (d+ax) I v


00

¢(z) = f q,(x)xZ- 1 dx o

a v+zB (v+l, z)

Re z > 0

Re z<-Re v

-v z -1 (ab) b (v-z)

F [v,v-z;l+v-z;-c/(ab)] 2 1

-1 F [l,v;l+v-z; (ab) (c+ab) ]

2 1

Rez<Rev

-v -1 -z c z b F (l,z;l+z;-ab/c)

2 1

Re z > 0

• (cd) ,>v+,>z-"B (z -2v-z) 1

O<Re z<-2Re v

00

¢ (x) ~(Z) = J ¢(x)xz - 1 dx

2.25

ac < bd 'f(l+v)f(z)sec(TIz)

O<Rez<1,

2.26

ac > bd

O<Rez<1,

2.27

ac < bd 'c1,z+1,vd 1,z-1,V-1 f (1+v)f(z)sec(TIz)

1, 1, .{p- v - z [(1- ac) l_P-v - z [-(1- ac) J}

l+v-z bd l+v-z bd

o < Re z < 3-'2

2.28 v -v- 3"

(c+bx) (d+ax) 2

ac > bd

'f(-v-1,)f(z)sec(TIz)

• [p3/2+v-z[(1_ bd) 1,l_p3-'2+v-z[_(1_bd) 1,l} v+z-1, ac v+z-1, ac

o < Re z < 3/2

00

¢ (x) <P (Z) f z-l = ¢(x)x dx

0

2.29 (c+bx) v (d+ax) 11 d l1 c V+ zb- zB(Z,-l1-V_z)

ac • 2Fl (-l1,Z;-l1-V;l-bd)

= c Va- Zd l1+ ZB(Z,_l1_V_Z)

· F (-v z'-l1-v'l_bd) O<Re z<-Re (V+l1) 2 1 ' I I ac

2.30 [(a-x) (c+bx)]v x<a r(l+v) (c+ab)v(ac/b)~V+~z

0 x>a -V-Z 'r(z)Pv [(c-ab)/(c+ab)]

Re v > -1 Re z > 0

2.31 (a-x)v(c+bx)-v-~ x<a r(1+V)2v+zr(z)c~z-~v-~

0 x>a • (a/b)~v+~zp-v-z[(l+ab/c)~J v-z

Re v > -1 Re z > 0

2.32 v -v- 3/.

(a-x) (c+bx) 2x<a r(1+V)2v+z(c+ab)-~r(z)

0 x>a 'c~z-~v-1(a/b)~v+~z

Re v > -1 -v-z ~ 'Pv-z+1 [(l+ab/c) ]

Re z > 0

a

Re v > -1, b > -cia

2.34 o x < a

[ (x-a) (c+bx) ] v x > a

Re v > -1

2.35 o x < a

Re v > -1

2.36 o x < a

v -v- 3/ (x-a) (c+bx) 2X > a

Re v > -1

ro

~(z) = J <P(x)xz-ldx o

F (-)l,Z;v+l+z;-ab/C) 2 1

• F (-)l,l+v;l+v+z; ab) 2 1 c+ab

Re z > 0

r(l+v) (c+ab)V(ac/b)~V+~Z

or(-2v-z)pv+z[(ab-c)/(ab+c)] v

Re z < -2 Re v

r(1+V)r(~_z)2~+v-za~z+~v-~

• z-v-1'[(1+ C )~] Pv+z-~ ab

Re z < ~

r (l+v) r (3-Z- Z) (c+ab)-~

= z-l <P(x) <!> (z) = f <P(x)x dx

0

2.37 0 x < a (c+ab)~az+vB(l+v,-~-v-z)

(x-a) v (c+bx) ~ x > a F [-~,l+v;l-~-z;c/(c+ab)l 2 1

Re v > -1 = b~az+v+~B(l+V;-~-v-z)

F [-~,-v-~-z;l-~-z;-c/(ab)l 2 1

Re z<-Re (v+~)

2.38 [x+(a2+x2) l,]-v -1 -v z v(v+z) a (l,a) B(z,,>V-'>z)

0 < Re z < Re v

2.39 (a 2+x2)-'>[(a 2+x2)'>+a]V (2a)v+z-1B (,>z,1-v-z)

0 < Re z<l-Re v

2.40 (a2+x2)-~[(a2+x2)'>+x]V -z v+z-l 2 a B(z,,>-,>z-,>v)

0 < Re z<l-Re v

2.41 (a 2+x2)-'>[(a 2+x2) '>-x] v 2- z a v+ z- 1B(Z,,>_,>z+,>v)

2.42 (b-x) v-l r (V)kAbV+l+Z r (z) [r (v+z) ]-1

• (xk+ak) A z z+l z+k-l x b z+v z+v+l z+v+k-l ;_bk/ak ) k=1,2,3,"'; Re(v,z»O

~'--k---"" , k

¢(x)

2.43

2.44

2.45 o x < a

2.46

x < a

o x > a

00

~(z) = f ¢(x)xz-ldx o

(2a)~z-lr(~z)r(1-v-z)

v+z-l -1 = (2a) [r (l-v-~z) 1

F (-~v-~z,1-3~v-~z;1-v-~z;~-~b/a) 2 1

O<Re z<l-Re v

2-~-~v-~zaV+z-lr(z)r(~_~v_~z)

• (b2_1)~+~V-~Z ~+~V-~Z(b) P~v-l

O<Re z<l-Re v

Re z<l±Re v

Re z>Max(O,-2 Re v)

00

z-l ¢(x) <P (x) = f ¢(x)x dx

0

2.47 [x+a+(x2+2ax)~]-V -v z -1 2va (~a) [r(l+v+z)] r(2z)r(v-z)

0 <Re z < v

2.48 (x+a) [(x+a)2+b2 ]-1 7fcsc(7fz) (a 2+b 2) ~z-~

'cos[(l-z)arctan(b/a)]

0 < Re z < 1

3 ~ (z-l) (2a) z-2v-2 [r (3/2 +V ) ]-1 2.49 (x+a) (x 2+2ax)-v- ~

·r (Z-V- 3/2) r (2+2v-z)

3/2 +Re v < Re z < 2+2Re v

2.50 [(x+a) 2_b2]-V-~ 7f~(2b)-v[r(~+v)]-1(a2_b2)~z-~v-~

a > b .r(z)r(1+2V-z)p-v [a(a2-b2)-~] v-z

0 < Re z < 1+2 Re v

2.51 (x+a) [(x+a)2_b 2]-V- 3-'2 (~b)-vr(1+v)B(z,2v+2-z)

a > b • (a 2_b 2) ~z-~V-lp -v [a (a 2_b2)-~ l+v-z

0 < Re z < 2+2 Re v

00

z-l ¢ (xl 1>(zl = I ¢(xlx dx

0

2.52 {x+a+[(x+al2-b2]~}-v vb -vr (z) r (v-z)

a > b .(a2_b2)~zp-v[a(a2_b2)-~1 z

0 < Re z < v

2.53 (x2+2ax)-~ -~ -v z-l 11 a (2a) r(~-z)r(z)

• [x+a+(x2+2ax)~]-v . r (1+ v- z) [r ( z+ v) ] -1

~ < Re z<l+Re v

2.54 (x 2+2axcosS+a 2)-1 1IcscSa z-2 csc(1Iz)sin[(1-z) S]

-11 < S < 11 0 < Re z < 2

2.55 (x 2+2axcoshy+a 2)-1 'ITcschy z-2 csc(1Iz)sinh[(1-z)y] a

0 < Re z < 2

2.56 (x 2+2ax cos8+a 2) -~ z-l csc(1Iz)Pz_1 (cosS) 1Ia

-11 < S < 11 0 < Re z < 1

2.57 (x2+2axcoshy+a2)-~ z-l csc(1Iz)Pz_1(coshy) 1Ia

0 <Re z < 1

<X>

cp(x) cjl (z) = f cp(x)xz-ldx 0

2.58 (x 2 +2axcos e + 2 -v a ) 2v-~r (~+v) (sin e)~-va z-2v

e ~-v e) -7f < < 1T "B(z,2v-z)P ~(cos z-v-

0 < Re z < 2Re v

2.59 (x 2 +2ax co shy + 2 -v a ) 2v-~r(~+v) (sinhy)~-va z-2v

~-v oB(z,2v-z)p ~(coshy) z-v-

0 < Re z < 2 Re v

2.60 x sin e z-l CSC(1TZ) sin(ze) 7fa

o (x 2+2ax cose+a 2 )-1 -1 < Re z < 1

-1T < e < 1T

2.61 sinhy z-l esc (1TZ) sinh (yz) x 1Ta

o (x 2+2ax coshy+a 2 )-1 -1 < Re z < 1

2.62 (a+xcose) z-l 1Ta CSC(1TZ)COS(ze)

o (x 2+2 ax cose+a 2 )-1 0 <Re z < 1

-7f < e < 1T

'" z-l cj>(x) <!>(z) = f qdx) x dx 0

2.63 (a+x coshy) 7fa z-l csc(7fz)cosh(zy)

• (x 2+2ax coshy + 2 -1 a ) 0 < Be z < 1

2.64 (x2+2al;x+a 2) -v v-~ z-2v 2 r(~+v)a B(z,2v-z)

I; not on the real '(1;2_l)I;,-~vp~-v (1;) z-v-~

axis between -1 and z-2v = a B(z,2v-z)

_00

• F (z,2v-z;~+v; ~-~1;) 2 1

0 < Be z < 2 Re v

1.3 Exponential Functions

3.1 -ax -z 0 e a r (z) Be z >

3.2 -bx -z 0 e x < a b y(z,ab) Be z >

0 x > a

3.3 0 x < a -z b r(z ,ab)

-bx x > e a

3.4 (b+x)-le-ax eabbz-lr(z)r(l_z,ab) Be z > 0

'" z-l <P(x) q, (z) = f <p(x)x dx

0

3.5 (b+x)ve-ax e~abb~z+~v-~a-~-~v-~z

.r(z)W~+~v_~z,~v+~z(ab)

Rez > 0

3.6 (b 2+x 2)-le-ax a~bZ_3/2r(Z)S~ ~(ab) -z,

Re z > 0

3.7 (b-x) veax x b · F (z;z+v+l;ab) 1 1

Re v > -1 Re z > 0

3.8 (b2-x2)-~exp[-a(b2-x2)~] ~rr~(2b/a)~z-~r(~z)

x b Re z > 0

3.9 0 x b

3.10 0 x b

1.3 Exponential Functions 27

00

z-l q,(x) <P(z) = f q,(x)x dx 0

3.11 (a 2+x 2 )-" n-"(2a/b) "Z-"r("Z)K"_"Z(ab)

'exp[-b(a 2+x 2 )"l Re z > 0

3.12 exp[-b(a 2+x 2 ) "1 n-"a(2a/b) "z-"r("Z)K"+,,z(ab)

Re z > 0

3.13 e -ax 2 -bx -"z 1 2 (2a) r(z)exp(~ la)

on [b(2a)-"1 -z Re z > 0

3.14 exp (-axP ) p > 0 p-la-z/Pr(z/p) Re z > 0

3.15 exp (-ax -p) p > 0 -1 zip par (-zip) Re z < 0

3.16 exp(-axP-bx-P ) 2p-l (b/a) "z/PK I [2 (ab) "1 z p

p > O·

3.17 l-exp(-axp)p > 0 -1 -zip -p a r (zip) -p < Re z < 0

3.18 l-exp(-ax-p)p > 0 -1 zip -p a r(-z/p) O<Rez<p

00

 (z) = f z-l <P(x)x dx

0

3.19 (eax_l) -1 -z a r(z)1;(z) Re z > 1

3.20 (e ax_l) -Ie -bx -z a r(z) 1; (z,l+b/a) Re z > 1

3.21 e-px (eax+l)-l (2a) -zr (z) [1; (z ,lz+Jw/a) -1; (z ,l+Jw/a) ]

Re z>O, p>-a

3.22 -ax -1 -z -1 1-z

e [J.,-x r(z) [1;(z,a)-J.,a +(l-z) a 1

+ (eX_I) -1] Re z > - 1

3.23 (eX_I) -2 r(z) [1;(z-l)-1;(Z)] Re z > 2

3.24 -ax -x -2

e (l-e) r(z) [1;(z-l,a-l)-(a-l)1;(z,a-l)]

Re z > 2

3.25 (ex _1;) -Ie -ax r(z)Y(1;,z,a+l)

larg 1;1 < 1T Re z > 0

3.26 (eax+l) -1 -z 1-z a r(z)(1-2 )1;(Z) Re z > 0

= cjl (x) z (a)

(x 2-1) -'> -J_,>z(a)Y,>_,>z(a»)

-1 'exp[-a(x-x »)x > 1

3.28 (1-t2 ) -~ x < 1 3/. '>

'>1f 2a [J,>z (a)Yl,z-l, (a)

'exp[-a(x -1

-x) ) -Jl,z_,>(a)Yl,z(a»)

0 x >1

3.29 e-ax [(ex _l)-l_x -l) -1 r(z) [I;(z,a)+(l-z) a l-z -z -a )

Re z > 0

3.30 (1+x2) -v-l ~(2a)-~v-~[r(1+v»)-lr(~z)

oexp[_a(1-x 2») .r(1+V-~Z)M~_~z+~v,~v(2a) 1+x2

0 < Re z < 2+2 Re v

3.31 (l+x 2) -v-l ~(2a)-~v-~e-a[r(1+v»)-lr(~z)

'exp(- ~) .r(1+v-~Z)M~_~z+~v,~v(2a) 1+x2

0 < Re z < 2+2 Re v

w

<P(x) (z) ! z-l = (x)x dx

0

3.32 (l_x 2 )-v-1 -kv-~ ~(2a) 2 r(~Z)W, +" , (2a)

--iV ''2-;:..zZ I ~\)

l+x 2 I Re 0 'exp [-a (--) ], x < z >

l-x 2

0 X > I

(l_x 2)-V-1 -kv-!:: a 3.33 ~(2a) 2 2e r(~z)W, +" , (2a)

~\) Yz- YzZ , ~\)

'exp (- 2ax 2 l-x 2

0 X > I

3.34 (l-x 2 ) -~ r(z)D 2 [(2a)~] -z

l+x I 'exp[-a(I_X)]' x < Re z > 0

0 x > I

3.35 (l-x 2) -~ e a r(Z)D 2 [(2a)~] -z

'exp (- 2ax) I-x '

x 0

0 x > I

3.36 (l-x2)-~ r(z)D {2~[b+(b2-a2)~]~} -z

'exp[_(2ax+b+bx 2)] l-x 2

'D {2~[b-(b2-a2)~]~} -z

x 0

0 x > I

1.3 Exponential Functions

3.37

3.38

3.39

3,40

q,(x)

ax+bx2 'exp [-2 (---) 1

l-x 2

x < 1

o x > 1

(b-x) v-l

o x > b

k = 2,3,4,,"

(b-x) v-1

'exp[a(b-x)k 1 x b

k = 2,3,4,-"

(b-x) v-l x b

Re (v, z) > 0

= ~(z) = f q,(x)x Z- 1 dX

o

Re z > 0

31

• F (~z+l ••• z+k-l,z+v z+v+l kkk' k' , k ' k' k '

Re (v, z) > 0

• F (~v+l ••• v+k-l z+v z+v+l kkk' k' , k ' k' k '

Re (v, z) > 0

• F (z;~,v+z;~a2b)+ar(~+z)r(v) 1 2

• F (!.2+Zi~2,!z+v+z;~a2b) 1 2

00

z-l ¢(x) q,(z) = J ¢(x)x dx

0

h 3.41 0 x < 1 D~_1 [(2a) 2] r (l-z)

(X 2 -1)-!, Re z < 1

oexp[_a(x+l)] x-I

x > 1

3.42 0 x 1

3.43 0 x < 1 r(l-z)D {2!,[b+(b 2 -a 2 )!,1!,} z-l

(x 2 -1) -!:z 0D {2!,[b-(b 2 -a 2 )!,1!,} z-1

2 oexp[_(2aX+b+bx )] Re z < 1

x 2 -1

x > I

3.44 0 x 1

00

z-l <jJ (x) <!>( z) = f <jJ(x)x dx 0

3.45 (1+x2)-V-1 ~(2a)-~v-~[r(1+v)]-lr(1+v-~z)

1-x2 .r(~Z)M~_~z+~v,~ (2a) 'exp [-a (--) ]

1+x2

0 < Re z < 2+2 Re v

3.46 (1+x2) -v-1 ~(2a)-~V-~e-a[r(1+v)]-lr(1+v-~z)

'exp(- ~) .r(~z)M~_~z+~v,~v(2a) 1+x2

0 < Re z < 2+2 Re v

3.47 _l--V- k

0 x < 1 ~(2a) '2 2r(1+v-~z)W, 1 1 1 (2a) ~Z-~\)-"2,YzV

(x 2_1)-v-1

x 2+1 Re z < 2+2 Re v 'exp[-a(--)]

x 2-1

x > 1

3.48 0 x < 1 -!zv-!z a ~(2a) e r(l+v-~z)

(x 2_1)-V-1 .W!zz-~v_!z,!zv(2a)

'exp(- ~) Re z < 2+2 Re v x 2-1

x > 1

34 I. Mellin Transfonns

1.4 Logarithmic Functions

00

z-l 4> (x) .p (z) = J .p(x)x dx

0

-1 a Z -1 Re z > 0 4.1 log x x < a z (loga-z )

0 x > a

4.2 0 x < a z -2 a z Re z < 0

log (x/a) x > a

4.3 -1 (x+a) logx waz-lcsc(wz) [loga-wcot(wz)]

0 < Re z < 1

4.4 -1 -1 0 <Re < 1 x log (l+x) wcsc(n) (l-z) z

4.5 log (l+a/x) -1 z wz a csc (wz) 0 <Re z < 1

4.6 log (l+ax) -1 -z wz a csc (wz) -1 < Re z < 0

4.7 -1 (b+ax) log (b+ax) _bl - V a -zwcsc (owz)

0 <Re z < 1 • [y+1j!(l-z)"';logb]

4.8 log[(a+cx)/(b+cx)] -1 -z Z Z WZ C csc (wz) (a -b )

0 < Re z < 1

1.4 Logarithmit Functions 35

00

z-l cjJ(x) ( x) = J <l>(x)x dx 0

4.9 10g11-axl -1 -z cot ('ITz) -l<Re z<O 'ITZ a

4.10 10 111+xl g l-x -1 'ITZ tan (~'ITz) -l<Re z<l

4.11 log I~I -1 csc ('ITz) [a 2 -b 2 cos ('ITz)] 'ITZ b-x

O<Re z<l

4.12 -1

"[\)I' (H~z) -\)I' (~z)] > 0 (x+l) 10gx x < 1 Re z

0 x > 1

4.13 -1

\)I' (z) (x-l) 10gx x < 1 Re z > 0

0 x > 1

4.14 -1 (a-x) 10gx z-l

'ITa [loga cot('ITz)-'ITcsc 2 ('ITz)]

Principal value O<Re z<l

4.15 -1 -1 (a+x) (b+x) 10gx -1 z-l 'IT (b-a) csc('ITz) [a 10ga

0 < Re z < 1 z-l z-l z-l -b 10gb-'ITcot('ITz) (a -b )]

4.16 log (l+x) x < 1 -1 z [10g2-~\)I (l+~z)

0 x > 1 + ~\)I(~+~z)]

Re z > -1

00

</> (x) <l> (z) f z-l = </>(x)x dx

0

4.17 0 x < 1 rrz -1

cot ('rrz)

log (l+x) x > 1 -1 < Re z < 0

4.18 log (l-x) x < 1 -1 -z [y+1jJ(z+l)] Re z > -1

0 x > 1

4.19 0 x < 1 -1 -1 z [z +1jJ(l-z)+y]

log (x-1) x > 1 Re z < 0

4.20 -1 2 rr3 csc 3 (rrz) [2-sin 2 (rrz)] 0 (l+x) (logx) < Re z < 1

4.21 (l+x) -11og(1+x 2 ) ~rrcsc(rrz){log4+(1-z)sin(~rrz)

-2 <Re z < 1 • [1jJ ( 3/4 - .. z) -1jJ ( .. - .. z) ]

- (2-z) cos (~rrz) [1jJ (l-"z) -1jJ (~-"z)]}

4.22 log (1+2x cos8+x 2 ) 2rrz-1cos(z8)csc(rrz)

-rr < 8 < rr -1 < Re z < 0

4.23 log[x+(a 2 +x 2 ) ~] -(~a)zz-lB(Z,~V-~z) -1 < Re z < 0

1.4 Logarithmic Functions 37

00

z-l CP(x) q. (z) '" J cp(x)x dx

0

4.24 (a 2+X2)-~ 2-zaz-1B(z,~-~z) [loga-~1Ttan(l~1Tz)]

·log[(a 2+x2) ~-x] 0 < Re z < 1

4.25 (a 2+X 2)-l:! 2- z a z - 1 B (z, ~-l:!z) [loga+l:!1Ttan (~1TZ)]

.log[(a2+x2)~+x] 0 < Re z < 1

4.26 log[l+(1-x 2)l:!] x < 1 z-l[~1T~ r (l:!z) -1 r(l:!+~z) - z ]

0 x > 1 Re z > 0

4.27 0 x < 1 ~ -1 -1 -l:!1T z f(-l:!z) [r(~-~z)]

log [x+ (x 2_1 ) ~] x > 1 Re z < 0

4.28 0 x < a ~ -1 z -1 -1T z a f(-~z) [f(~-~z)]

log x+ (x 2-a 2) ~

X > a Re z < 0 [ l:!] x- (x 2-a 2)

4.29 (y2_1)-~ 3" 3/ 2 2a 2b 2r(l:!z) [f( 3/2-~Z)]

·log[y+(y2-1) ~] • [r (l-l:!z) ]-1 (lb2-a2 1) -~Zq ~ 1 (b 2+a2 ) - z- Ib2-a21

y '" (2ab) -1(a 2+b 2+x2) 0 < Re z < 1

00

q, (x) <p(z) f z-l = q,(x)x dx

0

4.30 log[x+(1+x 2 ) 1,] -1 -J,z B (J,+J,z ,-~z) -l<Re z<O

4.31 (a 2+x 2+2ascos8)-1 -rrcos 8 dn [a z-2 csc(rrz)sin(z-l)8] --

dzn

o (logx)n O<Re z<2

-rr<8<rr

4.32 -x n dn r (z) > 0 e (logx)

dzn Re z

4.33 -ax -z -z > 0 log (l+e ) a (1-2 ) r (z) I;; (l+z) Re z

4.34 -ax -a-zf(z) I;; (l+z) > 0 log(l-e ) Re z

4.35 log [tanh (ax) ] - (2a) -zr (z) I;; (l+z) (2_2- z ) Re z > 0

00

4.36 log (1-2ae-xcos8+a 2e-2x) -2r (z) L -1 n (n+ 1) a cos (n 8)

1

0 < a < 1, -rr<8<rr Re z > 0

4.37 (log l/x)v x < 1 Z-v-lr(V+l)

0 x > 1 Re v > -1 Re z > 0

1.4 Logarithmic Functions 39

00

q, (x) 1> (Z) = f <jJ(x)xz-ldx

0

4.38 (a-x) v log (a-x) x < a a v+zB (v+l, z)

0 x > a • [loga+ljJ (1+v) -ljJ (1+v+z) )

Re v > -1 Re z > 0

4.39 0 x < a aV+ZB(v+l,-v-z)

(x-a) v log (x-a) x > a • [loga+ljJ(1+v)-ljJ(-v-z»)

Re v > -1 Re z < -Re v

4.40 (b+ax) -v b-v(b/a)zB(z,V-z)

-log (b+ax) • [logb+ljJ (v) -ljJ (v-z) )

0 < Re z < Re v

4.41 (l-x)vlogx x < 1 B(v+l,z) [ljJ(z)-ljJ(v+1+z)]

0 x > 1 Re z > 0

Re v > -2

4.42 (l-x)V(logx)2 x < 1 B (v+l, z) {[ljJ (z) -ljJ (V+1+z»)2

0 x > 1 +ljJ' (z) -ljJ' (v+1+z)}

Re v > -1 Re z > 0

4.43 e -at (logt) 2 r (z) a -z { [ljJ (z) -loga) 2 +ljJ I (z) }

Re z > 0

40

q, (x)

4.44 -at e logt

4.45

For a=l, -l<Re z<Re v

4.46

O<Re z<l+Re v

4.47

O<Re z<l-Re v

4.48

• {[a+ (a 2_X 2 ) ~l v log (a+ (a 2_X 2 ) ~l

~ v I~ +[a-(a 2-x 2 ) 1 log[a-(a 2 -x 2 ) 1

x < a

o x > a


'" w(z) = f q,(x)xz-ldx o

r(z)a-z[~(z)-logal Re z > 0

-v -1 z -a (v+z) (~a) B (z, ~v-~z)

o <Re z<Re v

• [loga+~~{~-~z+~v)

-~~(~+~z+~v) 1

• [loga-~~(~-~z-~v)

+~~(~+~z-~v) 1

• [log (2a) +~ (v+~z)

Re z>Max(O,-2 Re v)

1.4 Logarithmic Functions

¢ (xl

4.49 0 x < a

[x+(x2_a2)~]V

'log[x+(x2-a2)~]

_[x_(x2_a2)~]V

'log[x-(x2-a2)~] x > a

4.50 o x < a

4.51 2 2

log [c + (x+a) 2] c 2+(x+b)

O<Re z<l

00

f z-l W(zl = ¢(xlx dx o

~av[r(l_z)]-l(~a)z

r(~v-~z)r(-~v-~z)

Rez<±Rev

41

Rez<l±Rev

1.5 Trigonometric Functions

00

z-l ¢(x) <jJ( z) = f ¢(x)x dx

0

5.1 sin (ax) -z

a r(z)sin(~7fz)

-1 < Re z < 1

5.2 cos (ax) -z

a r(z)cos(~7fz)

0 < Re z < 1

5.3 sin (bx) < ~iz -1

a Z [ F (z; l+z; -iab) x a 1 1

0 X > a - F (z;l+z;iab) 1 Re z > -1 1 1

5.4 cos (bx) < ~z -1

a Z [ F (z;l+z;iab) x a 1 1

0 X > a + F (2; l+z;-iab) 1 1 1

Re z > 0

5.5 -1 (7fab)~(2a)z-1[r(~+~z) (a+x) sin (bx)

r (l-~z)

-1 < Re z < 2 2r (l+~z)

'S~_z,~(ab)- r (l,-l,z) s_~_z,~(ab) 1

5.6 -1 (a+x) cos (bx) ~ z-l r(~z)

(7fab) (2a) [r (~-~z) S~_z, ~ (ab)

0 < Re z < 2 - 2 r (~+~z)

S_~_z,~(ab) 1 r (-~z)

¢(x)

5.7

• sin (bx)

-1 < Re z < 3

5.8

• cos (bx)

0<Rez<3

5.9

• sin (bx)

-l<Re z<1+2 Re v

5.10

• cos (bx)

O<Re z<1+2 Re v

'" ~(z) = f ¢(x)x Z- 1 dx o

F (1;2-!;zz,3/2 -!;zz;!oa 2 b 2 ) 1 2

+!;zrra z - 2sec(!;zrrz)sinh(ab)

z-2 +!;zrra csc(!;zrrz)cosh(ab)

F (~+~Zi 3/2+~Z-V, 3/2 ;~a2.b2) 1 2

43

+2 z-2v-l rr !;zr (3/2 -!;zZ-v) [r (l+v-!;zz) rl

• F (!;zz;1+!;zz-v,!;z;!oa 2b 2 ) 1 2

!;z z-2v-l 2v-z -1 +rr 2 b [r (!;z+v-!;zz) 1 r (!;zz-v)

44

5.11

5.12

5.13

5.14

5.15

¢ (x)

v-I (a-x)

• sin (bx)

x < a

Ox> a

Rev>O

v-I (a-x)

• cos (bx)

o

Rev>O

'sin (bx)

o

'cos (bx)

o

• sin (bx)

x < a

x > a

x < a

x > a

x < a

x > a

(Principal value)


'" W(z) = J ¢(x)xz-ldX

o

\iB(z,v)az+ v- l [ F (z,v+z;-i ab) 1 1

- F (z, v+ z; iab 1 1 1

Re z > -1

\B(z,v)az+ V- l [ F (z,v+z;iab) 1 1

+ F (z,v+z;-iab)] 1 1

z+2v+ 1_ \a oB(v+l,\+\z)

Re z > -1,

Re z > 0

Re v > -1

\aZ+ 2VB(\Z,V+l) F (\z;\,l+v+\z; 1 2

Re z > 0, Re v > 1

z-2 k z-l -\1fa cos (ab) + (1fb/a) 2 (2a)

-1 . r (\+\z) [r (l-\z) 1 8\_z, \ (ab)

-1 < Re z < 3

1.5 Trigonometric Functions 45

00

<j> (x) <!J{z) = f <j> (x) x z - 1 dx 0

5.16 {a 2_x 2)-1 ~7fsin (ab) a z - 2 + (2 7fb) ~ (2a) z- 3/2

'cos (bx) -1

·r (~z) [r (~-"z)] S,,+z,~{ab)


5.17 (a+x) -1 7f csc{7fz)az{l-~7fa~b

k k k k k • cos [b (a+x) 2] • [J, {ba 2)H 1 (ba 2) -H, (ba 2) J , (ba 2) ]

~-z -~-z ~-z -~-z

O<Re z < 31'2

5.18 (a-x) v x < a a V+1B{z,V+l)

'cos [b (a-x)~] . F (V+l;1+v+z,~;-~ab2) 1 2

0 X > a Re z > 0, Re v > -1

5.19 -bx sin (ax) f{z) (b 2+a 2) -~zsin [z arctan{a/b)] e

Re z > -1

5.20 -bx

cos (ax) r{z) (b2+a2)-~Zcos[z arctan (alb) ] e

5.21 -bx sin (ax) < c ~i(b+ia)-Zy[z,c{b+ia)] e x

0 x > c -~i(b-ia)-Zy[z,c{b-ia)]

Re z > -1

co z-l cp(x) <l> (z) = J cp(x)x dx

0

5.22 -bx cos (ax) e x < c ~(b+ia)-Zy[z,c(b+ia)l

0 x > c +~(b-ia)-Zy[z,c(b-ia)l

Re z > 0

5.23 0 x < c ~i(b+ia)-zr[z,c(b+ia)l

-bx sin (ax) e x > c -~i(b-ia)-zr[z,c(b-ia)l

5.24 0 x < c ~(b+ia)-zr[z,c(b+ia)]

-bx cos (ax) e x > c +~(b-ia)-zr[z,c(b-ia)]

5.25 (eax_l) -1 ~ia-zr(z) [l;(z,l+ibja)-l;(z,l-ibja)]

'sin (bx) Re z > 0

5.26 (eax_l) -1 ~a-zr(z) [l;(z,l+ibja)+l;(z,l-ibja)]

'cos (bx) Re z > 1

5.27 (eax+l) -1 r(z) {b-zsin(~wz)+~i(2a)-z

'sin (bx) • [l;(z,~+~ibja)-l;(z,~-~ibja)

Re z > -1 -l;(z,~ibja)+l;(z,-~ibja)]}

co z-l CP(x) (z) = f CP(x)x dx

0

5.28 (eax+l) -1 r (z) {b -zcos (J,1TZ) +lz (2a)-z

'cos (bx) • [1;(z,J,+lzib/a)+1;(z,lz-lzib/a)

Re z > 0 -1;(z,J,ib/a) -1;(z,-lzib/a»)

5.29 -ax 2

sin(bx) lzba -lz- lzz r (lz+lzz) exp (-!' -1 F (-lzz; 'Y2 ;!.<b2/a ) 1 1

5.30 -ax 2

cos (bx) lza- lzz r(lzz)exp(-!. 0 F (-~Z+~i~i~b2/a) 1 1

5.31 -ax 2-bx -kz 1 -1 2 2 e -lzir(z) (2a) 2 exp[ /sa (b -c »)

o sin (cx) -k o {exp(-!.<ibc/a)D_ z [(2a) 2(b-ic»)

Re z > -1 -k -exp (!.<ibc/a) D_z [(2a) 2 (b+ic) ) }

-ax 2-bx -J,z 1 -1 2 2 5.32 e lz (2a) r( z) exp [/sa (b -c »)

'cos (cx) -~ • {exp (-!.<ibc/a) D_z [(2a) (b-ic»)

Re z > 0 _k +exp (!.<ibc/a)D_ z [(2a) 2(b+ic»)}

5.33 e-ax-b2/x ibz{(a+ic)-lzzK [2b(a+ic)lz) z

'sin (cx) -(a-ic)-lzzK [2b(a-ic)lz)} z

co

CP(x) <!>(z) = f CP(x)xz-ldx 0

5.34 -ax-b 2/x bZ{(a+ic)-~zK [2b(a+ic)~1 e z

'cos(cx) -~z ~ + (a-ic) K [2b (a+c) l} z

5.35 -ax~

sin (bx) i(2b)-zr(2z){exp[-i(~uz+Y8a2/b)1 e

Re z > -1 "D 2 [~ab-~(1-il-exp[i(~uz+l/8a2/b)1 - z

'D_2Z[~ab-~(1+i)1}

5.36 -ax~

e cos (bx) (2b)-zr(2z) {exp[-i(~uz+l/8a2/b) 1

Re z > 0 "D 2 [~ab-~(1-i)1+exp[i(~uz+l/8a2/b)1 - z

"D [~ab-~(l+i)l} -2z

5.37 sin [b (a 2+X2) ~l 1 ~ 1 -~u~a(2a/b) z-~r(~z)

0 <Rez<l "[Y~+~z(ab)sin(~uz)

-J~+~z(ab)cos(~uz)l

5.38 cos[b(a2+x2)~1 -~u~a(2a/b)~z-~r(~z)

0 <Rez<l • [J~+~z(ab)sin(~uz)+Y~+~z(ab)cos(~uz)l

ro z-l q,(x) <l> (z) = f q,(x) x dx

0

5.39 (a 2+x 2)-1; Y- ~z 1; (1;rrb/a) 2 (2a/b) r (1;z) J, , (ab) ~-~Z

.sin [b (a 2+x2) 1;] 0 < Re z < 2

(a2+x2) -1; ;." 1::z 5.40 -1;(1;rrb/a) 2(2a/b) 2 r(1;z)Y, , (ab)

~-~z

.cos [b (a 2+x2) 1;] 0 < Re z < 2

sin [b (a 2_x 2) 1;] 1.::: kz 5.41 x < a 1; ( 1;rr ab) 2 ( 2 a/b) 2 J, + ' ( ab )

~ ~z

0 x > a Re z > 0

5.42 cos [b (a 2_X 2) 1;] x < a ~z-!:z

-a (a/b) s 1;z- 3/2 , 1;z+1; (ab)

0 x > a Re z > 0

(a 2_x 2)-1; 1.: 1,,:z 3/ 5.43 1;rr 2(2a/b) 2 - 2r (1;z)H, , (ab)

;.zz-::-z

>< .sin[b(a2-x2) 2] x < a Re z > 0

0 x > a

(a 2_x 2)-1; k !'::;z 5.44 1;(1;rrb/a) 2(2a/b) 2 J, , (ab)

::-ZZ-Yz

.cos [(a 2_x 2) 1;] x < a Re z > 0

0 x > a

50

<jl (x)

5.45 o x < a

x > a

5.46 o x < a

Re z < 1

5.47 o x < a

x > a

5.48 o x < a

x > a

5.49 -u e cos v


00

~(z) = f <jl(x)xz-ldx o

·K~Z+~ (ab) Re z < 1

(kab) ~ (a/b) ~z [2~z-lr (kz) I (ab) 2 2 -~-~z

, -i2!:(z-l) +2':2e 4 s, 3 , +' (iab) J

~z- ;/2, ~z Yz

!-,; ~z-!-.. ~7f 2(~a/b) 2r (~z) [I, L (ab) -L, ,(ab) J

~-'"2Z ~z-~

Re z < 2

(~7fb/a) ~ (2a/b) ~z [r (l-~z) J -lK, ,(ab) ;.zZ-Yz

Re z < 2

Re z > 0

¢ (xl

5.50 -u . e Sln v

5.51 sin(a/x)sin(bx)

-2 < Re z < 2

5.52 sin (a/x) cos (bx)

-1 < Re z < 2

5.53 cos (a/x) sin (bx)

-1 < Re Z < 2

5.54 cos (a/x) cos (bx)

-1 < Re z < 1

q, (z) z-l = f <j>(x)x dx

o

k z-k -1 (2a) 22 2r (':;+':;z) [r (l-':;z) ]

• (a/b)-,:;z-~KLi [2(ab)':;] "2' z

Re z > -1

kZ ~ 1r(a/b) 2 csc(':;1rZ)

• {J [2 (ab) ':;-J [2 (ab) ':;] Z -z

~1r(a/b),:;zsec(':;1rz)

• {J [2 (ab) ':;]+J [2 (ab) ':;] Z -z

kZ ~1r(a/b) 2 sec(':;1rz)

• {J [2 (ab) ':;]+J [2 (ab) lz] z -z

kZ ~1r(a/b) 2 csc (lz1rz)

~ k • {J [2 (ab) 2]_J [2 (ab) 2] . -z Z

+21r- l sin(1rz)K [2 (ab) lz]} z

51

'" ¢(x) <!>(Z) = f ¢(x)xz-ldx

0

5.55 sin [a(x-b 2 Ix) ] 2bzsin(y,~Z)Kz(2ab)

-l<Re Z < 1

5.56 sin[a(x+b 2/x)] y,TTbzsec (Y,TTZ) [J (2ab) +J (2ab)] Z -z

-1 < Re Z < 1 -l<Re Z < 1

5.57 cos [a(x-b 2 /x)] 2bzcoS(Y,TTZ)K (2ab) Z

-l<Re Z < 1

5.58 cos[a(x+b 2/x)] y,TTbzcsc(Y,TTZ) [J (2ab) -J (2ab) 1 -z Z

-l<Re Z < 1

5.59 (l+x2) -Y, 3/ 1

- (Y,TT) 2 a'i [J -\-Y,Z (a)Y 3/4+y,Z (a)

-1 'sin[a(x+x )] +J 3,"4+y,Z (a) Y -\-y,z (a) ]

-l<Re Z < 2 0/ k:

= (Y,TT) 2a '[J _ 3/4-y,Z (a) J\+y,z (a)

-Y_3/4-y,Z (a)Y\+y,z (a)]

5.60

5.61

5.62

5.63

5.64

<P{x)

-1 'cos[a{x+x l]

-1 < Re z < 2

2 } -~ {l-X x <

-1 ·sin[a{x-x }]

0 x >

{1-x 2 } -~ x <

-1 'cos[a{x-x }]

1

1

1

o x > 1

o x < 1

'sin[a{x-x-1 }] x > 1

o x < 1

'cos[a(x-x-1 }] x > 1

00

¢(z) = f <jJ{x) x z - 1dx o

Re z > -1

h {~1Ta} 211 1 {a}K 1 {a}

~z-Yz ~z

Re z > -1

h {~1Ta} '1 1 1 {a}K 1 {a}

~-~z '"2Z

Re z < 2

h {~1Ta} 'I 1 {a}K 1 1 {a}

-;;zz Yzz-~

Re z < 2

53

54

5.65

5.66

5.67

5.68

q,(x)

(1+X2)-~sin(2ax ) 1+x2

'exp(- ~) 1+x2

-1 < Re z < 2

(1+x 2) -~cos (2ax ) 1+x2

'exp(- ~) 1+x2

O<Rez<l

0, x < 1

(x2_1)-~sin(2ax x 2 -1

0, x < 1

2 1 -~ 2ax (x -) cos (----) x 2 -1

x > 1


00

~(z) = ! q,(X)Xz-ldx o

-~ -b 2 (lTa) e r (l-~z)f (~+~z)

-~ -b ~(lTa) e r(~-~z)r(~z)

Re z < 2

Re z < 1

5.69

5.70

5.71

5.72

</>(x)

(1+x2) -~cos ( 2ax) 1+x2

1-x 2 'exp [-b (--) ]

1+x2

O<Rez<l

1-x 2 'exp [-b (--)]

1+x2

-1 < Re z < 2

o x < 1

(x 2-1) -lzcos( 2ax) x 2 -1

x 2+1 ·exp[-b(----)], x > 1 x 2 -1

o x < 1

x > 1

00

$(Z) = f </>(x)x Z- 1dx o

.n {2~[(a2+b2) lz+b]lz} z-l

'M [(a 2+b 2) lz_b] ~-~z,-~

Re z < 1

55

56

5.73

5.74

5.75

5.76

 1

1+x 2 ·exp[-b(----)], x < 1

l-x 2

o x > 1


00

~(z) = f <P(x)xz-ldx o

Re z > 0

Re z > -1

00

<P(x) <P (z) = f <P(x)xz-ldx 0

5.77 log x sin (ax) -z a r (z) sin (J,rrz) [1jJ(z) -log a

-1 < Re z < 1 +J,rrcot (J,rrz)]

5.78 cos (ax) log x -z a r(z)cos(J,rrz) [1jJ(z)-log a

0 < Re z < 1 -J,rrtan (J,rrz) ]

5.79 -bx sin (ax) r (z) (a 2 +b 2 ) -J,zsin (z arctan (a/b)] e

·log x • {1jJ(z)-J,log(a 2+b 2)+arctan(a/b)

Re z > -1 • cot [z arctan(a/b)]}

5.80 e -bx cos (ax) r (z) (a 2 +b 2) -J,zcos [z arctan (a/b) ]

·log x • {1jJ(z)-J,log(a 2+b 2)-arctan(a/b)

Re z > 0 ·tan[z arctan (a/b) ] }

5.81 -x

sin (x+ax2) (2a)-J,zr(z)e-2/asin(~rrz) e

Re z > -1 .D (a-J,) -z

5.82 sin(a log x) x < 1 -a(a 2 +z 2 ) -1

0 x > 1 Re z > 0

5.83 -x

sin(a log x) I r (z+ia) I sin [argr (z+ia) 1 e

Re z > 0

00

z-l ¢ (x) <l> (z) = f ¢(x)x dx 0

5.84 -x

cos (x+ax 2 , (2a)-~zr(z)e-2/acOs(~TIz) e

Re z > 0 °D (a-~) -z

5.85 costa log xl x < 1 z(a 2 +z 2 )-1

0 x > 1 Re z > 0

5.86 -x cos (a log xl I r (z+ia) Icos [argr (z+ia) 1 e

Re z > 0

5.87 arcsin (x/a) x < a ~TIZ -laz{I_TI-~r (~+~z) [f(l+~z) ]-l}

0 x > a Re z > -1

5.88 arccos (x/a) ~ -1 z -1 x < a ~TI z a r (~+~z) [r (l+~z) ]

0 x > a Re z > -1

5.89 arctan (ax) -~TIz-la-zsec(~TIz) -1 <Re z < 0

5.90 arccot(ax) ~TIz-1a-zsec(~TIz) 0 < Re z < 1

5.91 arctan (x/a) x < a ~z-laz[TI+1J!(~+~z) -1J!( 3/4+~Z)]

0 X > a Re z > -1

00

z-l <jl (xl a Re Z > 0

5.93 -x arctan(ae ) Re Z > 0 2-z-1r(z)aY(-a2,z+1,~)

5.94 (a2+x2)-~V z-v -1 . 1f) a [r(v)] r(z)r(v-z)sln(Iz

osin[v arctan (x/a) ] -1 < Re z < Re v

5.95 (a2+x2)-~V z-v -1 1f a [rev)] r(z)r(v-z)cos(Iz)

'cos[varctan(x/a)] 0 < Re Z < Re v

5.96 (a2_x2)-~ 1f2-z a z - 1 r (z)

·cos[v arccos (x/a) ] .[r(~+~z-~V)r(~+~z+~V)]-l

x < a Re z > 0

0 x > a

5.97 0 x < a 1f2 z-la z-l r (l-z)

(x 2_a 2) -~ • [r(l-~z-~V)r(l-~z+~V)]-l

'cos[v arccos(a/x)] Re z < 1

x > a

00

z-l <jJ (x) (z) = ! <jJ(x)x dx 0

5.98 arcsin[a(a2+x2)~] -1 z sin (~7fz) z a

0 < Re z < 1

5.99 arccos[x(a2+x2l-~] -1 z sin(~7fzl z a

0 < Re z < 1

5.100 (a2_x2)-~ 7f2- za z- l r (zl

cos [\I arccos(x 2ja 2-ll] . [r (~+\I+~zl r (~-\l+~zl ]-1

x < a Re z > 0

0 x > a

1.6 Hyperbolic Functions 61

1.6 Hyperbolic Functions

00

<jl (x) (z) J z-l = <jl(x)x dx

0

6.1 sech(ax) a-z21-zr(z)Y(_1,z,~)

= a- z 21 - 2z r(z)

• [r;(z,,.)-r;(z,~")l Re z > 0

6.2 csch(ax) -z -z 2a (1-2 )r(z)r;(z) Re z > 1

6.3 tanh (ax) 21-za-z(21-z_1)r(z)1;(z)

-1 < Re z < 0

6.4 I-tanh (ax) 2 (2a) -zr (z) (1_2 1 - 2 ) 1; (z)

Re z > 0

6.5 coth(bx)-l 2 (2b) -zr (z) 1; (z) Re z > 1

6.6 -1 -csch x -z -1 1 x 2(2 -l)r(z)(;(z) < Re z <

6.7 sech 2 (ax) 2-z -z 2-z 2 a (1-2 )r(z)(;(z-l)

Re z > 0

6.8 CSCh2 (ax) 2-z -z Re z 2 2 a r(z);:;(z-l) >

00

z-l ¢ (x) b • [~(z,~+~a/b)-~(z,3~+~a/b)1

Re z > 0

6.10 -ax csch (bx) -z e 2(2b) r(z)~(z,~+~a/b)

-a > b Re z > 1

6.11 -ax e tanh (ax) 21-2zb-z r (z)

• [~(z,~a/b)-~(z,~+~a/b)l

-a-zr(z) Re z > -1

6.12 -ax coth(bx) b -zr (z) [21-z~ (z ,~a/b) - (b/a) zl e

Re z > 1

6.13 [coth(bx)-l]e -ax 2(2b)-zr(z)~(z,1+~a/b) Re z > 1

6.14 sinh(ax)csch(bx) (2b) -zr (z)

a 0

1.6 Hyperbolic Functions 63

00

z-l 1

6.16 sinh (ax) sech (bx) -2z -z 2 b r(z) [;;:(z,lo-loa/b)

a -1

6.17 cosh(ax)sech(bx) -2z -z 2 b r(z) [1; (lo+loa/b)

a 0

6.18 [cosh (ax) +cos e] -1

lz(2n/a)zcsc6csc(lznz)

-n < 6 < n 6 6 • [I:; (l-z ,J.;-2n) -I:; (l-z ,lz+2n)]

Re z > 0

6.19 cosh (lzax) [cosh(ax)+cos6] -1 nZ22z-3a-zsec(lz8)csc(lznz)

-n < 6 < n • [I:; (1-z,lo+\8/n) +1:; (1-z,lo-lo6/n)

-I:; (l-z, 3 .... 4 +lo6/n) -I:; (l-z, 3"-4-\6/n)]

Re z > 0

64

6.20

6.21

6.22

6.23

6.24

¢ (x)

-1 sinh (~ax) [cosh (ax) +cos8]

-7f < 8 < 7f

-ax -1 e [cosh(ax)+cos8]

-7f < 8 < 7f

a > b

log [tanh (ax)]

arctan[sinh(ax)]


'" ~(z) = J ¢(x)xz-ldx o

• [<; (l-z, ' .... ,+\6/7f) -<; (l-z, ~,-\8/7f)

+<;(1-z,\-\8/7f)-<;(1-z,\+\8/7f)]

Re z > 0

(27f/a)zcsc8CsC(7fZ)

• [cos (8+~7fz)] (l-z ,~+~8/7f)

Re z > 0

o [<;(-z,\-\a/b)-<;(-z,\+\a/b)

+r; (-z, ~,+\a/b) -<; (-z, ~-\a/b) ]

Re z > 0

-1 z 7fZ csc(~7fz) (~7f/a)

Re z > '0

-7fz-lsec(~7fz) (27f/a)z

• [<; (-z, ~,) -<; (-z,\)]

-1 < Re z < 0

6.25

6.26

6.27

6.28

<jl (xl

(l+x2) -~sinh (2ax ) 1+x2

'exp(- ~) 1+x2

-1 < Re z < 2

(1+x2)-~COSh(2ax ) 1+x2

.exp(-~) 1+x2

O<Rez<l

0, x < 1

(x2_1)-~sinh(2ax ) x 2 -1

b > a

0, x < 1

(x2_1)-~cosh(2ax ) x 2 -1

b > a

x > 1

x > 1

00

~(z) = f <jl(x)x z - 1 dX o

-!z -b 2 (1Ta) e r(l-~z) r(~+~z)

Re z < 2

Re z < 1

65

66

6.29

6.30

6.31

6.32

<P(x)

(1+x2) -lzcosh (2ax ) 1+x 2

l-x2 'exp [-b (--) 1

1+x2

O<Rez<l

(1+x2) -lzsinh ( 2ax) 1+x 2

l-x 2 'exp[-b(--) 1

1+x 2

-1 < Re z < 2

o x < 1

(x 2_l)-lzcosh(2ax) x 2 -1

b > a

o x < 1

(x 2_l)- lzsinh(2ax) x 2 -l

x 2+l 'exp [-b (--) 1 x 2 -1

b > a


<!>(z) z-l

= f <p(x)x dx o

Re z < 1

Re z < 2

6.33

6.34

6.35

6.36

<jl(x)

(1+x2)-~cosh( 2aX) 1+x 2

l-x 2 'exp [b (--) 1

1+x2

O<Rez<l

(1+x2)-~sinh( 2ax) 1+x 2

l-x 2 °exp [b (--) 1

1+x2

-1 < Re z < 2

(l-x 2) -~cosh( 2ax) l-x 2

l+x2 "exp [-b (--) 1,

l-x 2

0

b > a, Re z

(l-x 2 ) -\inh( 2ax) l-x 2

l+x2 "exp [-b (--) 1

l-x 2

0

b > a

x <

X >

> 0

x <

X >

1

1

1

1

00

~(z) = f <jl(x)xZ-ldx o

67

"D {2~[b+(b2-a2)~1~} -z

oM" ,[b- (b 2 _a 2 ) ~l ~-~z,~

Re z > -1

1.7 The Gamma Function and Related Functions

00

z-l cjJ (x) <P (z) = f cjJ(x)x dx

0

7.1 y+\jJ (x+l) -TrCSC (Trz) ;;; (l-z) -l<Re z<O

7.2 \jJ' (l+z) Tr(1-z)csc(Trz);;;(2-z) O<Re z<l

7.3 .\jJ(l+x)-log x -Trcse (Trz) i',; (l-z) O<Re z<l

7.4 \jJ' (l+x) - (l+x)-l -1 -Tr(l-z)cse(Trz) [(z-l) -;;;(2-z)]

O<Re z<2

7.5 log (l+x) -\jJ (l+x) -1

Trese (Trz) [;;; (l-z) +z ] O<Re z<l

7.6 \jJ h+~x) -log (~x) z-l

2csc (Trz) (2 -1) ;;; (z) O<Re z<l

7.7 \jJ (a+x) -\jJ (b+x) TrCSC (Trz) [;;; (l-z,b) -;;; (l-z,a)]

O<Re z<l

7.8 log[x ~ f (x)

f (~+x)] -z -2z

~Tr 2 see(~Trz)f(z)

• (2 1 + z _1) I:; (l+z) O<Re z<l

7.9 ;;;(v,a+ax) -1 -z

[f(v)] a f(z)f(v-z)l:;(v-z,a)

Re a > 0, Re v > 1 0 < Re z < -1 + Re v

1.8 Legendre Functions


CP(x)

8.1 x < 1

x > 1

8.2 P v (a+x) x<l-a

o x>l-a

-1 < a < 1

8.3 qv (a+x)

a > 1

o < Re z < l+Re v

8.4 pv (a-x) x < a-I

o x > a-I

a > 1

8.5

larg(a-l) I < n

00

~(z) = f CP(x)x Z- 1dx o

69

n~2-zr(z) [r(~-~v+~z)r(l+~v+~z)l-l

Re z > 0

Re z > 0

r(z) (a2_1)~ZeinZq~z(a)

= (~n)~r(z)r(V+l-z) (a2_1)~z-'"

2 ~z -z (a -1) r (z)pv (a)

= (~n) -~r (z) (a 2_1) ~z-'"

in(v+~) -v-~[ ( 2 1)-~1 ·e qz-~ a a -

Re z > 0

o < Re z < l+Re v

70

$(x)

8.6 pv (x+a)

a > 1

-Re v O<Rez<{

l+Re v

8.7 o x < l+a

o <Re z < { Re (ll-V)

8.8 (a2_x2)-~1l

opll(x/a) v x < a

0 x > a

8.9 Pv(2x 2 /a 2 -l) x < a

0 x > a


00

~(z) = f $(X)Xz-ldx o

:k _ 3,." = _22TI 2s in(TIv)r(z)r(1+v-z)

f(ll-V-Z) r (l+ll+V-Z) [r(l-z) ]-1

Re z > 0

-1 • [r (l+v+~z) r (~z-v)]

Re z > 0

<j,(x)

8.10 {a2_x2)-~]l

.p]l (2x2/a2-1) x < \I

0 X >

8.11 {a 2+x2)\1

]l a 2-x 2 .p (--)

\I a2+x2

Re ]l<Re z < Re (2\1±]l)

8.12 {a2+x2)~\1

2a 2+x 2 .p ( J)

\I 2a (a 2+X2) 2

8.13 P (l+!;;x 2/a 2) \I

1+\1 0 < Re z <Re {

-\I

8.14 q (l+~x 2/a 2) \I

0 <Re z < 1+2 Re \I

a

a

co

~(z) = f ¢{x)xz - 1 dx o

71

Re z > -Re ]l

• r (~]l-\l-~z)

-2 Re \I

O<Rez<{ 1+2 Re \I

·r{~z)r{l+\1-!;;z)r{-\l-~z)

• [r{l+V+~z) ]-1

72

8.15

8.16

8.17

8.18

q, (x)

-2 Re v O<Rez<{

2+2 Re v

o < Re z < 2+2 Re v

-l-Re v < Re z < l+Re v

o x < a+b

-1 2 2 2 pv[(2ab) (x -a -b )]

x > a+b


1>(z) z-l

= f q,(x)x dx o

-1 -~n sin(nv)r(~z)r(l+v-~z)

-~,., k = -n "sin(nv) (ab) 2r (y,z)

-1 y,[r(l-y,z)] r(l+v-y,z)r(-v-Y,z)

k -1 = (ab/n) 2[r(1-y,z)] r(l+v-Y,z)

8.19

8.20

8.21

 (x)

(A2-1) J,J.lpJ.l(A) v

A = (2ab)-1(a 2+b 2+x 2)

O<Rez<{ 2+2 Re (v-J.l)

-2Re (v+J.l)

e-inJ.l(A2_1)-J,J.lq J.l(A) v

A = (2ab)-1(a 2+b 2+x 2)

o < Re z < l+Re (v+J.l)

-Re v O<Rez<{

l+Re v

(z)

• -v-J, (a 2+b 2 ) PI I

'iZ-J.l-'i I a 2_b 2 1

= J,(2ab) J.ll a 2_b 2 1J,z-J.lr (J,z)

r (l+v-z) r ( -v-z)

• F (l+v-z,-v-z;l-].l-Z;J,) 2 I

73

74

8.22

8.23

8.24

8.25

q, (x)

.p~(~+x/a)

o < Re Z < l-Re ~

-Re (2\J+~) Re~<Rez<{

2+Re (2\J-~)


00

~(z) = f q,{x)xZ-ldX o

F (1- z+!zv- ~ll, !z-z- !z'J- !zl1i 1-z -11 r-1 2 1

Re{~-2\J) Re~<Rez<{

2+Re (~+2\J)

± Re ~ < Re z < 2+Re{~+2\J)

<P(x)

8.26 o x < a

Re]1<l

8.27 0 x < a

(x 2_a 2)-Y,]1

.p]1 (2x 2 /a 2_1) x > a v

Re ]1 < 1

8.28 0 x < a

(x 2_a 2 )-Y,

op]1 [(x2/a2_1) y,] v

x > a

8.29 o x < a

x > a

00

W(z) = ! <P(X)xz- 1dx o

75

Re z < Re (]1-V, ]1+v+ 1)

o [r (l+Y,]1-y,z) r (l-y,]1_y,z)]-l

Re (]1-2v) Re z > {

Re (]1+2 v+ 2)

l-Re v -Re ]1 < Re z < {

2+Re v

± Re ]1 < Re z < 2+Re v

76

8.30

8.31

8.32

</lex)

.p]l (l+x) v

• (a+x) -Pp]l (l+x) v

Re (p-]l-v) O<Re z<{

l+Re (v+P-]l)

X-~]l(2+x)-~]l

• (a+x) -Pp]l (l+x) v

Re (P+]l-v) Re ]l<Re z<{

l+Re (P+]l+v)


00

~(z) = f </l(x)xz-ldX o

2 ]l+zr (z) r (-]l-V-z) r(l+v-]l-z)

• [r(l-]l+v) r(-]l-v) r(l-]l-z) ]-1

-Re (]l+v) O<Rez<{

l-Re (]l-v)

2]l-P+Z r (z-p) r (p-V-]l-z) r (l+P-]l+v-z). r(l+v-]l)r(-v-]l)r(l+p-]l-z)

p,P-V-]l-z,l+p+v-]l-Z; • F ( ~a)

3 2 p+l-z,l+p-]l-Z;

Z,-]l-v,l+v-]l; • F ( ~a)

3 2 l-p+z, l-]l;

[r (p) r (l-]l) ]-lr (z-]l) r (P+]l-z) aZ-]l-P

Z-]l,-v,l+v; • F ( ~a)

3 2 l-]l-p+z,l-]l;

-TI- l sin(TIv)2 Z- P-]l[r(1+p_z)]-1

'r(z-]l-p)r(p+]l-V-z)r(p+]l+v+l-z)

P,P+]l-V-Z,p+]l+v+l-z; • F ( ~a)

3 2 l+p-z,l+p+]l-Z;

8.33

8.34

8.35

8.36

8.37

¢(x)

Xlz11(2+X)lzl1

'e -axp 11 (l+x) v

Re z > 0

x- lz11 (2+x)-lzl1

'e -axp 11 (l+x) v

Re z > Re

xlz11(2+x)lzl1

-ax q11(l+x) 'e v

x- lz11 (2+x)-lzl1

·e-aXq 11 (l+x) v

o Re z > {

Re 11

11

'pA(l+ax) x b

Re 11 > 0, ab < 1

00

~(z) = f ¢(x)xz - 1dx o

[r(-l1-V)r(l-11+V)]-l a - 11 -ze -a

31 I 1+11, 1 'G23 (2a )

l1+ Z ,-v,l+v

31 I 1, 1-11 'G23 (2a

z-l1,l+v,-v

22 I 1+11, 1 • G2 3 (2 a )

z+l1,l+v,-v

'G22 (2a 23 I 1-11, 1

Z-l1,l+V,-V)

77

.r(z_lzA)b11 - lzA -l+z F (-v,l+v, 3 2

z-lzA;l-A,l1-lzA+z;-lzab)

Re z > lz Re A

(z) z-l

= f <p(x) x dx o

8.38 (1-x 2 ) -~j.l

apj.l (x) \)

x < 1

0 x > 1 Re z > 0,

8.39 (1_x 2 )K

apj.l (x) \)

x < 1 . [r (1-j.l) r (l+K-~j.l+~z) ]-1

0 x > 1

Re(K-~j.l»-l, Re z > 0

~+~v-~~,-~v-~~,l+K-~~i • F ( 1)

3 2 I-j.l,l+K-~j.l+~z;

8.40 (1-x 2 ) -~j.lsin (ax)

.p j.l (x) \)

x < 1

0 x > 1

Re z > -I, Re j.l < 1

~+!zz,l+~z; a F ( -loa 2

2 3 3/2,1-~\)-~1l+~Z, 3/2+~V-!2'\.1+~Zi

8.41 ~ j.l-z -1 1T 2 r (z) [r (l+~\)-~11+~Z) r (~-~\)-~11+~Z)]

1

(1-x 2 ) -Yz11cos (ax)

x < 1

o x > 1

~z, ~+!zZ; a F ( -loa 2)

2 3 ~,~-~\)-~ll+~z,l+~v-~ll+~Zi

Re z > 0, Re 11 < 1

q, (x)

x < 2

x > 2

8.43 (I-x) -~

-+ P~[-(I-x)~l}x < 1

o x > 1

1 1

8.44 (1-x)-~(I-a2+a2x)~lJ

• {p v [a (l-x) ~l

+Pv [-a(l-x) ~l}x < 1

0 x > 1

8.45 P~(x) x < 1

0 x > 1

Re lJ < 2, Re z < 2

8.46 [ (b+x)2_11~lJ

'q-lJ (b+x) v

79

'" ~(z) = f q,(x)x z- 1 dX o

2 Zr 2 (Z) [r (Z-V) r (l+v+z)]-l

Re z > 0

• [r (~+~v+z) r (z-~v) r (1-~lJ+~v)

.r(~-~lJ-~v)l-1

Rez>~IRelJl

• F (-~V-~lJ,~+~v-~lJ;~+z;a2) 2 I

-1 < a < I, Re z > 0

• F 3 2

o < Re z < I+Re(v-lJ)

80

8.47

8.48

8.49

8.50

q, (x)

.pll [a(1+x) l-z] v

1+l-zRe (V-Il) O<Re z<{

l-z-l-zRe (V+Il)

-i1T1l Jl k .e q [a(1+x) 2] v

O<Re z<l+l-zRe(v+jl)

Re z > 0

.pA(1-2x) \!

o

x b

ReJl>O, b<l


'" ~(z) = f q,(x)xz-ldx o

F (l-z+l-zV-l-z1l,1+l-zv-l-zIl-Z;1-Jl-z;1-a-2 ) 2 1

-1 [r(l-A)r(jl-l-zA+z)] r(jl)

·r (Z_l-zA)bjl-l-l-zA+z. F (-v,1+v, 3 2

Re z > -l-z Re A

8.51

8.52

8.53

8.54

8.55

q, (x)

e-i7f)lq)l[(1+x 2/a 2 ) ~l v

±2Re )l < Re z < ~+Re v

-i7f)l )l 2 2 ~ 'e q [(l+x/a) 1 v

±Re )l < Re z < 3-1+Re v

00

¢(z) = f q,(x)x Z- 1dx o

o < Re z < l+Re (v+)l)

Re )l < Re z <f-Re v b+ Re v

81

2)l-laz-1r(~z-~)l)r(1-~z+~v)r(~-~z-~v)

·[r(l-~)l-~z)r(l-~)l+~v)r(~-~)l-~v)l-l

(l-Re v

Re)l<Rez< 2+Re v

'r(-~)l-~z) r(~)l-~z) [r(l+~v+~z)

.r(~+~v-~)l)r(~+~v+~)l)l-l

82

8.56

8.57

8.58

8.59

8.60

¢ (x)

<{-2 Re v 2Re II < Re z

2+2 Re v

ei2rrll{qll[(1+x2/a2)~1}2 v

±2Re II < Re z < 2+2Re v

(a2+x2)-~pll[(1+x2/a2)~1 v

.pll [(l+x2/a2)~1 -v

2Re II < Re z < 2±2 Re v

p~ [(l+x2 /a 2) ~l

'eirrllq-ll[(1+x2/a2)~1 v

1+2Re II < Re z < 2+2Re v

o }< Re z <

2Re II

1 {

2+2Re v


<!> (z)

-1 • [r (~+~z) r (l+v+ll) r (l+v+~z) 1

-1 • [r(l-~z)r(l-Il-~Z)r(l-Il+V)r(l-Il-V)l

-1 • [r(l-ll-~Z)r(l+v+~z)l

• r(l+v-~z)r(l+v+ll)

• [r(l+V-ll)r(l+v+~z)r(l-~Z-Il)]-l

1.9 Orthogonal Polynomials


<P(x)

9.1 e-~en(x)

Re z > 0 if n even

Re z > -1 if n odd

9.2 e -a~e [(2x) lz] n

Re z > 1 if n even

Re z > lz if n odd

9.3 e-ax[He (xlz)]2 n

Re z > 0

9.4

·Hen (Sx)Hen (yx) •••

co


[!zn] n! L

m=O

-1 [m! (n-2m) !]

hZ(a)n!r(z) where

co

= L h~(a)tn n=O

See Appel, P. and

83

M.J. Kampe de Feriet,

1926: Fonctions hypergeo

mitriques et hyperspheri-

ques. Polynomes

d'Hermite. Gauthiers-

Villars, p. 343.

Erdelyi, A., 1936: Math.

z. 40, 693-702.

00

z-l ¢ (x) $(z) = f ¢(x)x dx 0

9.5 -k

(1-x 2 ) 2Tn (x) x < 1 Il2- z r (z) [r (l~+~z+lm) r (~+~z-lm) ]-1

0 x > 1 Re z > 0

9.6 (1-x2 ) -~T (l/x) x n < 1 2z-2[r(z)]-lr(~n+~z)r(~z-lm)

0 x > 1 Re z > 0

-k Il2~+2n-z(n!)2r(2z) 9.7 (2-x) 2Tn (x-I) x < 2

0 x > 2 • [(2n) ! r (~+z+n) r (~+z-n) ] -1

Re z > 0

9.8 k 1T2~+2n-z[(n+l) !]2 r (2z-1) (2-x) 'Un (x-I) x < 2

0 x > 2 [(2n+2) !f(Z+3/2+n ) r(z-~-n) ]-1

Re z > -~

9.9 (2-x) vT (I-x) x < 2 22n+v+z(n!)2 r (1+V)r(z)

n (2n) !r(l+v+z)

0 x > 2 • F (-n,n,z;~,l+v+z;l) 3 2

Re z > 0, Re v > -1

9.10 (2-x) Vu (I-x) x < 2 2l+2n+v+z [(n+l) ! ]2r (l+v) r (z)

n (2n+2) !f (l+v+z)

0 x > 2 • F (-n,n+l,z; o/2,1+v+z;1) 3 2

1.9 Orthogonal Polynomials 85

00

z-l <jJ (x) <l> (z) = f <jJ(x)x dx 0

9.11 (2-x) \)T (x-l) x < 2 22n+\)+z (n!) 2 r (1+\) r (z)

n (2n) !r(1+\)+z)

0 x > 2 F ( -n, n, 1 + \) ; J" 1+ \)+ Z ; 1) 3 2

Re \) > -1, Re z > 0

9.12 (2-x) \)U (x-l) < 2 22n+1+\)+z [(n+l) !]2 r (1+\) r (z)

x (2n+2) ! r (l+\)+z) n

0 x > 2

Re \) > -1, Re z > 0 F (-n,n+l,l+\); %,l+\)+z;l) 3 2

9.13 (2-x) -~ rrJ,2 z -J,r(z)r(J,+n-z) [r (J,-z) r (J,+n+z) 1 -1

Tm (l-x) Tn (l-x) . F (-m/m,z/~+zi~,~+n+z,~-n+zil) ~ 3

Re z > 0

( -1) n ( J,- J,z) 9.14 P 2n (x) < 1 n

Re > 0 x 2 (i,z)n+l

z

0 x > 1

(-1) n (l-J,z) n

Re > -1 9.15 P2n+l (x) < 1 z x 2 (J,+J,z) n+l

0 x > 1

9.16 (2-x) \)p (x-l) x < 2 [f(1+\)+z)]-12 \)+zr(\)+1)r(z) n

0 x > 2 F (-n,l+n,\)+l;l;\)+l+z;l) 3 2

Re \) > -1, Re z > 0

00

z-l ¢(x) (z) = f ¢(x)x dx 0

9.17 (2-x) vp (I-x) x < 2 [r (l+v+z) ]-12 v+zr (v+ 1) r (z) n

0 x > 2 . F (-n,l+n,z,l,l+v+z,l) 3 2

Re v > -1, Re z > 0

9.18 P (1-2x 2) n x < 1 ~(_1)nr2(~z) [r(~z+n)r(~z-n)]-l

0 x > 1 Re z > 0

9.19 Pn (I-x) x < 2 z -1 2 r (z) r (l+n-z) [r (l-z) r (l+n+z)]

.Pm (l-z) F (-m,m+l, z, z, 1 ,1+n+z, I-n+z, 1) 4 3

0 X > 2 Re z > 0

9.20 (1_x2)v-~ x < 1 2 2v- 1 - z r(n+2V)r(z)

ocv (x) -1 n • in! r (v) r (~+~n+v+~z) r (~+~z-~)]

0 x > 1 Re v > -!z , Re z > 0

9.21 (1_x2)v-~ x < 1 2 z - 1 r(n+2v) r (~z-~n) r (~z+v+~n)

'Cv (l/x) -1 n • in! r (v) r (z+2v)]

0 x > 1 Re v > -!z, Re z > 0

9.22

9.23

9.24

9.25

9.26

<P(x)

Pn(l-yb+yx) x 0,

(2-x) V-J,

·cv (l-x) n

o

x > b

x b

x < 2

x > 2

x < 2

x > 2

Re 13 > -1

x < 2

x > 2

'" r z-l <P(z) = , tj>(x)x dx o

-1 z n! [r(n+1+z) 1 r(z)b

p(z,-z) (l-yb) n

Re z > 0

• F (-n,n+l,z;l,]J+z;J,yb) 3 2

He(]J,z) > 0

87

• [nlf(2V)f(J,-V-n+z)f(J,+V+n+z)]-1

He v > - J" He z > 0

f(1+S)2 13+z f(n+2v)f(z)

• [n! f (2v) r (1+13+z) ]-1

• F (-n,n+2v,z;J,+V,1+13+z;1) 3 2

v+z-~ 2 'f (y,-I-V) r (m+2]J) r (n+2v) r (z) r (J,+v+n-z) mlnlr(2v)r(2]J)r(J,+v-z)r(5~+v+n z) .

-IlI,m+2]J,z, J,-v+z; • F ( 1)

~ 3 J,+v+n+z,J,+]J,J,-v-n+z;

He z > 0, Re v > -J,

88

¢ (x)

9.27 (b-x) \1-1

A • Cn (l-yx) x b

2A+O,-1,-2,'"

9.28 (b-x) \1-1

'C A (yx'-Z) 2n x b

9.29 (b-x) \1-1

A " 'C2n+l (yx 2) x b

Re \1 > 0

9.30 (b-x) A-l:;

A • Cn (l-yb+yx)

x b

Re A > -1, 2A+0,-1,-2," •


00

¢(z) = f ¢(x)xz-ldx o

F (-n,n+2A,z;l:;+A,\1+z;l:;yb) 3 2

Re(\1,z) > 0

F (-n,n+A,z;l:;,\1+Z;y2b ) 3 2

Re (\1,Z) > 0

Re z > -rz

(2A) r (l:;H) [r (l:;H+n+z) ]-1 n

r (z) bA-l:;+zP (a, S) (l-yb) n

a = A-l:;+z, S = A-l:;-z, Re z > 0

9.31

9.32

9.33

9.34

9.35

 0

x b

x b

x b

00


[n!f(l+a-z)]-lf(a+n+l-z)f(Z)

Re z > 0

-1 [(11) [n!f(l+a)f(l1+z)] f(z)

f(a+n+l)b z + I1 - 1 F (-n,z;a+l, 2 2

l1+z;Sb)

89

Re(I1,Z) > 0

-1 f (l+a+n) [nlf (l+a) f (Hz)]

'f(A)f(z)bHz- l

• F (-n,A;l+a,A+Z;Sb) 2 2

Re(A,z) > 0

r (l+a+n) [f (l+a+n+z) ]-lba + z

Re a > -1, Re z > 0

-1 • F [-n,l+v-z;l-n-z;a(a-b) ] 2 1

90

9.36

9.37

9.38

9.39

9.40

<jJ (xl

L~(SX)

o

'L~[S(b-X) ]

o

-x a e L 1 (A xl

m I 1

a "'L n (A xl

In n n

(2-x) S

.p (a, S) (l-x) n

0

x b

x b

x < 2

x > 2


00

f z-l ~(z) = <jJ(xlx dx

o

-1 r(l+a+n) [n!r(l+a)r(I1+Z)]

F (a+n+l,z;a+l,I1+Z;-Sb) 2 2

Re(I1,z) > 0

r (l+a+n) [n!f (l+a) r (Hz) ]-1

• F ( 1 + a+n , A ; a+ 1 , A + z ; - Sb ) 2 2

Re(A,Z) > 0

Buchholz, H., 1953:

Die konfluente hypergeometrische

Funktion. Springer Verlag.

Erdelyi, A., 1936:

Math. Z. 40, 693-702.

[n!r(1+a-zlr(1+S+n+z)]-12 S+ z

·r(z)r(l+S+n)r(l+a+n-z)

Re S > -1, Re z > 0

9.41

9.42

9.43

9.44

¢ (xl

(b-x) 13

.p(a,S) (yb-l-yx) n

x b

(b-x) a

.p (a,S) (l-yb+yx) n

x b

(b-x) ]l-l

.p (a,i3) (l-yx) x b

(b-x) ]l-l

.p(a,i3) (yx-l) x b

00

w(z) = f ¢(x)xz-ldx o

[f(S+n+l+z)]-lf(S+n+l)f(z)

Re 13 > -1, Re z > 0

-1 [f(a+n+l+z)] f(a+n+l)f(z)

'ba+zP (a+z, i3-z) (l-yb) n

Re a > -1, Re z > 0

-1 [n If (l+a) f (]l+z) ] f (a+n+l)

91

'f(]l)f(z)bz+]l-l F (-n,l+n+a+i3, 3 2

z; l+a,li+Z; lzyb)

Re(]l,z) > 0

'f(]l)f(z)bz+]l-l F (-n,l+n+a+i3, 3 2

z;l+i3,]l+z;lzyb)

Re(]l,z) > 0

92

9.45

<jl(x)

(b-x) ]1-1 (l-\zyx) 13

p (a, 13) (l-yx) x b


<l> (z) f z-l <jl(x)x dx o

-1 [n!f(l+a)f(]1+z)] f(l+a+n)

'f(]1)f(z)b z+ ll - l F (l+a+n,-n-S, 3 2

z;l+a,ll+z;\zyb)

Re(ll,z) > 0

1.10 Bessel Functions 93

1.10 Bessel Functions

co

4> (x) 4J(z) = f 4>(X)xz- l dx 0

10.1 J v (ax) 1:!(l:!a)-zr(l:!v+l:!z) [r(l+l:!v-l:!z)]-l

-Re v < Re z < ~2

10.2 Yv (ax) -n-12z-la-zcos[l:!n(z_v)]

± Rev <Re z < 3-'2 ·r(l:!z-l:!v)r(l:!v+l:!z)

10.3 -1 (2 ) z-l [r(l:!z+l:!v) Sl_z,v(ab) (a+x) J v (bx) a r (l+z+l:!v)

-Re < Re z < 5/2 -2 r (1:!+l:!z+l:!v) S_z,v(ab)] v r 11:!+l:!v-l:!zj

10.4 (x2-a2) -lJ (bx) z-2 v -I:!na Yv (ab)

Principal value _2(2a)z-2 r (I:!v+l:!z) Sl_z,v(ab) r 11+I:!v-l:!z)

Rev <Re z < ~2

10.5 (x 2_a 2) -ly (bx) I:!naz- 2J (ab)+2z-ln-lcos[l:!n(z-v)] v v Principal value .r(l:!z-l:!v)r(l:!z+l:!v)Sl_z,v(ab)

~2 <Re z <±Rev

10.6 Jv(ax) x < 1 (I:!a)v[(z+v)r(l+v)]-l

0 x > 1 . F (~v+l:!z;v+l,1+~v+l:!z;-~a2) 1 2

Re z > -Re v

00

cjl{x) <l> (z) f z-l = cjl{x) x dx

0

10.7 e-axJ (bx) (a 2 +b 2 ) -,>zf{v+z) v

Re z > - Re V .p-v [a{a 2+b 2 ) -'>1 z-l

v -1 -z = ('>b/a) [f (l+v)] a f (z+v)

. F (,>z+,>v,,>+,,,z+"'v;v+1;-b 2 /a 2 ) 2 1

10.8 exp (_b 2 x 2 ) J)ax) -1 1-z [ar(l+v)] b r(",v+",z)

Re z > -Re v 'exp (_1/0 a 2 /b 2 ) M (laa 2 /b 2 ) 8 ",z-"',"'v

v-1 -z 10.9 sin(ax)Jv(ax) 2 a r("'-z)r(,>+",v+'>z)

r(l+v-z)r(l-",v-,>z)

-l-Re v < Re z < ",

v-1 -z 10.10 cos (ax)Jv(ax) 2 a r(",-z)f(,>v+",z)

r(",-,>v-",z)r(l+v-z)

-Re v < Re z < ",

10.11 sin (ax) Y v (ax) 2z-1TI-~a-zsin[~TI(z-v)]

-l±Re v < Re z < ~ . r(~+~z+~v)r(~+~z-~v) f(l-~z+~v)r(l-~z-~v)

10.12 cos (ax) Y v (ax) 2z-1TI-~a-zcos[~TI{z-v)l

±Re v < Re z < ~ r(~z+~v)r{~z-~v)

r(~-~z+~v)r(~-~z-~v)

+.10 Bessel Function

4>(X)

10.13 cos (ax) J v (bx)

10.14 cos (ax)Jv(bx)

10.15 cos (a-x)Jv(x)

x < a

o x > a

Re z > -Re v

95

00

f z-l ~(z) = 4>(x)x dx

o

= (~TIb)-~cos[~TI(v+z)] (a2_b2)~-~Z

.e-iTI(z-~)qZ-~(a/b) v-~

= cos[~TI(v+z)]r(v+z)

.(a2_b2)-~Zp-V[a(a2_b2)-~] -z

a > b

• F (~v+~z,~z-~v;~;a2/b2) 2 1

= b -~2Z-V-~2r (v+z) r (~+~z-~v)

.[r(1+~v_~Z)]-1(b2_a2)~-~z

• [p~:~(a/b)+p~:~(-a/b)]

a < b

-1 z z (v+z) a J v (a)+2a

~ n -1 • L (-1) [(v+z) 2n+l] (v+l-z) 2n-l n=l

• (v+2n) J v + 2n (a)

96

10.16

10.17

10.18

<P(x)

sin (ax) J v (bx)

sin (ax) J v (bx)

sin(a-x)Jv(x) x < a

Ox> a


00

¢(z) = J <P(x)x z - 1 dx o

v -z (J,b/a) a sin [J,'Jf (v+z) 1

-1 r (v+ z) [r ( 1+ v) ] F (J,+ J, v+ J, z ,

2 1

= sin[J,'Jf(v+z)]f(v+z)

a > b

F (J,+J,v+J,z,J,-J,v+~z;3/2;a2/b2) 2 1

= -b -J,2 z - v - 3/2r (v+z) r (~z-J,v)

a -Re v • (v+2n+l) (v+1-z) 2nJ v+2n+l (a)

1.10 Bessel Function

cp (x)

10.19 (b-x) ),

'Jv [a (b-x)] x b

Re (Hv) > -1

10.20 e±iaz(b_x)]l-l

'Jv(ax) x b

Re ]l > 0

10.21 (b-x) Ae ±iax

'Jv [a (b-x) ] x b

Re (V+A) > -1

10.22 (b-x) A

k 'J2v [a(b-x) 2] x b

Re (A+V) > -1

97

'" ~(z) = f CP(X)Xz- 1dx o

.[r(l+v)r(l+v+]l+z)]-l F (~+~A+~V, 2 3

Re z > 0

.[r(l+v)r(v+]l+z)]-l F (v+z,~+V; 2 2

v+]l+z,2v+1;±i2ab)

Re z > -Re v

.[r(l+v)r(l+v+Hz)]-l F (l+v+A, 2 2

~+v;1+v+A+z,2v+l;±2iab)

Re z > 0

.[r(l+2v)r(1+A+v+z)]-1

F (A+l+v;1+2v,1+A+v+z;-~a2b) I 2

Re z > 0

'" <P( z) f z-l

q, (x) = q,(x)x dx 0

10.23 J~ (ax) ~(~a)-zr(l-z)r(v+~z)

-2Re v<Rez<l -2 -1

• [r (l-~z) 1 [r (l+V-~z) 1

10.24 J ll (ax) J v (ax) -z

~(~a) r(l-z)r(~V+~ll+~z)

-Re (V+ll) < Re z < 1 [r(l+~V-~ll-~Z)r(l+~v+~ll-~Z)

.r(l+~ll-~V-~z)l-l

10.25 Jv(ax)Jv(bx) 2z-1[r(1-~z)1-lr(v+~z)

-2Re v < Re z < 2 '(la2_b21)-~zp-v (a2+b 2 ) -~z la2-b 2 1

5 10.26 J2 (ax) +y2 (ax) 7r - -'icos (7rv) a -zr (~z) r (~-~z)

v v

±2Re v < Re z < 1 ·r (v+~z)

10.27 J v (ax)Yll (ax) -z

~sin [7r (V-ll) 1 (~a) r (l-z)

-Yv (ax) J ll (ax) ·r(~z+~V+~ll)r(~z-~V+~ll)

Re(±ll±V) <Re z < 1 ·r (~z+~V-~ll) [r (l-~zHv+~ll) 1 -1

$(x)

O<Rez<l

0<Rez<2

10.30

Re z < 1

10.31

Re(±~-v)<Re z < 1

10.32

+J -v (ax) Y v (bx)

±2 Re v<Re z<2

00

J z-l ~(z) = $(x}x dx

o

-1 . Ir (l+v-l,z) r (l-v-l,z)]

_ 3/, -z -l,~ 2a cos[~(v+l,z)l

-1 • [r (l+v-l,z) ]

-1 z-l -z -~ 2 a cos[l,~(v-~+z)l

99

100

¢ (x)

10.33 Yv(ax)Jv(bx)

-2 He 0 V} <He z<2

-2 He 0 v} <He z<2

OJjl (ax) J v (ax)

Re z > -Re (jl+v)


co

<!> (z) f z-l = ¢(x)x dx o

-v b 2+a 2 '{COS(l,1TZ)p L (--)

-..,z b 2 _a 2

+ 21T- l e -i 1Tvsin [1T(v+l,Z) ]

b 2 2 • V (~)}, q-l,z 2 2

b -a

-v b 2+a 2 • {cos [IT(v+l,z)]p L (--) -.,z 2 2

b -a

a < b

-1 -ilTV +2lT e cos (lTv) sin [IT(v+l,z)]

a < b

°r(l,z+l,jl+l,v) F (l,+l,jl+l,v,l+l,jl+l,v, 3 3

l,z+l,jl+l,v;jl+l,v+l,]l+v+l;-a 2/b2)

tjl (x)

10.36 Jjl (ax) J v (bx)

-Re(jl+V) < Re z < 2

10.37

10.38

00

~(z) = f tjl(x)XZ- 1 dx o

101

a b

Eason, G. Noble, B.

and Sneddon, I.N., 1955:

Phil. Trans. ROy. Soc.

London (Al, 247, 529-

Bailey, W. N., 1936:

Proc. London Math. Soc.

40, 37-49. J. London Math.

Soc. 11, 16-20.

102

q, (x)

10.39 J v (bx)Yjl (ax)

Re(-V±jl) Re z < 2

10.40 Y (ax)Y (bx) jl .... c~ v

IRe(jl±v) I < Re z < 2

10.41 H(2) (ax)H(2) (bx) jl v

IRe jll+IRe vl<Re z<l

F H(l)H(l) or v jl

change i into -i


00

J z-l ~(z) = q,(x)z dx

o

-1 -1 -v -z -'"I [f (l+v) J a (1,a)

00

cos[1,rr(jl-v-z)]f(1,z+1,jl+1,v)

'f(1,z-1,jl+1,v) F (1,z+1,jl+1,v, 2 I

- J o

{J (ax)Y (bx)+4rr- 2 jl v

b < a

'sin[1,rr(z-V-jll]K (bx)K (ax)}x-zdx v jl

a < b

J {J (ax)J (bx)+4rr-2cos[1,rr(z-v-jl)] o jl v

• F (1,z+1,jl+1,v,1,z-1,jl+1,v;z;1-b 2 ja2 ) 2 I

q, (x)

10.42 (1_x 2 )aJ (ax)J (bx) ]1 v

o

10.43 (1_x 2 )a J (ax) ]1

x < 1

x > 1

.J [b(1-x 2 )!;;] v

x < 1

o x > 1

·Yv (x)

I Re v I <Re z<2 Re ]1+ 3/2

00

~(z) = f q,(X)Xz - 1 dX o


Quart. J. Math., Oxford

Ser. 9, 141-14 7

as before

Watson, G. N., 1922:

103

A treatise on the theory of

Bessel functions, Cambridge,

p. 436

00

 (x) = <j>(x)x dx 0

00

n -1 10.46 J~(x)Jv(a-x) x < a 2 z L (-1) [n!r(~+n+l)J

n=O

0 x > a • r (~+n+z) (z)nJ~+v+2n+z (a)

Re z >-Re ~, Re v > -1

10.47 (a-x) -1 (va) -1 2 z

00

L n -1 (-1) [n!r(~+n+l)J

n=O J v (x) J v (a-x) x < a

• (z)nr (n+~+z) (z+~+v+2n) 0 x > a

Re v > 0, Re z > -Re v 'Jz+~+v+2n(a)

10.48 0-1 (a-x) J~ (x) Bailey, W. N. , 1930:

'Jv (a-x) x < a Proe. London Ma th. Soc. (2)

0 x > a 30, 422-421 and 31, 200-208.

10.49 0-1 (a-x) JA(bx)

'J~(ex)Jv(a-x) as before

x < a

0 x > a

rp(x)

10.50

10.51

10.52

a < Re z < ~2 - Re v

10.53 (a2+x2)-~V

.y [b (a 2+X2) ~l v

10.54 (a 2_X 2) ~v

'J [b (a 2_X 2) ~l V

x < 0.

a x > a

Re v > -1

00

f z-l ~(z) = ~(x)x dx

a

• [Jv+~z(ab)cos(~nz)

-v ~z ~a (2a/b) r (~z) J 1.. (ab) v--,z

105

Re z > a

106

 a

10.56 0 x < a

(x2_a 2)l:iV

.J [b(x2-a2)~] v x >

Re v > -1

10.57 0 x < a

(x2_a2)-~V

.J [b (x 2_a 2) l:i] v x > a

10.58

Ox> a

Re v > -1

a


<jl (z)

Re z > 0

Rez<3/2 - Rev

al:iz+vb -l:iZ[2l:iZ-l r (l:iZ)Yl:iz+v(ab)

+~-laV+lr(l+V)SL 1 L + (ab)] -,z- -v,-,z v

Re z > 0

10.59

10.60

10.61

10.62

10.63

¢ (x)

Ox> a

Re v < 1

Jv(aX)J~[b(l-x2)~] x < 1

Ox> 1

o x > 1

Re ~ > -1, Re z > -Re v

J v (u) J v (v)

+ Yv(u)yv(v)

u = b [(a 2+x 2) lz±a] v

-Jv (u) Yv (v)

u = b [(a2+x2) lz±a] v

00

¢(z) = I ¢(x)Xz-ldx o

'S (ab) ~z+v-l, ~z-v

107

-2~z-lr(~z)csc(TIv)J,- (ab)] .,z-v

Re z > 0

Bailey, W. N. 1938:

Quart. Journal of Math.

9, 141-147.

"B (~+1, ~z+~v)

_ 3/ kz -TI 2COS (TIv) (a/b) 2

• r (lz-lzz) Y 1 (2ab) ':!z

± 2 Re v < Re z < 1

r (lz-lzz)J (2ab) ~z

±2Rev<Rez<1

00

¢ (x) (z) J z-l = ¢(x)x dx 0

10.64 JV(v)J_V(u) ~TI~r (v+~z) r Ui-!,Z) [r (l+v-~z) ]-1

u = b[(a2+x2)~±a] (a/b)~Z[J 1 (2ab) COS (TIV) v -~z

-2 Re v < Re Z < 1 - Y_~z(2ab)sin(TIv)]

10.65 J v (u) J v (v) ~TI -~r (v+!,z) r (~-!,z) [r (l+v-~z) ]-1

u = b [(a 2+X2) !,±a] (a/bl~zJ 1 (2ab) v -~z

-2 Re v < Re z < 1

10.66 J v (v)Y_ v (u) ~TI~r (v+~z) r (~-~z) [r (l+v-!iZ) ]-1

u = b [(a 2+x2) ~±a] hz v (a/b) 2 [J_~z(2ab)sin(TIv)

-2 Re v < Re z < 1 + Y_!,z(2ab)cos(TIV1]

10.67 J v (v)Y v (u) ~TI-!'r (v+~z) r (!'-!,z) [r (l+v-!,z) ]-1

u h (a/b) !'zY = b [(a 2+x2) 2±a] .

1 (2ab) v -~z

-2 Re v < Re z < 1

cjl (x)

10.68 (b_x))l-l

Ox> b

Re )l >0

10.69 (b-x) A

'J2[a(b-x)~] x b

Re (A+V) > -1

10.70 (b_x))l-l

10.71

J (ax~)J (ax~) x b

A-I k (b-x) J v [a (b-x) 2]

oJ [a(b-x)~] x b

00

~(z) = J cjl(x)xz- 1dx o

109

F (Z+V,~+V;Z+v+)l,v+1,2v+1;-a2b) 2 3

Re z > -Re v

(~a)2vr(1+A+v)r(z)bA+V+z

• [r 2 (l+V) r (Hv+l+z) ]-1

F (A+v+1,~+v;A+v+1+z,v+1,2v+1;-a2b 2 3

Re z > 0

-1 ['lTVr(z+)l)] F (~,z;l+v, 2 3

Re ()l ,z) > 0

r (A) sin ('lTv)r (z )bA-l+Z

['lTVr (v+z)]-l F (~,A;l+v, 2 3

Re(A,z) > 0

110

4> (x)

-Re v<Re z<~2+2Re P

10.74 (b 2+X2)A

.J [a(b2+x2)~] v

oJll (ax)Jv (ax)

10.76 (b_x)ll-l

o

x b

Re 11 > 0, Re z > -Re v


'" ~(z) = f 4> (x)xz - 1dx o

v-z • (2a) r (~+v-z) F (~+V-z; 1+2v-z; 2iab)

1 1

o < Re z < ~+Re v

o < Re z < ~-2Re A

r (z+v) r (11) (~a) Vb ll+v-1+z

o [r(1+v)r(V+ll+Z)]-l F (~v+~z, 2 3

~+~v+~z;v+1,~Il+~V+~Z,~+~V+~Il+~Z;

1.10 Bessel Function III

ex> z-l .p (x) q, (z) = J .p(x)x dx

0

10.77 Jv[a(x -1 -x) ] x < 1 K (a)I 1 (a)

~V-~Z ~V+==jZ

0 x > 1 Re v > -1, Re z > - 3-'2

10.78 0 x < 1 K (a)I (a) ~\)+~z ~v-~z

-1 Jv[a(x-x )] x > 1 Re v > -1, Re z < ~2

10.79 -1 Jv[a[x-x [] I~v_~z(a)K~v+~z(a)

Re v > -1 + I~v+~z(a)K~v_~z(a)

Pricipal value -~2 < Re z < l--2

10.80 (x+x-1) -1 a-1[r(1+v)]-2r(~+~v+~z)

-1 ·r(~+~V-~z)M, 1 (~a)M 1 1 (~a) 'Jv[a(x+x )]

~Z/~\) -~v,~z

-l-Re v < Re z < l+Re v

10.81 -1 J 2v [a(x+x l] -~TI[Jv_~z(a)Yv+~z(a)

_3/2 < Re z < 3/2 + Jv+~z(alYv_~z(a)]

10.82 -1 Y2v [a(x+x l] ~TI[Jv_~z(alJv+~z(al

- 3/2 < Re z < 3/2 -Yv_~z(a)Yv+~z(al]

112

(jl(x)

Principal value

10.84 v -v (a+bx) (b+ax)

10.85 v -v (a+bx) (b+ax)

10.86 (1+x2)-lJ (2ax) v 1+x 2

l-x2 ·exp [-b (--) 1

1+x2

-Re v < Re z < 2+Re v

10.87 (1+x2)-lJ (2ax) v 1+x 2

°exp[- ~) 1+x 2

-Re v < Re z < 2+Re v


co

~(z) = J (jl(x)xz-ldx o

-~2 < Re z < ~2

¢(x)

l-x 2 'exp [b (--) 1

1+x2

-Re v<Re z<2+Re v

2bx 2 'exp(- --)

1+x2

-Re V<Re z<2+Re v

1+x2 'exp [-b (--) 1 ,

l-x 2

x < 1

o x > 1

Re z > -Re v

2bx 2 'exp(- ---) x < 1

l-x 2

Ox> 1

Re z > -Re v

00

~(z) = f ~(x)xz-ldx o

113

114

<j> (x)

10.92 a x < 1

Re z < 2+Re v

10.93 a x < 1

(x2-1) -lJ (2ax ) v x 2-1

°exp(- ~), x 2-1

Re z < 2+Re v

10.94 J]1 (ax) J v (b/x)

x > 1


 (z) z-l = f <j>(x)x dx

a

-1 -1 ~a [f(l+v)] f(l+~v-~z}

1 1 -z 20 1 2 21' 1 1 1 1 • 1 ) ~(~a) G (-- a b Yzv,Yzz+~~,YzZ-Yz~/-YzV 04 16

- 3/2 -Re ]1 Re z > }

3/2 + Re \!

1.11 Modified Bessel Function 115

1.11 Modified Bessel Function

00

z-l <jl(x) <l>(z) = f <jl(x) x dx 0

11.1 Kv (ax) z-2 -z 2 a r(~z+~v)r(~z-~v)

Re z > ± Re v

11. 2 (b 2+X2) -IK (ax) (2b)z-2r(~z-~v)r(~z+~v) v

Re z > ± Re v -Sl-z,v (ab)

11. 3 -bx v (ax) r(v+z) (b2_a2)-~Zp-v [b(b2-a2)-~1 e I

z-l

b > a Re z >-Re v

11.4 eaxK v (ax) rr-~cos(rrv) (2a)-zr(~-z)

±Re v < Re z < ~ ·r (z+v) r (z-v)

1l.S e-a~ v (ax) rr~(2a)-z[r(~+z)1-1

Re z > ± Re v • r (z-v) r (z+v)

11.6 -b~ e v(ax) k

(~rr/a) 'r(z-v)r(z+v)

k !':z k z (a 2 _b 2) 4-, p'- (b/a) -a<b<a v-~

k (~rr/a) 'r(z-v)r(z+v)

(b2_a2)~-~Zp~-~(b/a) b > a v-~

116 I. Mellin Transform?

00

z-l <p<x) 1> (z) = J ¢(x)x dx 0

11. 7 -bx 2 2 a-lb~-~Z[r(l+v)l-lr(~Z+~V) e I (ax) v

Re z > -Re v oexp (l/ea 2/b) M" , ('.!a 2/b) ~-~z,~v

-b 2 -1 1:-1:z 11. 8 e x K (ax 2) ~a b 2 2 f(~z+~v)r (~z-~v) v

Re z > ± Re v exp ( 1,.. a 2/b) W ('.!a 2/b) 8 ~-~z, ~v

11. 9 (l+x2) -II (~) a-l[r(1+v)1-2r(~z+~v)r(1-~z+~v) v 1+x 2

-Re v<Re z<2+Re v ·M~z_~,~v(a)M~_~z,~v(a)

11.10 sin(ax)Kv(ax) 2z-2a-zr(~+~z+~v)r(~+~z-~v)

Re z > -1 . F (~+!:zz+!:2v,~+~z-!zv; ~ ;-1) 2 1

11.11 cos (ax)Kv(ax) 2z-2a-zr(~z+~v)r(~z-~v)

Re z > 0 • F (~z+!zv,!zz-!zv;~;-l) 2 1

11.12 sin (bx) Kv (ax) -krrsee [~Tr (z-v) 1 r (v+z) (a 2+b 2) -~z

Re z>-l ± Re v o{p-v [b(a2+b2)-~1_P-v [-b(a2+b2)-~1} z-l z-l

11.13 cos (bx) Kv (ax) '.!Trese [~Tr(z-v) 1 r (v+z) (a 2+b 2) -~z

o{p-v [b(a2+b2)-~1+P-v [-b(a2+b2)-~1} z-l z-l

00

<j> (xl <l> (z) I z-l = <j>(xlx dx

0

11.14 -v

(a+x) e -bx (2TIab)-~a-v(2b/a)-~zr(z)

• Iv [b (a+x) ] • [r (l+2v-z) ]-lM 1 1 (2ab) -;.zZ,V-~Z

0 < Re z < ~ +Re v

11.15 -v (a+x) e

-bx (2ab/TI)-~a-V(2b/a)-~Zr(Z)

oKV [b (a+x) ] 'W_~Z,V_~Z(2ab) Re z > 0

11.16 (a+x)-vebx (2ab/TI) -~a -v r (z) [r (~+v) ]-1 (2b/a) -~z

'Kv [b (a+x) ] ·r(~+V-Z)W, 1 (2ab) ~Z ,V-;.zZ

0 <Re z < ~ +Re v

11.17 -x ~ e sin (ax ) Kv (x) (~TI) ~a2-Zr (~+V+Z) r (~-V+Z) [f(l+z) ]-1

Re z > I Re v I-~ • F (~+v+z,~-v+z; 3/z,1+z;_1/eaZ) z z

11.18 -x ~ e cos (ax 2) Kv (x) ~ -z -1

TI 2 [r(~+z)] r(z+v)r(z-v)

Re z > IRe vi · F (v+z, z-v; ~, ~+z; -l/aa Z) Z Z

00

z-l <p (x) ¢ (z) = J <P(x)x dx 0

1 22 11-Z'~ 11.19 (b+x) -p rr-~[r(p)]-lbz-PG (2ab ) 23 P-Z,\),-V

-ax (ax) -Re v < Re < ~+Re e I Z P v

11.20 (b+x) - P e -ax 1 1 31 11-Z'~ rr"[r(p)]- bZ-PG (ab ) 23 p-z,v,-v

oK\) (ax) Re Z > ± Re v

11. 21 (b+x) -P 1 1 32 11-Z'~ rr-~[r(p)J- cos(rr\))bz-PG (2ab )

23 p-z,v,-v

eaxK v (ax) ± Re v < Re Z < ~+Re p

11. 22 (b+x)-Ae-ax rr-~(2a)Ar(z)bzeab

21 I~-A'O oIv[a(b+x)] °G (2ab )

23 -Z,V-A,-V-A

0 < Re Z < ~+Re A

11.23 (b+x) - Ae -ax rr~(2a)Ar(z)bzeab

30 10'~-A I Kv[a(b+x)] °G (2ab ) 23 -Z,V-A,-V-A

Re Z > 0

1.11 Modofied Bessel Function

<jl (x)

11.24 (b+x) -Ae ax

Kv[a(b+x)]

0 <Re z <

11.25 (b 2+X2) A

'K [a(b 2+x2) ~l v

Re z > 0

11.26 (b2_X2) ]1-1 x b

Re ]1>O,Re z>-Re v

11.27 x b

Re ]1>0, Re z>±Re v

co

~(z) = f <jl(x)xz-ldX o

119

~-~(2a)Acos(~V)bzr(z)e-ab

31 I~-A'O • G23 (2ab -Z,V-A,-V-A)

o < Re z < ~+Re A

-1 • [r(l+v)r(~z+~v+]1)]

• F (~z+~v;1+v,~z+~v+]1;~a2b2) 1 2

f (v) = (~ab) -V r (v) r (~z-~v) [r (~z+]1-~v) ]-1

120

¢ (x)

11. 28 x b

Re A>-l-lzRe v;Re z>O

11.29 (b-x) \1-1 x b

11. 30 0 x b

Re v > -~

11.31 (b_x)A-le ±ax

'Iv[a(b-x)] x b

Re (A+V) > 0


<l>( Z) z-l

= f ¢(x)x dx o

v -1 lz (lza) [r (l+v) r(l+ Hlzv+lzz) ]

or(lzz)r(1+A+lzv)b2A+ z+ v

v -1 (lza) [r(l+v)r(z+)1+v)] r(\1)r(z+\1)

ob Z+\1+v-l F (lz+v,z+v;2v+l,\1+v+z;±2ab) 2 2

Re \1 > 0, Re z > -Re v

o (2a) -lz-lzzb -lz+v+lzzW, (2ab) YzZ, V+!;zZ

Re z < lz-Re v

(lza) Vr(Hv) [r(l+V) r(Hv+z) ]-1

A+v-l+z "b F (!;z+v,A+v;2v+l,A+v+z,±2ab) 2 2

Re z > 0

co

<j> (x) ~(Z) = J <j>(x)xz - 1dx

°

11. 32 ° x 'l.-2-Re )1

11.33 ° x °

11.35 ° x b 30 I ° "G (\a 2b ) 13 -ll,lzv-l+z,z-l-lzv

122

11.36

11. 37

11.38

11.39

q,(x)

o x -1

o (x-b) I.e -ax

KV[a(x-b)]

Re(Hv) > -1

o

(x-b) A ax e

"KV[a(x-b)]

Re(Hv) > -1

o

(x-b) A

x b

x b

x b

Re(Hv) > -1


00

~(Z) = f q,(x)x z- 1 dX o

22 I-A,l:! 'G (2ab )

23 -z-A,v,-v

Re Z < l:!-Re A

31 I-A,l:! 'G (2ab )

23 -z-A,v,-v

22 I-A,l:! 'G (2ab )

23 -z-A,v,-v

Re Z > l:!-Re A

00 z-l

¢ (x) ( z) = f ¢(x)x dx 0

11. 40 J (ax) K (ax) z-3 -zr r -1 v v 2 a (J,z) (J,v+\iz) [r(l-\iz+J,V)]

Re z (:2 Re v

11. 41 J)bx)Kv(ax) 2 z -2 r (J,z)r(v+J,z) (a 2+b 2)-J,z

>t 2 b 2

Re z p-v (~)

-2 Re v -J,z a2+b2

11.42 Y v (ax) Kv (ax) _~-12z-3a-zr(J,z)r(\iz+~v)

Re z >t ·r("z-~v)cos[~~(~z-v)]

±2 Re v

11.43 Iv(ax)Kv(ax) \i~-~a-zr(J,z) r(J,-J,z) r(v+J,z)

o } <Re z<1 -1 -2 Re v

• [r (1-J,z+v) ]

2z-2a-zr(J,z+J,~+J,V) [r (1-~z+~VI-~~)] -11.44 K~(ax)Iv(ax) 1

Re (-V±~) < Re z < 1 "B (l-z, ~z-J,~+ ~v)

11. 45 K~(ax) \i~~a-zr(~z+v)r(J,z-v)r(J,z)

Re z >C 0 • [r(J,+~z) ]-1 ±2 Re v

11.46 K~ (ax) Kv (ax) a-z2z-3[r(z)]-lr(~v+J,~+~z)

Re z > Re(±~±v) ·r(~v-~~+~z)r(~~-~v+~z)

• r (~z-~v-~~)

124

<j>(x)

a>b, Re z >f \-2Re v

11.48 KV(aX)KV(bX)

{o

Re z > ±2 Re v

11. 49 Kj.l (ax) J v (bx)

Re (a±ib) > 0

Re z > Re(-V±j.l)

11. 50 Kj.l (ax) Iv (bx)

b < a, Re z > Re(-V±j.l)


00

W(z) = J <j>(x)xz-ldx o

-v a 2+b 2 'p (--)

-l,z a2_b2

=(~ab)-1,2z-2(a2_b2)1,-I,Zr(I,Z)

'e-il,~(z-l) I,z-l,(a 2+b 2 ) qv-l, 2ab

z-2 = 2 r(l,z)r(v+l,z)

·r (l,v-l,j.l+l,z) F (l,v+l,j.l+l,z, 2 1

11.51

11.52

11.53

¢ (x)

Kj.l(ax)KV(bx)

Re(a+b) > 0

Re z > Re(±j.l±V)

±2 Re v < Re z < 1

J (ax)K (bx)K (cx) v j.l p

00

¢(z) = f ¢(x)Xz-ldx o

125

F (~v+~~+~z,~v-~~+~ziz;1-b2/a2) 2 I

• r (y,z+v)r (y,z-v)

-1 • [1' (l-v-y,z) l' (l+v-y,z) 1


Proc. London Math. Soc.

40, 37-48.

Journal London Math. Soc.

11, 16-20.

126

cjl(x)

11.54

o x > a

Re v > -1

11.55

11. 56

11. 57

x < a

o x > a

Re z > Max(O,Re v )

11.58 o x < a

x > a


co

¢(z) = J cjl(x)xz-ldx o

Re z > 0

-\} 1.:z ~b (2b/a) 2 r (~z) KV_~Z (ab)

Re z > 0

~bv(2b/a)~zr(~Z)K +' (ab) v '2Z

Re z > 0

.r(~Z)I~Z_V(ab)

\} !':z (2b) r (l+V) (b/a) 2 5, 1 '+ (a b)

~z- -\}, ~z V

Re v > -1

co

¢ (x) <jl (z) = J z-1 ¢(x)x dx 0

11.59 J v (v) Kv (u) 2 z - 1 r (1:2v+1:2z) [r (1+1:2v-1:2z) J-1

u (2b)1:2[(a 2+x2)1:2±a]1:2 (a/b)1:2z k = · K z [2 (ab) 2] v

Re z >-Re v

11. 60 Yv(V)Kv(u) _~-12z-1cos[1:2~(v-z)]r(1:2z+1:2v)

u (2b)1:2[(a 2+x2)1:2±a]1:2 kz K [2 (ab) 1:2] = · r (1:2z-1:2v) (a/b) 2 v z

Re z > ± Re v

11.61 Iv(V)Kv(u) 1:2~ -1:2 r (v+1:2z) r (1:2-1:2z) [r (1+v-1:2z) ]-1

u k (ajb) 1:2zK , (2ab) = b [(a 2+x2) '±a] · v 'iZ

-Re v < Re z < 1

11.62 Kv (v) Kv (u) k -1

1:2~'r (1:2z-v) r (1:2z+v) [r (1:2+1:2z)]

u k (a/b) 1:2zK, (2ab) = b [ (a 2+X 2) '±a] v 'iZ

Re z > ± 2 Re v

00

z-l q,(x) cp (z) = J q,(x)x dx 0

11. 63 K (a I x-x -11 ) 0

\1T2(J 2 (a)+y 2 (a)] !.zz !2Z

Principal value

11. 64 Kv[a(x -1

-x) ] < 1 \1T l CSC (1TV) [J, +' (a) Y , +' (a) x ~\) ~z -~\):::2Z

0 x > 1 -J_~v+~z(a)Y~v+~z(a)]

-1 < Re v < 1

11.65 0 x < 1 '41TlCSC(1TV) [J, , (a)Y, , (a) ~\.l-~Z -7zV-:-ZZ

-1 1 -J_~v_~z(a)Y~v_~z(a)] Kv[a(x-x )] x >

-1 < Re v < 1

11. 66 -1 K2v [a (x+x )] Kv+~z(a)Kv_~z(a)

11. 67 v -v (a+bx) (b+ax) 2Kv+z (a) Kv_z (b)

oK2V{[al+bl+ab(x+x-1)]~}

11. 68 (l+xl) -II ( 2ax) ~a-1[r(1+v)]-2r(1+~V-~z)r(~v+~z) v l+xl

I-xl '-< oexp (-b(--)] oM" , [(bl+al ) "+b] l+xl 7z- ~z ,7z\.l

-Re v < Re z < 2+Re v oM [(bl+a l ) ~-b] ~Z-!:2/!.zV

rjl(x)

l-x 2 'exp [b (--) 1

1+x2

-Re v<Re z<2+Re v

2bx2 'exp(- --)

1+x2

-Re v<Re z<2+Re v

1+x2 'exp [-b (--) ],

l-x 2

x < 1

Ox> 1

b > a, Re z>-Re v

2bx 2 'exp(- ----), x < 1

l-x2

o x > 1

b > a, Re z>-Re v

DO

w(z) = f rjl(x)xz - 1dx o

129

130

11. 73

11. 74

11.75

11. 76

1> (xl

'exp(- ~) 1+x 2

- Re v < Re 2 < 2+ Re V

o x < 1

x 2+1 'exp [-b (--)], x > 1 x 2-1

b > a, Re 2 < 2+Re v

o x < 1

x > 1

Re 2 < 2+Re v, b > a

o x < 1

Re 2 < 2 ±Re v


00

f 2-1 <!> (2) = 1> (xl x dx

o

-1 -1 b lza [r (1+v) 1 r (1+lzV-~22) e

'" cj>(x) 4>(z) = f cj>(x)x z- 1dx 0

11. 77 (1-x2) -lK (~) ~a-lr(~z-~v)r(~z+~V) v l-x 2

1+x2 1 .W~ ~ ~ [b-(b2-a2)~1 ·exp [-b (--) 1 , x <

l-x2 - z, V

0 X > 1 ·W [b+(b2-a2)~1 ~-~z,~v

Re z > ±Re v

11. 78 (1-x2) -lK ( 2ax) v l-x2

~a-lebr(~z-~v)r(~z+~v)

2bx 2 .W~_~z,~v[b-(b2-a2)~1

·exp (- --), x < 1 l-x 2

·W [b+(b2-a2)~1 ~-~z,~v

0 x > 1 Re z > ± Re v

'" z-l q, (x) <jJ (z) = f q,(x)x dx 0

11. 79 0 x < 1 -1 b

~a e r(I-~v-~z)r(I+~v-~z)

(x2-1) -IK ( 2ax) v x 2-1

oW, " [b-(b2-a2)~1 ~z-~,~v

'exp(- ~), x > 1 'W, " [b+ (b 2_a 2) ~l x 2-1 '2Z-~,:-zv

Re z < 2±Re v

11. 80 K Il(ax) J )b/x) ~(~~-zG:~ (i6 a2b21~v,~z+~Il,~Z-~Il,-~v)

Re z > IRe 11 I - 3/2

11.81 K (ax) Y (b/x) ~(_I)m+lU2a)-z 11 v

I ~-~v-m 0G'o (~2b2 )

15 16 ~V,-~V,~Z+~~,Yzz-~~,~-~v,m

m integer, Re Z > - 3/2+ IRe III

11.82 KIl(ax)Kv(b/x) !.(~a) -zG'o (L 8 04 16

a2b21~v,-~v,~z+~~/~z-~~)

1.12 Functions Related to Bessel Function 133

1.12 Functions Related to Bessel Function

co

q,{x) <!J (z) = [ q,(x)xz - 1dx 0

12.1 Rv (ax) z-l -z -1

2 a r (lzz+lzv) [r (1+lzv-lzz) ]

-l-Re v < Re z < { 3/2 ·tan [lz'1T(z+v) 1 l-Re v

12.2 IIv (ax) -Yv (ax) 2 za -zr (lzz-lzv) [r (l-lzz-lzv) ]-1

±Re v < Re z < l-Re v cos('1Tv)csc['1T(v+z)]

12.3 Iv(ax)-Lv(ax) z-l -z

2 a sec[lz'1T(v+z)]

-Re Re l-Re -1

v < z < v • r(lzz+lzv) [r (1+lzv-lzz)]

12.4 Iv(ax)-L_v(ax) z-l -z 2 a cos('1Tv)sec[lz'1T(v-z)]

-Re v < Re z < l+Re v • r(lzz+lzv) [r (1+lzv-lzz) J -1

2 2 '1T-lz2-v-1a-lzv-lz-lzz[r(~+v)]-1 12.5 e -a x H (bx)

v

Re z > -l-Re v 'b v+l r (lz+lzv+lzz)

· F (l,lz+lzv+lzz; '!.-2, 3/2+V ;-\b 2ja 2) 2 2

12.6 H [a(b 2 +x2) lz] 1:zz 0

lz(2bja) r(lzz)

0 < Re z < 1 • [Hlzz (ab) sec (lz'1Tz)

+J_lzz(ab) tan (lz'1Tz)

134

12.7

12.8

12.9

$ (x)

'{I [a(b 2+x2)lz] -v

_[, [a (b 2+X2) lz] } v

Re v < ~


00

(z) = f $(X) x z - 1dx o

'r(lzz) r(lz-v-lzz) r(lz+v+lzz)

[Hlzz+v (ab) -Ylzz+ v (ab) ]

o < Re z < 1-2 Re v

O<Rez<l

V :kz lzcos (7fv)b (2b/a) 2 f(lzz)

'sec [7f (v+lzz)]

Re z > 0

1.12 Functions Related to Bessel Function

12.11

12.12

12.13

12.14

q, (x)

1-2Re v O<Rez<{

o

o

o

i-Rev 2

x > b

x > b

x > b

12.15 s (ax) \l,V

00

~(z) = J q,(x)xZ- 1 dx o

v ~z !:ib sec [1f(v+!:iz) 1 (2b/a) 2 f(!:iz)

-hZ v+hz...._ !:i(!:ia) 2 f(!:iz)b 2 H. +' (ab) v :.zz

-v -1 (2b) r(!:i+v) [f(!:i+!:iz)] f(!:iz)

kZ (b/a) 2 s +' ,(ab)

'J ~Z,V-::-.2Z

-hZ v+kZ-!:i(!:ia) 2 f(!:iz)b 2"" +' (ab) v YzZ

-l-Re \l < Re z < l-Re \l

-1 • [f(l-!:iv-!:iz)f(l+!:iv-!:iz)]

135

Re z > 0

Re z > 0

Re z > 0

00

z-l rjl (x) 1> (z) = f rjl(x)x dx 0

12.16 (b 2+X2) \V v kz \r(\z)b (bja) 2

• s [a (b 2+X2) l;;] · { r (~-~\)-~1l-~2Z) S\z+\l,\z+v (ab) \l,V r (\-\v-\\l)

3-Re v _2\l-1+\zr (l;;+\\l+\v) r(\+\\l-l;;v) O<Re z < {

_3/2-Re(v+\l) (sin[\rr(\l+v}]J_v_$ab)

-cos[\rr(\l+v}]Y_v_\(ab}}}

12.17 (b2+X2) l;;v \b v (bja) \zr (\z) r (~-~Z-~lJ-~$ r (lz-!,v-!'\l)

'S [a(b 2+x2} \] \l,V • S\z+\l, \z+v (ab)

O<Re z<l-Re (v+\l)

12.18 Jv (ax) -Jv (ax) -z l;;sin(rrv} (\a) csc(rrz}

0 < Re z < 1 • r (l;;v+\z) [r (l+\v-\z) ] -1

12.19 J (ax) +17 (ax) -z v -v rrcos(\rrv} (\a) csc(\rrz}

0 < Re z < 2 • [r (l-l;;z+\v) r (l-\z-\v) ] -1

12.20 J v (ax) -iT-v (ax) -z rrsin (\rrv) (\a) sec (\rrz)

-1 < Re z < 1 • [r (l-\v-\z) r (l+\v-\z) ]-1

1.12 Functions Related to Bessel Function 137

12.21

12.22

12.23

12.24

¢ (xl

o < Re z < 1- Re v

'{J [b(a2+x2)~] v

+iJ [b (a 2+x2) ~J} -v o < Re z < 3/2 - Re v

'{J [b(a2+x2)~ v

-J [b(a2+x2)~} -v o < Re z < l-Re v

ker 2 (ax)+kei 2 (ax) v v

00

~(z) = f ¢(x)xz - 1 dx o

-1 v ~z n sin(nv)a r(~z) (a/b)

kZ -~ncsc(~nv)2 2 [cos(~nv)J 1 (ab)

-v-~

+ sin(~nv)Y 1 (ab)]} -\)-='2Z

n -lsin (nv) a vr (~z) (a/b) ~z

k Z -~n2 2 sec (~nv) [sin (~nv) J 1 (ab)

-v-~

-cos(~nv)Y 1 (ab)]} -:.oz-'J

Re z > ±2 Re v

1.13 Whittaker Functions and Special Cases*

00

¢(x) (z) f z-l = ¢(x)x dx

0

Erf (ax) -., -1 -z

13.1 -rr z a r(.,+.,z)

-1 < Re z < 0

13.2 exp(-a 2 x 2 )Erf(iax) .,irr 2 a- z [r(1-.,z)]-lsec(.,rrz)

-1 < Re z < 1

13.3 Erfc(ax) rr-.,z-la-zr(.,+.,z)

Re z > 0

13.4 exp(a 2 x 2 )Erfc(ax) -z .,a sec(.,rrz)r(.,z)

0 < Re z < 1

13.5 exp(b 2 x 2 )Erfc(ax) rr-.,z-la-zr(.,+.,z)

b < a, Re z > 0 • F (.,z,.,z+.,;1+.,z;b 2 /a 2 ) 2 1

13.6 sin (bx) Erfc (ax) rr-"(z+l)-lba-z-lr(l+.,z)

Re z > -1 • F ("+"z,l+.,z; 3/2 ;.,z+3/z;-lob 2 /a 2 ) 2 2

13.7 cos (bx) Erfc (ax) -., -1 -z rr z a r (.,+I,z)

Re z > 0 . F (.,z,.,+l,z;.,;1+.,z;-lob 2/a 2 ) 2 2

* The Fresnel, exponential, sine, cosine and error-integrals; incomplete gamma and parabolic cylinder functions.

1.13 Whittaker Functions and Special Cases 139

co z-l q,(x) iP (z) = f q,(x)x dx

0

13.8 exp(a2x 2 )Erfc(ax+b) ~-~(2a)-zr(z)r(~-~z,b2)

0 < Re z < 1

13.9 Ei (-ax) -1 -z -z a r(z) Re z > 0

13.10 eaxEi (-ax) -~a-zr(z)csc(~z)

0 < Re z < 1

13.11 Ei[-b(a+x)] -azr(z)r(-z,ab) Re z > 0

13.12 e-a~i (ax) -z -~a r(z)cot(~z)

0 < Re z < 1

13.13 e ax [Ei(-2ax) -z ~a r (z) [1jJ (l-~z) -1jJ (~-~z)]

-Ei(-ax)] 0 < Re z < 1

13.14 e ax [Ei (-ax-bx) -1 l-z b (alb) b r (z)Y (-a,l,l-z)

-Ei (-ax) ] 0 < Re z < 1

13.15 e ax [Ei(_ax)]2 r (z) [2~csc (~z) cot(~z)

Re z > 0 -~1jJ' (l-~z)+~1jJ' (~-~z)]

co z-l q, (x) <P (z) = f q,(x)x dx

0

13.16 Ei (-ax) Ei (ax) -1 -z 'lTZ a r(z)cot(1,'lTZ)

0 < Re z < 2

13.17 Si (ax) -1 -z sin(1,'lTz)r(z) -z a

-1 < Re z < 0

13.18 si (ax) -1 -z sin (l~'lTz) r (z) -z a

0 < Re z < 2

13.19 Ci (ax) -1 -z cos (1,'lTz) r (z) -z a

0 < Re z < -2

13.20 Ci (ax) sin (ax) 1,'lTa -z

sec(1,'lTz)r(z)

-si(ax)cos(ax) -1 < Re z < 1

13.21 Ci (ax) cos (ax) -1,'lTa -z

cs c (1, 'IT z) r (z)

+si(ax)sin(ax) -1 < Re z < 1

13.22 [Ci(ax)]2+[si(ax)]2 -1 -z 'lTZ a r(z)csc(1,'lTz)

0 < Re z < 2

13.23 1: k i'lT-\z-l tan (\'lTz)r(\+z) ,-\<Re Erf (ix 2) Erfc (x 2) z< 1

1.13 Whittaker Functions and Special Cases

¢ (x)

13.24 sin [b (a 2+X2) J,]

'si [b (a 2+x2) J,]

+cos [b (a 2+X2) J,]

'ci [b (a 2+X2) J,]

13.25 sin[b(a 2+x 2 )J,]

'Ci [b (a 2+X2) J,]

-cos [b (a 2+X2) J,]

'si [b (a 2+x 2 ) y,]

13.26 (a 2+x2)-J,

• {sin [b (a 2+x2) J,]

'si[b(a2+x2)J,]

+cos [b (a 2+x2) J,]

oCi[b(a2+x2)J,]}

13.27 (a 2+x2)-J,

• {sin [b (a 2+X2)~]

'ci [b (a 2+x2) y,]

-cos [b (a 2+x2) J,]

'si [b (a 2+x2) J,]}

ro

~(z) = f ¢(x)x z - 1dx o

O<Rez<2

O<Rez<l

O<Rez<3

O<Rez<2

141

00

z-l ¢ (x) <l> (z) = f ¢(x)x dx 0

13.28 S (ax) _(2rr)-~z-la-zr(~+z)

_3/2 < Re z < 0 • sin [~rr (Hz) ]

13.29 C (ax) -(2rr)-~z-la-zr(~+z)

-~ < Re z < 0 'cos [~rr (~+z)]

13.30 cos (ax) C(ax) 2-~rr2a-Z [r (l-z) ]-lsec [~rr (~-z)]

+sin(ax)S(ax) -~ < Re z < 1

13.31 sin(ax)C(ax) _3/2 2 -z 2 rr a -1 r(l-z)] csc[~rr(~-z)l

-cos(ax)S(ax) -~2 < Re z < 1

13.32 [~-S (ax) 1 cos (ax) 3

2- /2a- Zr (z) csc [~rr (~-z) 1

-[~-C(ax)lsin(ax) 0 < Re z < 3

3 13.33 [~-C (ax) 1 cos (ax) 2- /2a -zr (z) sec [~rr (~-z) 1

+[~-S(ax)lsin(ax) 0 < Re z < 3

13.34 [~-c(ax)12 -~ -1 -z ~rr z a sec(~rrz)r(~+z)

+[~-S(ax)12 0 < Re z < 1

13.35

13.36

13.37

13.38

¢ (x)

_(a 2+x 2 )-!:lsin[b(a 2+x 2 )lz]

• {lz_C[b(a 2+x 2 )lz] }

00

$(z) = f ¢(xlx z - 1dx o

k kZ ",(!:lab/TI) '(a/b)' r(!:lz)r('\-!:lz)

3 -1 • [r( I'"ll S!:lz-l,!:lZ+!:l(ab)

3;. 1::z !:l (!:lab/TI) 2 (a/b) 2 r (!:lz) r (",-!:lz)

Q<Rez<lz

-k 1::z 5 (2TIa/b) '(a/b)' r ( Y,,-lzz) r (lzz)

~ 1:2z lz(2TIa/b) - '(a/b) r(lzz) r (~ .. -lzz)

o < Re z < ~2

'" <P(x) cp (z) = f <P{X)X z- 1 dx

0

13.39 -1 -z Re > 0 r (v,ax) z a r(v+z) z

13.40 eaxr{v,ax) a- z [r{l-v)]-lr (z)r(v+z)

0 < Re z < l-Re v ·r{l-v-z)

=[r{l-v)]-l wa-zr {z)csc[w{v+z)]

13.41 -bx e r (v,ax) z-la-z{l+b/a)-z-vr (z+v)

Re z > 0 . F [l,v+z,l+z:b/{a+b)] 2 I

13.42 -bx y{v,ax) v-1 (1+b/a)-v-z r (z+v) e

Re z > -Re v . F [l,v+z:v+l: (l+b/a) -1] 2 I

13.43 e -2b~i (u-2ba) \, '+' -w (a/b)'1 '1Zr{~z)cot{~'lfZ)K,+, (2ab) '1 '1Z

+e2b~i (-u-2ba) u = 2b (a 2+x2) \ 0 <Re z < 1

13.44 -2ab _w-\'z-l(a/b)~+~Zr(~Z)K~, (2ab) e Ei (-u+2ba) '1Z

+e 2abEi (-u-2ba) u = 2b{a 2 +x 2 )\ Rez > 0

13.45 D [(ix~)] ~[r(-v)]-lr(Z)B(-~V-~z,~+z) v

~ 0 < -Re v °Dv[(-ix) ] < Re z

13.46

13.47

13.48

13.49

¢ (xl

e -bxD (ax~l v

e - ~ax~_ (ax) --:K,]1

-~-Re ]1 < Re z < Re K

00

@(zl = I ¢(xlxz-1dx o

o < Re z < -~Re v

. r (2 z) Re z > 0

4b-a 2 F (-~v,z;~+z-~v; -----)

2 1 4b+a2

Re z > 0

r (1+2]1) [r (~+]1+K) r (~+]1-z) ]-1

-z 'a r(K-z)r(~+]1+z)

Re z > ± ~Re a

146

$(X)

13.50 e-bxM __ (ax) -><, Il

eRe Z > -l:!-Re Il

b > a

13.51 . l:!alc e WK ,1l (ax)

-l:!±Re Il < Re z < -Re K

13.53 -bx._ e wK,Il(ax)

Re z > -l:! ± Re Il

b > a


co

~(z) = f $(x)x Z- 1dx o

a l:!+llr (l:!+Il+z) (b-l:!a) -Il-l:!-z

• F [l:!+Il+z,Il+K+l:!ll+2Ill (l:!-b/a)-l] 2 1

= a l:!+ll r (l:!+Il+z) (b+l:!a) -l:!-Il-Z

• F [l:!+Il+z,Il-K+l:!ll+2Ill (l:!+b/a)-l] 2 1

·r(l:!+Il+z)B(l:!-Il+z,-K-z)

Re z > -l:! ± Re Il

1-K+z; l.!-b/a)

= al:!+ll r (l:!+Il+z)r(l:!-Il+Z)

• [r (l-K+z) ] -l(b+l:!a) -Il-z-l:!

• F (l:!+Il+Z l:!+Il-K .1-K+z. 2b-a) 2 1 ' , '2b+a

00

<j>(x) w(z) = J q,(x)xZ- 1dx 0

13.54 (b+x) -Pe -~ax r (1+2>1) [r (p) r (~+K+>I) j-1b z-p

~, >I (ax) 22 11-Z,1-K eG (ab )

23 P-Z,~+~,~->J

-~-Re >I < Re Z < Re (p+K)

13.55 (b+x) -Pe -~ax 31 11-Z l-K

oWK, >J (ax)

[r(p)]-lb z-PG (ab' )

23 p-z,~+~,~-~

Re Z > -~±Re ~

13.56 (b+x) -Pe~ax [r(p)r(~-K+>J)r(~-K->J)]-l

wK,>J(ax) 32 11-Z,1+K ebz-PG (ab )

-~±Re >J<Re 23 p-z,~+>J,~->J

z<Re(p-K)

148

13.57

13.58

13.59

13.60

13.61

¢ (x)

-WK ,:\ [a(b+x) 1

o < Re z < J,-Re (K+:\)

(x+b)-aeJ,ax

'W [a (b+x) 1 IJ,V

o < Re z < Re(a-lJ)

(x+b)-ae-J,ax

'W [a (b+x) 1 IJ,V

o x b

Re IJ > 0

o x b

'G


00

¢(z) = f ¢(x)xz-ldX o

31

23 /

l+lJ,a

(ab a-z,J,+v,J,-V)

bz-ae" r (z) G (ab ) lab 30 la,l-1J

23 0-Z, !z+v 1 !z-v

Re z > 0

31 "G

23 /

l+K,l-Z

(ab I-IJ-z, J,+A, J,-:\)

Re Z < l-Re (K+IJ)

30 blJ-l+zr(IJ)G

23 /l-Z,l-K

(ab 1-1J- Z , J,+A, J,- A)

RelJ>O

00

¢(x) <p(Z) = f ¢(xlxz - 1dx 0

13.62 0 x b Re Z < J,+Re (K-]..I)

13.63 -1 0 x b 23 -A-Z,J,+]..I,l-z-]..I

Re Z < Re (K-A)

13.64 0 x b 23 -Z-A,l-z+]..I,J,-]..I

Re A > _3/2 ±Re ]..I

13.65 0 x b 23 -Z-A,J,+]..I,J,-]..I ,]..I

A - 3/2 ±Re Re Z < -Re (K+A) Re > ]..I

00

z-l tjl (x) iP(z) = J tjl(x)x dx

0

13.66 (l+x)-V-J,e-J,ax r(l+2V)r(z)a-J,Z

oM [a(l+x)] -1

jl,V • [r(2v+l-z)r(jl+v+~)]

O<Re z<~+Re (v+jl) ° Mjl_~Z, v-~z (a)

13.67 (l+x) v-~e -~ax -~z a 2 r (z) W, +' (a) jl-'1Z,V '1Z

-Wjl,V [a(l+x)] Re z > 0

13.68 D2V-~ [ (ax)~] (21T) ~ (l.ia) -zr (2z)

"D [(ax)~] · [r (~4+~Z+V) r (3/4+~Z-V) ]-1 -2v-~

Re z > 0

13.69 M_jl,v(ax)wjl,v(ax) ~r(1+2v)a-zr(v+~+~z)

· [r(v+~-~z)r(v+~-jl)]-l -1-2Re V<Re z<-2Re jl

· B(-jl-!,z, l+z)

13.70 wjl,v(ax)W_jl,v(ax) ~a-zr(l+z)r(~+~z+v)r(~+~z-v)

(-1 -1 · [r(l+J,z+jl)r(l+~z-jl)]

Re z > -1±2 Re v

q, (xl

13.71 sin (cx~)

'e-~~ (x) 11,V

13.72 [x~+ (a+x) ~12 0-

'e -~~ (x) 11,V

13.73

largal<7f

13.74

largal<7f

13.75

I arga I <7f

00

~(z) = f q,(x)x z- 1dx o

Re z > -l±Re v

largal<7f,

33 1~'l'l+l1+Z 'G (a )

34 ~+v+z,~-v+z,-o-,o-

-!,±Re v < Re z < -Re(l1+o-)

33 IO'~'~+l1+z • G (a

34 -o-,l1+z,Z-l1,o-

-~±Re v < Re z < ~-Re (11+0-)

1 a 32 IO'~'~-l1+Z 7f-':ia G (a 34 -o-,z+v,z-v,o-

Re z > -~±Re v

152

13.76

13.77

13.78

13.79

¢ (x)

k sin (ex 2)

elz~ (x) ]l,V

k cos (ex 2)

'e -lz~ (x) ]l,V

k cos (ex 2)

Re Z > -lz-Re(A±V)


¢ (Z)

22 Ilz+V-Z,lz-V-Z .G (~e2 )

23 lz,-]l-z,O

-liRe v < Re Z < lz-Re ]l

-1 [r (l-]l+z) 1 r (J,+v+z) r (lz-v+z)

F (¥v+z, !.2-\)+z i !:2,l-l1+z; _~C2) 2 2

Re Z > -lz±Re v

22 Ilz+V-Z,lz-V-Z 'G (~e2 )

23 O,-]l-z,lz

-lz±Re v < Re Z < lz-Re ]l

(lza)2A r(lz+A+V+Z)r(lz+A-V+Z) r(l+]l+a) r(l+A-a) r(l+A-]l+Z)

1+ A, ~+ A, lz+ A+V+Z, ~+ A-V+Z; • F ( -a 2)

4 4 1+A+a,1+A-a,1+2A,1+A-]l+Z;

¢(x)

13.80

13.81

Re p>-l±Re ll±Re v

13.82 exp [-J,x(a+S)]

Re p>-l-Re ll±Re v

13.83 exp [J,x(a+S)]

'WK (ax)W, (Sx) ,ll ",v

'" ~(Z) = f ¢(x)x z- 1dx o

Re Z > { -J,-Re (Hv)

-J,-Re (ll±v)

·r(l+ll+v+z)r(l-ll+v+z) r(-2v)

• F (1+1l+V+z,1-1l+v+z,J,-A+V;1+2v,3/2-K+V+z;1) 3 2

+ [r (J,-Hv) r (3/2- K- v+ z ) ]-1

·r(l+ll-v+z) r(l-ll-v+z) r(2v)

• F (1+1l-v+z,1-1l-v+z,J,-A-v;1-2V'~2-K-v+z;1) 3 2

aJ,+ll s-ll-J,-z F (J,+K+ll, 3 2

1+1l+v+z,1+1l-v+z;1+21l'~2-A+1l+z;-a/S)

33 1J,+ll,J,-ll,l+A+Z 'G (S/a )

33 J,+v+z,J,-v+z,-K

-l±Re ll±Re v<Re z<-Re (K+A)

154

13.84

13.85

13.86

13.87

13.88

¢ (x)

exp [ -lzx ( a- S) 1

exp [-lzx(a+S) 1

exp [-!z(x/a+S/x)]

exp [!z(x/a-S/x)]

.WK,jJ(x/a)WA,v(S/x)

Re S >0, / arga / <37[/2

exp[!z(x/a+S/x) ]

'wK,jJ(x/a)WA,v(S/X)

/ arg (a, S) / <31T12


00

~(Z) = J ¢(x)xZ-ldx o

-Z -1 S [f (!z-A+v) f ('rA-V) 1

23 j1ftjJ, !z-jJ,l+A+Z ·G (S/a )

33 1ftv+ Z , !z-v+ Z , K

Re Z > -l±Re jJ±Re v

22 j1ftjJ,!rV,l-A+Z S-zG (S/a )

33 !z+v+ Z, !z-v+ Z, K

Re Z > -l±Re v±Re jJ

40 jl-K,l-A-Z

SZ G24 (S/a !z+jJ, !z-jJ, !z+v-z, !z-V-Z)

Re(a,S) > 0

SZ[f(!z-K+jJ)f(!z-K-jJ)]-l

41 jl+K,l-A-Z ·G (S/a )

24 ~+J.l,~-J.l,1z+v-z,1z-'J-z

Re z<-!z-/ Re v /-Re K

Z -1 S [f(!z-A-V) f(!z-K+jJ) f(!z-K-jJ) fC!z-A+v)]

42 /l+K,l+A-Z

.G24 (S/a !z+jJ,!z-jJ,!z+V-Z,!z-V-)

-!z+Re A+ / Re jJ / <Re z<!z-/ Re v I-Re K

1.14 Elliptic Integrals and Elliptic Functions 155

1.14 Elliptic Integrals and Elliptic Functions

00

<P(x) 1 Re z > 0

14.2 K[(1_x 2/a 2)J,j x < a \lIa z r 2 (J,z) [r (J,+~z) ]-2

0 x > a Re z > 0

0 x < a \lIa2r2(-~z) [r(J,-~z) ]-2

14.3 K[ (1-a 2/x 2) J,] x > a Re z < 0

0 X < a k z-k -z- 5/2 2

1I 2a 22 r (\-J,z)

14.4 (a+x) -J,K [ (~~:) J,] x > a • [r (l-z) ]-1 Re z < J,

14.5 (a+x) -lK ( I a-x I) 1/8 11 -1 aZ-l[r(J,z)r(~_J,z)]2 a+x

0 < Re z < 1

14.6 h (a2+x2) - 2 \az-lr2(J,z)r(~_~z)

'K [a (a 2+X2) -J,] [r ( ~+ J,z) ] -1 0 < Re z < 1

14.7 (a 2+X2) -~ [x (a 2+X 2) -J,] \az-lr(~z)r2(J,-J,z)

0 < Re z < 1 • [r (l-J,z) ]-1

14.8 (a 2+x2)-J, !aa z-l zr2 (J,z) r (J,-J,z)

'E [a (a 2+X 2) -J,] ·[r(J,+~z)] -1

0 < Re z < 1

156

14.9

14.12

14.13

<P(x)

O<Rez<1

2 2 1 'K{ [x + (a-b) 2] '2}

x 2+(a+b)

k (a2+x2) - 2

o x < a+b

2 2 -~ [x - (a-b) ]

x > a+b


00

W(z) = I <P( ) x z - 1dx o

-z-l -1 1T2 r(lz-'iZ)r(Z) [r(lz+lzz)]

O<Rez<l

a 2 +b 2

'q-lz-lzz ([ 2 2[) a -b

O<Rez<1

• [r (l-z) r (h+lzz) ]-1

O<Rez<l

[ 2 2[~Z-~ a 2 +b 2 • a -b q 1 1 ( )

-'2Z-'2 [ 2 2[ a -b

Re z < 1

<jJ(x) <jl (z) = 7 z-l <jJ(x) x dx 0

14.14 e (0 I ax) 21T -2za -z (22z_1 ) r (z) 1;(2z) Re z > !o 2

14.15 e (0 I ax)-l -2z -z 21T a r (z) 1; (2z) Re z > J, 3

14.16 e (J, I ax) 21T-J,a- z r(!o-z) (2 1 - 2z _1) 1;(1-2z) 3

Re z > 0

14.17 e (0 I ax) 2a-z1T-2z(21-2z_1)r(z)1;(2Z) 4

Re z < 0

14.18 e (y I x) 1T-J, 22z-l r (J,_z) 1

-~ < Y < J, • [1;(1-2z,~-J,y)-?;(1-2z,~+J,y)

+1; (1-2z, 3/4 +J,y) - 1; (1-2z, 3/4-~Y) J

- 1T-~22z-lr(J,-z) 14.19 e (y I x) 2

0 < Y < 1 • [1;(1-2z,~y)-1;(1-2z,J,+~y)

-?;(1-2z,~-J,y)+1;(1-2z,1-J,y)J

14.20 e (y Ix) 3

-~ 1T r (~-z) [1; (1-2z ,y) +1; (1-2z,1-y) J

0 < Y < 1 Re z < 0

00

<P(x) ¢(z) = ! ¢(x)xz - 1 dx 0

14.21 e (y Ix) 1T -':,r (':,-z) [z; (1-2z ,':,+y) +1:; (1-2z,':,-y)] 4

-" < Y < J, Re z < 0

14.22 e (ylx) 1T -':, 22z-l r (J,-z) [z; (1-2z ,,,,+':,y) - z; (1-2z, 3/4 +J,Y) 1

-J, < Y < J, +1:; (1-2z,,,,-J,Y)-1:; (1-2z, 3/4 -"y) 1

Re z <%

14.23 e (y I x) 1T -':, 2 2z- 1 r (,,-z) [I;; (1-2z, J,y) -I;; (1-2z, J,+W) 2

0 < Y < 1 +Z;(1-2z,"-"Y)-1;;(1-2z,1-':,y)] Re z<-'1i:

14.24 e (y Ix) 1T -"r (.,-z) [I:; (1-2z ,y) -Z; (1-2z ,1-y) ] 3

0 < Y < 1 Re z < 0

14.25 e (y Ix) 1T -"r (.,-z) [z; (1-2z, .,+y) -I;; (1-2z,,,-y)] 4

-., < Y < " Re z < 0

2 1+2z 1T -2z r (Z) 00

(_1)n(2n+l)-2z 14.26 e 1 (y I x) I 0

-" < y < ." 'sin[(2n+l)1TY] Re z > 0

21+2z1T-2zr(z) 00

-2z 14.27 e (ylx) I (2n+l) cos [(2n+l)1TY] 2 0

0 < y < 1 Re z > 0

'" z-l q, (x) cp (z) = f q,(x)x dx 0

00

27T- 2z r (z) -2z 14.28 e (ylx)-l L n cos (27Tny) 3

1

0 < Y < 1 Re z > 0

14.29 e (yl x) -1 4

27T -2zf(z) '" L n -2z (-1) n cos (27Tny) 1

-~ < Y < Y, Re z > 0

160

1.15 Hypergeometric Functions

15.1

15.2

15.3

15.4

¢(x)

(l_x)13-y-n

· F (-n,13;y;x) x < 1 2 1

0 X > 1

n = 0,1;2,···

(l-x) y-1

F (a.,13;y;l-x) x < 1 2 1

0 X > 1

(l-x)P-1

• F (a., S;y; I-x) x < 1 2 1

o

o

(X-l)y-1

• F (a.,S;y;l-x) 2 1

x > 1

x < 1

x > 1


'" ~(z) = f ¢(x)xz- 1dx o

[r(y+n)r(y-z)r(l+13-y+Z)]-1

·r(y)r(l+13-y)r(z)r(y+n-z)

Re(13-y) > n-l Re z > 0

[r(y-a.+z)r(Y-13+z)]-1

·r(y)r(z)r(y-a.-13+z)

Re Y > O,Re z > { o

Re (a.+13-y)

r(p)f(z) [r(p+z)]-1

F (a.,S,p;y,p+z;l) 3 2

Re P > O,Re z > { o

Re (a.+S-y)

-1 [r(l-z)r(l+a.+S-y-z)]

·r(y)r(I+a.-y-z)r(l+13-y-z)

Re (l+a.-y) Re y > O,Re z > {

Re (l+S-y)

1.15 Hypergeometric Functions

15.5

15.6

15.7

15.8

15.9

</lex)

(l-x) 0'-1

F (o.,S;y;xz) x < 1 2 I

0 X > 1

(l-x)y-le xo

F (a.} S;y; l-x) x < 1 2 1

0 X > 1

Re Y > a

a x < a

(x-a)y-l

• F (a,S;y,a-x) x > a 2 I .

Re Y > a

F (a,b,c;-x) 2 I

Re(z,a-z,b-z) > a

-x e F (a ••• a • p q l' , p'

b ••• b . ax) l' 'q'

00

$(z) = f </l(x)x z - 1 dx o

[f (O'+z) ]-I f (0') f (z)

• F (a,S,z;y,O';z) 3 2

161

Re 0' > O,Re z > 0, I arg(-z) 1<11

-1 [f (y-a+z) f (y-S+z)] f (y) f (z)

'f(y-a-S+z) F (z,y-a-S+z; 2 2

y-a+z,y-S+z,o)

a Re z > {

Re (o.+B-y)

-1 [f(l-z)f(l+a+S-y-z)]

·f(y)f(l+o.-y-z)f(l+S-y-z)

F (l+o.-y-z,l+B-y-z,l+a+S-y-z,l-a) 2 I

Re(l+a-y) Re z > {

Re (l+B-y)

f(c)f(a-z)f(b-z)f(z) f (a) f (b) f (c-z)

c+ 0,-1,-2,'"

f(z) F ( z a ••• a'b '" b 'a) p+1 q , l' , P' l' , q'

p < a, Re z > 0

00

z-l <jJ(x) 1>(z) = J <jJ(x)x dx 0

15.10 v-l B(v,z) +IF +l(a ,"',a ,z; (l-x) F (a ••• a .

pq l' 'p' P q 1 P

b 1,···,bq ;ax) x < 1 b 1,···,bq ,z+v;a)

0 x > 1 Re v > 0, Re z > 0

15.11 K (2x!:2) F (a ••• a . v p q l' , p' !:2r(z+!:2v)r(z-!:2v)p+2Fq(!:2v+z,

b 1"" ,bq;ax) z-.!zv , a 1 I ..... lap i b 1 ' ..... ,b qi a

Re z > ~IRe vi p < 9 - 1

15.12 KA (ax)K]1(ax) Sinha, S. , 1943:

• F (a ••• a 'b ... Bull. Calcutta Math. Soc. Pq l' 'P'1

, , 35, p. 37-42

b 'bx 2 ) q'

Part II. Inverse Mellin Transforms 163

2.1 General Formulas

00

z-l ¢ (z) = f x q,(x)dx q, (x) 0

(2'JTi) -1 c+ioo

1.1 ¢ (z) f -z ¢(z)x dz c-ioo

1.2 -z

q,(ax) ; > 0 a ¢(z) a

1.3 ¢ (z+a) xaq,(x)

1.4 ¢(pz) p-l<jl(x1 / p ) ; p > 0

1.5 <jl (-pz) p -lq, (x -lip) ; p > a

1.6 -z p-l(ax)a/pq,[(ax)l/Pj, a <jl (pz+a) p,a > 0

1.7 -z a <jl(a-pz) p-l(ax)-a/pq,[(ax)-l/Pj;p,a> 0

1.8 <jl (n) (z) q, (x) (log x) n

For further formulas see Part I, 1.1.

164 II. Inverse Mellin Transforms

2.2 Algebraic Functions and Powers of Arbitrary Order

00

$(2) J 2-1 ¢ (x) = x ¢(x)dx

0

2.1 -1 2 1 < 2 a x a

Re 2 > 0 0 x > a

2.2 -1 z 0 < 2 a x a

Re 2 < 0 -1 x > a

2.3 -1 2 (x/a)\) (\)+2) a x < a

Re 2 > -Re \) 0 x > a

2.4 -1 z 0 (\)+2) a x < a

Re 2 < -Re \) (x/a)\) x > a

2.5 -2 2

-log(x/a) < 2 a x a

Re 2 > 0 0 x > a

2.6 -2 2 0 < 2 a x a

Re 2 < 0 log (x/a) x > a

2.7 -2 z - (x/a) \)log (x/a) (\)+2) a x < a

Re z > -Re \) 0 x > a

00

z-1 cp (z) = J x cfJ(x)dx cjJ(x) 0

2.8 -2 z

(v+z) a 0 x < a

Re z < -Re v (x/a) v 10g (x/a) x > a

2.9 [(z+a) (z+13)] -1 -1 a S (S-a) (x -x ) x < 1

Re z > -Re(a,S) 0 x > 1

2.10 -1 -1 a 1 [(z+a) (z+S)] (8-a) x x <

S -1 S 1 -Re a < Re z < -Re (8-a) x x >

2.11 [(z+a) (z+8)] -1

0 x < 1

< -Re(a,S) -1 8 a > 1 Re z (il-a) (x -x ) x

2.12 (Z2+ a 2) -1 -1 sin[a log (1/x) ] < 1 a x

Re z > JIm aJ 0 x > 1

2.13 (z2+a 2) -1 0 x < 1

Re z < JIm aJ -1

a sin(a logx) x > 1

2.14 (Z2+ a 2) -1 -~ia -1 -ia < 1 x x

-Im a Re < Im -~2ia -1 ia > 1 < Z a x x

Im a > 0

00

<1>(z) = f z-l x 1 1m a < 0

2.16 z (z2+ a 2)-1 cos (a log x) x < 1

Re z > 11m al 0 x > 1

2.17 z(z2+a 2)-1 0 x < 1

Re z > 11m al -cos (alog x) x > 1

2.18 z(z2+ a 2)-1 !zx -ia < 1 x

1m a > 0 -!.zx ia x > 1

-1m a < Re z < 1m a

2.19 z (z2+ a 2)-1 .!.2X ia

< 1 x

1m a < 0 -!zx -ia x > 1

1m a < Re z < -1m a

2.20 z-1(z2+a 2)-1 2a -2 sin2[~a1og(1/x)1 < 1 x

Re z > 11m al 0 x > 1

'" ¢(z) f z-l

<t> (x) = x <t>(x)dx 0

2.21 z (Z2+ a 2) -1" ~a-llog(1/X)sin[al0g(1/x)1 x < 1

Re z > 11m al 0 x > 1

2.22 -1 -~ a -~2Erf{ [a 1< z (z+a) log (l/x) 1 '} x < 1

Re z > 11m al 0 x > 1

2.23 (Z2+ a 2) -~ -1< [~1Tlog (l/x)] 'cos [ (alog (l/x) ]

• [(z2+a2)~+z]~ x < 1

Re z > 11m al 0 x > 1

(z2+a2)-~ -1< 2.24 [~1Tlog (l/x)] 'sin [ a log(l/x)]

• [(Z2+ a 2) lz_z] ~ x < 1

Re Z > 11m al 0 x > 1

2.25 -v Z [r(v)]-l[log(a/x)]V-l x < a z a

Re z > 0, Re v > 0 0 x > a

2.26 -v z 0 (-z) a x < a

Re Z < o ,Re v > 0 [r(v)]-l[log(x/a)]v-l x > a

00

<!l(z) J z-l

<)lex) = x <)lex) dx 0

2.27 -1 -v a-v [r(vl]-ly[v,alog(l/x)] < 1 z (z+a) x

Re v > 0 0 x > 1

Re z > (O,-Re a)

2.28 -1 -v z (z+ a) -v -1 -a [rev)] r[v,a1og(l/x)] x < 1

Re v > 0 -a -v x > 1

-Re a < Re z < 0

, [( z+a) 'z_a'z] v kv -1 ~a 2.29 'zV(X 2 [log(l/x)] x 2 I, ['zalog(l/x)]

'2V Re z>-Re a,Re v>O x < 1

0 x > 1

2.30 [(z+a) l-i+S'z] v -(2/c)'zV[21og(1/x)]-1-'zvx a-'zS

k } Re z>-Re a,Re v<O 0DV_1 {[2S1og(l/X)] 2 x < 1

Re S > 0 0 x > 1

2.31 (z+a) -Yz (2/TI)'z[210g(1/x)]-'z-'zvx a-'zS

[(z+a) 'z+S'-i] v oDv {[2S10g(1/x)]'z} x < 1

Re z>-Re a,Re v<l 0 x > 1

Re S > 0

'" 1>(z) = f z-l x q,(x)dx q, (x)

0

2.32 z -v [(z+a) lz_alz] v k kv-l ka (2/'JT) 2V [2log(l/x)] 2 x 2

Re z>-Re a,Re V>O oD_ v_1 {[2a1og(1/x)]lz} x < 1

0 x > 1

2.33 -v z (z+a) -~ (2/'JT)lz[21og(1/x)]lzv-lzx Yz a

[(z+a) lz_a Yz ] v k .D_v {[2a1og(l/x)] 2} x < 1

Re z>-Re a, 0 x > 1

Re v > -1

2.34 (alz+zYz) v -(2'JT)- lzV2- Yzv [1og(l/x)]-1-lzvx -lza

Re z>O, Re v>O ·D 1{[2a10g(l/X)]lz} x < 1 v-

0 x > 1

2.35"'" [(z+a) (z+S) ]-v rr lz [r(v)]-1(a-S)lz-v[1og(l/X)]v- Yz

Re z>-Re (a, b) k(a+B) 1 ox' I 1 [lz(a-S)log(l/x)] x <

\J-~

Re v > 0 0 x > 1

2.36 (z_a)v(z+a)-Il (2a)Yzv-lzll [r(ll_v)]-1[log(l/x)]YzIl-YzV-l

Re z>IRe a/ °Mlzll+lzv ,lzll_lzv_lz[2alog(1/x)]x < 1

Re (Il-v) > 0 0 x > 1

00

<!l(z) ! z-l <P(x) = x <P(x)dx

0

2.37 (z-a) j.l(z-S)-v (a_S)~j.l-~V[r(v_j.l)J-lx-~a-~S

Re z>Re(a,S) 'M~j.l+~V,~V_~j.l_~[(a-S)log(l/x)]

x < 1

0 x > 1

2.38 (a-z) j.l(a+z)-v (2a)~j.l-~V[r(V)]-1[lOg(1/x)]~V-~j.l-l

Re z> I Re al ·W~j.l+~V,~V_~j.l_l[2alog(1/x)] x < 1

Re (v-j.l) > 0 0 x > 1

2.39 (z-a)j.l(S-z)-v [r(_j.l)]-l(S_a)~j.l-~VX-~a-~s

Re a<Re z<Re S ~v-kj.l-l • [log (l/x) ] 2

Re (V-j.l) > 0 'W -~j.l-~V, ~+~j.l-~V [ (S-a) log (l/x) ,]

x < 1

[r(v)]-l(S-a)~j.l-~v[log(l/x)]~V-~j.l-l

-!za-!zS 'X W, +' , +' , [( S-a) logx]

~}.l ~\J,Yz ~}.l-~V

X > 1

2.40 (Z2+ a 2) -v TI~(2a)~-v[r(v)]-1[log(1/x)]V-~

Re z 11m al 'Jv_~[a log(l/x)] x < 1

Re v > 0 0 x > 1

2.41

2.42

2.43

2.44

2.45

ro

~(z) = f XZ-l~(x)dx o

Re Z>IRe al

Re v > 0

-Re a<Re z<Re a

Re v > 0

[z+ (Z2+ a 2) ~]-v

Re z> I 1m a I,Re v>O

(Z2+ a 2) -~

[z+ (Z2+ a 2) ~]-v

Re z>I1m al,Re v>-l

(Z2_ a 2) -~

• {[z+ (z 2_a2)~] v

_[z_(z2_a2)~]V}

Re z > -IRe al

~ (x)

1V_~[a log(l/x)]

o

'KV_~ [a log (l/x) J

x < 1

x > 1

x < 1

x > 1

-v 1-1 va [log(x)] J v [alog(l/x)] x < 1

Ox> 1

x < 1

o x > 1

-1 v 2~ a sin(~v)Kv[alog(l/x)] x < 1

o x > 1

-1 < Re v < 1

cjJ (z) J Xz-l~(x)dx ~(x) o

2.46

Re z > 11m al

x < 1

o x > 1

2.3 Exponential and Logarithmic Functions 173

2.3 Exponential and Logarithmic Functions

00

1>(z) = J xz-l<p(x)dx <P(x) 0

3.1 etZ 2 -'< _hy2 let e lz(net) '2e 4

Re et > 0

3.2 V etZ 2 -l<v -l< -l:--ay2 I et Z e (2et) 2 (net) 2e

-l< Re z > 0, Re et > 0 .Dv [(2et) 2y]

z -lze et/z -k k 3.3 (ny) 2cosh [2 (ety) 2] x < 1

Re z > 0 0 x > 1

3.4 z -lze -et/z -k k: (ny) 2COS [2 (ety) 2] x < 1

Re z > 0 0 x > 1

-3.5 z -v-Ie Cl./z (Ylet)lzV 1v [2(ety)lz] x < 1

Re z > 0, Re v > -1 0 x > 1

3.6 z-V-1e -et/ z k\) 1:: (y let) 2 J v [2 (ety) 2] x < 1

Re z > 0, Re v > -1 0 x > 1

Y = log (l/x)

'" z-l ¢I(z) = f x 0, Re a 2 > 0 0 x > 1

3.8 z -~e -az~ _~ _!;;.a2./y (7fY) e x < 1

Re z > 0, Re a 2 > 0 0 x > 1

3.9 -1 -az~ Erfc (~ay -~) < 1 z e x

Re z > 0, Re a 2 > 0 0 x > 1

~ Erf (~ay -~) 3.10 z-l(l_eaz ) x < 1

Re z > 0, Re a 2 > 0 0 x > 1

k -v-~ -~ -v-l _1/8a2/y 3.11 v -az 2

z e Z 7f Y e

a 2 _k

Re z > 0, Re > 0 oDZv+1 [a(Zy) 2] x < 1

0 x > 1

3.1Z 1/

exp(-3z 3) !:; -1 _ 3/. -~

3 27f Y 2K 1. (Zy 2) /3

X < 1

Re Z > 0 0 x > 1

y = log (l/x)

'" <l>(z) = f X Z- 1 ¢(x) ¢(x)

0

_I" 1/ 3~( 1lY) -lK (2y -~) 3.13 z 3exp (-3z 3) x < 1

2/3

Re z > 0 0 x > 1

_ 2/ 1/

1T -1 e"W) -~ (2y -~) 3.14 z 3exp (-3z 3) x < 1 1"3

Re z > 0 0 x > 1

3015 (Z2+ a 2) -J, k -b J o [a(y2_b2) 2] X < e

·exp[-b(z2+a2)~] 0 x > e -b

Re z > 11m 0.1

3.16 (Z2_ a 2) k -b I [a(y2_b 2) 2] X < e 0

k -b ·exp [-b (Z2_ a 2) 2] 0 X > e

Re z > IRe 0.1

3.17 (z2+a2)-J,[(z2+a2)~_z]V \) !":V a [(y-b) / (y+b) ] 2

k k -b °exp [-b (Z2+ a 2) 2] oJv [a(y2-b 2) 2] x < e

Re z > 11m ai, Re v > -1 0 x > -b e

y = log (l/x)

176

3.18

3.19

3.20

3.21

3.22

00

~(z) = f xz-lq,(X)dx o

Re z > IRe ai, Re v > -1

Re z > 0

Re z > 0

-1 z log z

Re z > 0

z -l:ilog z

Re z > 0

y = log (l/x)

II. Inverse Mellin Transforms

q, (x)

a v [(y-b) / (y+b) ]l:iv

.I [a(y2_b 2 ) l:i] v

o

'log (y/a)

o

-b x < e

-b x > e

x < 1

x > 1

k ~ -l:i1TY [2 (ay) 2]-l:iJ [2 (ay) ] o 0

'log(y/a)

o

-y-log Y

0

-l< - (1TY) 2 [log

0

y+y+log

x < 1

x > 1

x < 1

x > 1

4] x<l

x > 1

00

z-l <l>(z) = f x q,(x)dx QJ (x) 0

3.23 -n-1 (n!) -lyn (1+lz+' " "+ 1 -y-1og y) z log z n

n = 1,2,3,·" x < 1

Re z > 0 0 x > 1

3.24 -n-lz z log z 'IT - lz22n (n!) [(2n) ! ]-lyn-lz

n = 1,2,3, ••• • [2 (1+ !. +". "+ 2~_1)-Y-10g(4Y)] 3

Re z > 0 x < 1

0 x > 1

3.25 -\I z log z [r(\I)]-ly\l-l[~(\I)_log y] x < 1

Re z > 0, Re \I > 0 0 x > 1

3.26 -1 z log (1+z/a) -Ei (-ay) x < 1

-Re z > 0 0 x > 1

3.27 -1 z log(z/a-1) -Ei (ay) x < 1

Re z > a 0 x > 1

3.28 log(l-a/z) -1 a Y (I-x) x < 1

Re z > a 0 x > 1

Y = log (l/x)

'" <!>(z) f z-l q,(x) = x q,(x)dx

0

3.29 10 (~) -1 -a a < 1 g z-a y (x -x) x

Re z > a 0 x > 1

3.30 10 (z+b) -1 a b < 1 g z+a y (x -x ) x

Re z > -(a,b) 0 x > 1

3.31 (z+a)-llog(z+a) _xa (y+log y) x < 1

Re z > -a 0 x > 1

3.32 -1 (z+a) log (z+b) x a {log(b-a)-Ei[-y(b-a)]} x < 1

b > a, Re z > -b 0 x > 1

3.33 -1 z log (l +a 2/ z 2) -2Ci (ay) x < 1

Re z > 0 0 x > 1

3.34 z-110g(z2+a 2) 2 log a-2Ci(ay) x < 1

Re z > 0 0 x > 1

3.35 (z2+a 2)-110g z a-1 {cos(ay)Si(ay)+sin(ay)

Re z > 0 • [log a-Ci(ay)]} x < 1

0 x > 1

Y = 10g(1/x)

'" z-l oj> (z) = f x <P(x)dx 0 0 x > 1

3.37 10g(1-a 2/z 2) 2y-l[l-cosh(ay)] x < 1

Re z > a 0 x > 1

2 2 -1 3.38 10g(~) 2y [cos (by) -cos (ay)] x < 1 z2+b 2

Re z > 0 0 x > 1

3.39 10g[c 2+(z+a)2] 2y -1 cos (cy) (xa_xb ) x < 1 c2+(z+b) 2

Re z > 0 0 x > 1

z2_a 2 -1 1 3.40 log (--) 2y [cosh (by) -cosh (ay) ] x <

z2_b 2

Re z > (a,b) 0 x > 1

3.41 (Z2+a 2)-1 -1 sin (ay) [y+log (~y fa) +ci (2ay) ] -a

·log (z 2+a 2) -1 +a cos (ay)Si(2ay) x < 1

Re z > 0 0 x > 1

Y = log (l/x)

00

<l>(z) f z-l ¢(x) = x ¢(x) dx 0

J, -1 a 2y k 3.42 109(z,+a) ye Erf( ay 2) x < 1 z~-a

Re z > 0 0 x > 1

3.43 (Z2+a 2)-J, Jo(ay) log a-J,1TYo (ay) x < 1

'log[z/a+(1+z 2/a 2)J,] 0 x > 1

Re z > 0

3.44 (Z2+ a 2) -J, -J,1TYo (ay)-Jo (ay) [y+log(2y/a)]

'log(z2+a 2) x < 1

Re z > 0 0 x > 1

3.45 (Z2+a 2) -J, -J,'fTHo (ay) x < 1

k 'log[a/z+(1+a2/z2) 2] 0 x > 1

Re z > 0

3.46 (Z2_ a 2)-J, Ko(ay)-Io(ay) [y+log(2y/a)]

-log (Z2_a 2) x < 1

Re z > a 0 x > 1

y = log (l/x)

'" z-l (z) = f x q,(x)dx q,(x)

0

3.47 (Z2_ a 2)-lz I o (ay)log a+Ko (ay) x < 1

k 0log(z+(Z2-a 2) 2) 0 X > 1

Re z > a

3.48 (Z2_ a 2)-lz Ko(ay) x < 1

'log[z/a+(z2/a 2-1)lz) 0 x > 1

Re z > -a

3.49 -1 2 z (log z) (y+1og y)-1T 2/6 x < 1

Re z > 0 0 x > 1

3.50 -v 2 z (log z) [f (v) )-ly V-1

Re z > 0 • {[1/1(v) -log y)2-1/1' (v)}

x < 1

0 x > 1

2.4 Trigonometric and Hyperbolic Functions

00

(z) = f xz-lcp (x) dx cp(x) 0

4.1 z CSC(lIZ) (_1)n ll-1(x/a)n(1+x/a)-1 a

-n < Re z < I-n,

n = O,±1,±2,···

4.2 z sec (lIZ) (_l)nll-l(x/a)n+~(l+ x/a)-l a

-n-~ < Re z < -n+!z,

n=O,±1,±2,'"

4.3 z cot(lIZ) lI- l (x/a)n(l-x/a)-l a

-n < Re z < l-n,

n=O,±1,±2,'"

Principal value

4.4 aZtan (lIZ) -1 n+~ -1 -11 (x/a) (I-x/a)

-n-~ < Re Z < ~-n

n=O, ±I, ± 2, II • 10 ;

Principal value

4.5 -1 Z -1 Z a CSC(lIZ) 11 log (l+a/x)

0 < Re Z < 1

2.4 Trigonometric and Hyperbolic Functions 183

'" 1>(z) = f Xz- 1 q,{x)dX q, (x) 0

4.6 -1 -z csc (-1Iz) -1 Z a 'IT log(l+ax)

-1 < Re Z < 0

4.7 -1 -1 -1 (l-z) csc{'1Iz) '11 x log(l+x)

0 < Re Z < 1

4.8 Z-lc-z{az_b z ) '11 -llog [(a+cx) / (b+cx) J

'csc('1Iz) 0 < Re Z < 1

-1 -z sec (!z'1lz) -1 4.9 Z a -2'11 arctan (ax)

-1 < Re Z < 0

4.10 -1 -z sec (!z'ITz) 2'11 -1 arccot(ax) Z a

0 < Re Z < 1

4.11 -1 -z cot ('ITz) logll-axl -1 < Re < 0 Z a Z

4.12 -1 -11 ll+xl -1 1 Z tanh (!z'1lz) 'IT og I-x < Re Z <

co

<jJ (Z) f z-l <p (x) = x <P(x)dx

0

4.13 z CSC(1TZ) -1 sin8 x(x 2+2ax+a 2)-1 a 1T a

'sin (z8) -1 < Re z < 1

-1T < 8 < 1T

4.14 -1 cos(z8)CSC(1TZ) -1 COS8+X2) z !z1T log(1+2x

-1 < Re z < 0 -1T < 8 < 1T

4.15 csc 2 (1TZ) -2 n -1 1T x (x-l) log x

-n < Re Z < l-n,

n=O,±1,±2,"',

Principal value

4.16 sec 2 (1TZ) -2 n+'" -1 1T x 2(x-l) log x

-n-!.z < Re Z < !z-n,

n=O,±1,±2,···j

Principal value

4.17 csc 3 (1TZ) !z1T-3 (-x)n(1T 2+1og 2 x)/(1+x)

-n < Re Z < l-n,

n=O,±1,±2,· .. ·

00

z-l <p(z) = f x cp (x) dx cp (x) 0

4.18 a Zcsc(1Iz) -1 a (a+xcos8) (x 2 +2ax cos8+a 2 )-1 11

cos(z 6) 0< Re z < 1

-11 < 8 < 11

4.19 z-'zsin(a/z) (1IY) -'zsinh [ (2ay) 'z] sin [ (2ay) 'z] x < 1

Re z > 0 0 x > 1

4.20 z -'zcos (a/z) -k k k (1IY) 'cosh[(2ay) 2]cos[(2ay) 2) x < 1

Re z > 0 0 x > 1

4.21 -v sin (a/z) ~v-~ k z (y/a)' '{sin (311V/4+1I/4)berV_1 [2 (ay) ']

Re z > o , -cos (311V/4+1I/4)bei v _1 [2 (ay)'z] }

Re v > 0 x < 1

0 x > 1

4.22 -v cos (a/z) kv-k h z - (y/a) 2 '{cos (311v/4+1I/4)berv _ 1 [2 (ay) ']

Re z > 0, +sin(3 11v/4+1I/4)bei v _ 1 [2(ay)'z]}x < 1

Re v > 0 0 x > 1

y = log (l/x)

00

z-l 0 0 x > 1

4.24 -~e -az!z (~)-~exp[-~(a2-b2)/YJcos(~ab/y) z

·cos (bz~) x < 1

a ,;, b, Re z > 0 0 x > 1

4.25 z-vcos(az-~) [ f(v) J -ly v-I F ( iV, !.2;-~a 2y) < 1 x

0 2

Re z > 0, Re v > 0 0 x > 1

4.26 z-vsin(az-~) a [f (~+v) ]-1 F ( ; !z+v , ~2; -!za 2y) x < 1 0 2

Re z > o ,Re v > -!z 0 x > 1

4.27 arctan (a/z) -1 sin(ay) < 1 Y x

Re z > 0 0 x > 1

4.28 -1 arctan (z/ a) z -si (ay) x < 1

Re z > 0 0 x > 1

Y = log (l/x)

00

<!> (z) = f x z - 1 <j> (x) <j>(x) 0

4.29 arctan[2az/(z2+b 2-a 2 )] 2y -1 sin (ay) cos (by) x < 1

Re z > 0 0 x > 1

4.30 (z2+a 2)-1 ~a -1 cos (ay) [ei (2ay) -y-1og (2ay) ]

'arctan (a/z) +!za -1 sin(ay)Si(2ay) < 1 x

Re z > 0 0 x > 1

4.31 log(z2+a 2) -2y -1

sin (ay) (y+1ogy) < 1 x

'arctan (a/z) 0 x > 1

Re z > 0

4.32 (z2_a2)-~ Ko (ay) x < 1

'arccos (a/z) 0 x > 1

Re z > -Re a

4.33 (Z2_a 2) -~ ~7fLo (ay) x < 1

'arcsin (a/z) 0 x > 1

Re z > Re a

4.34 sin(z2/a ) -L k

~7f 2a2sin(~ay2-~7f)

y = log (l/x)

00

z-l <l> (z) = f x <P(x) dx <P(x)

0

4.35 COS(Z2/a ) ~~-~a~cos(~ay2-~TI)

4.36 aZcsc(TIz)sinh(bz) -1 sinh b(x 2+2ax coshb+a 2)-1 TI ax

-1 < Re Z < 1

4.37 aZcsc(TIz)cosh(bz) -1 a(a+xcosh b) (x 2+2ax cosh b+a 2)-1 TI

0 < Re z < 1

4.38 z -~sinh (a/z) _h ~ k

~(TIY) 2{cosh[2(ay) 2]-cos[2(ay) 2]}

Re z > 0 x < 1

0 x > 1

4.39 z -~coSh (a/z) k k k ~(TIY) 2{cosh[2(ay) 2]+cos[2(ay) 2]}

Re z > 0 x < 1

0 x > 1

4.40 -v

sinh (a/z) hV-~ k k

z ~ (y/a) 2 2{IV_1 [2 (ay) 2]-JV _ 1 [2 (ay) 2]}

Re z > 0, x < 1

Re v > -1 0 x > 1

4.41 -v

cosh (a/z) ~(y/a)~v-~{I 1[2(aY)~]+J 1[2(aY)~1} z v- v-

Re z > 0, 0 x < 1

Re v > 0 0 x > 1

Y = log(l/x)

00

z-l cj>(z) = f x <P(x)dx <P(x) 0

z -~sinh (~ab/z) -k ~ k 4.42 ('flY) 2S in (ay 2) sin (by 2) x < 1

'exp[-~(a2+b2)/zJ 0 x > 1

Re z > 0

-k -~ k k 4.43 z bosh (~ab/z) (ny) 2COS (ay 2) cos (by 2) x < 1

'exp[-~(a2+b2)/zl 0 x > 1

Re z > 0

4.44 (z2+a 2)-1 4a'lT -2 1 sin (ay) I x < 1

coth(2az/'lT) 0 x > 1

Re z > 0

k -2 a I -2 4.45 sech (az 2) -a ['1)\j e (l,v ya ) 1 -0 x < 1 1 v-

Re z > 0 0 x > 1

k -2 a e 4 (l,v I y a - 2 ) 1 v= 0 4.46 csch (az 2) -a ['1)\j x < 1

Re z > 0 0 x > 1

-1: k -1 -2 4.47 z 2tanh (az 2) a e (Olya ) x < 1 2

Re z > 0 0 x > 1

y = log (l/z)

'" (z) J z-l

¢ (xl = x ¢ (x) dx 0

4.48 z -~coth (az~l -1 (0 Iya -2) < 1 a e x 3

Re z > 0 0 x > 1

4.49 sinh(vzY,)csch(azY,) -1 a -1 -2 a [a-v e (y,va Iya )] x < 1

4

-a < v < a 0 x > 1

Re z > 0

1,; 1: -1 a -1 -2 4.50 cosh(vz2)sech(az2) a [a-v e 1 (y,va I ya )] x < 1

-a < v < a 0 x > 1

Re z > 0

y = log (l/z)

2.5 The Gamma Function and Related Functions 191

2.5 The Gamma Function and Related Functions

'" 0 -x z e

5.2 r (z) -1 < Re < 0 -x -1 z e

5.3 sin(az)f(z) -xcosa sin(x sin a) e

Re z > -1 -~7T < Re a < ~7T

5.4 cos (az) f (z) -xcosa cos(x sin a) e

Re z > 0 -~'TT < Re a < ~rr

5.5 sin (az) f (z) -xcosa sin(x sin a) e

-m < Re z < I-m m-l + 2 (-l)rsin(ar)xrjr!

m = 2,3, ••• r=l

-!z1T < Re a < ~'IT

5.6 cos(az)f(z) -xcosa cos(x sin a) e

-m < Re z < I-m m-l - I (-l)rcos(ar)xrjr!

m = 1,2,' .. r=O

-!zIT < Re a < !zIT

5.7 sec(rrz)f(z) eXErfc (xl:!)

0 < Re z <!:i

00

z-l (z) = f x q,(x)dx q,(x)

0

5.8 sec (1Iz) f(z) e~rfc(xt,)

n-lz < Re z < n+~ -1 n-1 ( -1) r r (t,+ r) x - t,--11 I r

n = 1,2,3,··· r=O

5.9 -1 z r(t,+t,z) -1I~rf(ax)

-1 < Re z < 0

5.10 -1 Re 0 -Ei (-x) z r (z) z >

5.11 r(z)csc(1Iz) -1 eaxEi (-x) -11

0 <~ Re z < 1

5.12 r (z) cot (1Iz) -1 -ax-. -11 e E~ (x)

0 < Re z < 1

5.13 z-l r (z) cot (t,1Iz) 1I-1Ei (-x)Ei (x)

0 < Re z < 2

5.14 -1 sin(t,1Iz)r(z) -Si (x) z

-1 < Re z < 0

00

z-l <l> (z) = f x <j>(x)dx <j>(x)

0

5.15 -1

cos (~1Iz) f (z) -Ci (x) z

0 < Re z < 1

5.16 sec (~1Iz) f (z) 211-1 [Ci(x) sinx-si (x) cos xl

-1 < Re z < 1

5.17 csc(~1Iz)f(z) -211-1 [Ci(x) cosx+si (x)sin xl

-1 < Re z < 1

5.18 -1 f (~+z) 211 -"arccos (x") < 1 z f (1+z) x

Re z > -!:i 0 x > 1

5.19 f (z) f (v-z) f(v) (1+x2)-~vsin(varctan x)

'sin(~1Iz) -1 < Re z < Re v

5.20 f(z)f(v-z) f(v) (1+x2)-~vcos(V arctan x)

'cos (~1Iz) 0 < Re z < Re v

5.21 f (z) [f (~+~z-~v) 211-1(4-x2)-~cos[varccos(~x)J

-1 < 2 ·f (~+~z+~v) J x

Re z > 0 0 x > 2

'" z-l <!>(z) = f x <P(x)dx 0

5.23 -1 sec(~7Tz) r(~+z) 27T~{[~-C(ax)12+[~-s(ax)]2} z

0 < Re z < 1

5.24 -1 CSC(~7TZ)r(Z) 7T- l [Ci 2 (x)+si 2 (x)] z

0 < Re z < 2

5.25 -1 z r (v+z) r(v,x)

Re z > 0

5.26 r(Z)CSC[7T(V+Z) ] -1 x 7T r(l-v)e r(v,x)

0 < Re z < l-Re v

5.27 [r(z)/r(~-z) ]2 -1 ~ ~ 27T Ko (4x )-Yo (4x )

0 < Re z < ~8

5.28 r(~-z)r(~+~v+~z) I-v r(l+v-z) r(l-~v-~z) 2 sin x J v (x)

-l-Re v < Re z < ~

00

~(z) J z-l <p(x) = x <p(x)dx

0

5.29 f(l:;-z) f (l:;v+l:;z) I-v x JV (x) r(l:;-~v-l:;z)f(l+v-z) 2 cos

-Re v < Re z < l:;

5.30 f(l:;-z) f (z+v) n!ze -~xI v (l:;x) r (l+v z)

-Re v < Re z < l:;

5.31 f (l:;-z) f (z+v) r (z-v) ~ !.:x

n'sec(nv)e' Kv(l:;X)

± Re v < Re z < l:;

5.32 f (z-v) f (z+v) 7r -~e -!axK v (l:;x) f (~+z)

Re z > ± Re v

5.33 f (2z) f (v+z) 2J2V(2X~)K2V(2X~) f (l+v-z)

Re z > 0, -Re v

5.34 f(z)f(l:;-z)f(v+z) ~

l:;n 2 sec (nv)

±Re v < Re z < !2 • [J2 (xl:;) +y2 (Xl:;)] V v

5.35 f(a+z)jf(S+z) [f(s_a)]-l x a(l_x)S-a-l

Re z > -Re a,Re(S-a) > 0 x < 1

0 x > 1

00

<1>(z) f z-l rp(x) = x rp(x)dx

0

5.36 f(a+z)f(S-z) f(a+S)x a (l+x)-a-S

-Re a < Re z < Re S Re (a+S) > 0

5.37 f(a-z)jf(S-z) 0 x < 1

Re z < Re a, Re (S-a) > 0 [ f(B_~]-lx1-B(x_1)S-a-1

x > 1

5.38 f(a+z)jf(S-z) xlz+~ia-}2S J (2xlz) a+S-1

-Re a<Re z <-~+lzRe (S-a)

5.39 f(a+z) f(S+z) 2xlza+ lzSK S (2xlz) a-

Re z > -Re(a,S)

5.40 f(1-2z) f(a+z) J a + S (2X lz )Ja_s (2x1:2)

o[f(l+S-z)f(l-S-z)f(l+a-z)]-l -Re a < Re z < 1:2

5.41 f (1-2z) f (z+a) f (z+S) CSC(21TS) [Ja + S (2x1:2)y a _s (2x1:2)

f(z-S)jf(l+a-z) -Y a + S (2x1:2)J a _ S (2x1:2)]

-Re a } < Re z < 1:2

±Re S

DO

z-l <jJ(z) = J x <P(x)dx <P(x) 0

5.42 cos [1T( S+z) r (1-2z) h !,:

-1TJ, +'S(2x 2 )y, 'S(2x 2 ) ~a. Yz ~a-;:.z

·r(z+a) r(z+S) -Re(a,S) < Re z < ~

• [r(l+S-z) r(l+a-z)] -1

5.43 r(1-2z) r(z+a) r (z+S) 1:: 1::

2Ia+S (2x 2)Ka_S (2x 2)

• [r(l-z+a) r(l-z+S)] -1 -Re(a,S) < Re z < l,

5.44 -1 [r(2z)] r(z+a)r(z-a)

!-::: 1:: 4Ka_S (2x 2) Ka+S (2x 2)

"r(Z+S) r(Z-S) Re z > Re(±a±S)

5.45 r (a+z) r(a+S) [r(2a)]-le-l,XMS ,(x) I CJ.-~

'r(S-z)jr(a-z) -Re a<Re z < Re S,

Re(a+S) > 0

5.46 r (l-a+z) e -l,~l S ,(x) - Ict-~

• r (a+z) jr (S+z)

-l+Re a Re z > {

-Re a

5.47 r (l-a+z) l<~ r(a+S)r(l-a+S)e 2 -S,a-l,(X)

·r (a+z) r (S-z) -Re a } < Re z<Re S

-l+Re a

ro z-l (z) = J x ¢ (x) dx ¢ (x)

0

5.48 f(v+~+z)f(-~-z)f(1+2z) f (~+v-~) M (x~) W (xl-i) f(v+~-z)f(l+z-~) f (1+2v) -~,v ~,v

-~-Re v < Re z < -Re ~

5.49 f(1+2z)f(~+z+v)f(~+z-v) W (x~) W (x~) f (l+z+~) f (l+z-~) ~,v -~,v

Re z > { -~

-~±Re v

5.50 f(a+z)f(S+z) [f(y+6_a_S)]-lxa (1_x)y+6-a-S-l

-1 (o-S,y-S;y+o-a-S;l-x) • [f(y+z) f(6+z)] . F

2 1

Re z > -Re (a, S) x < 1

Re (y+6-a-S) > 0 0 x > 1

5.51 f(a-z)f(S-z) f(a)f(S) [f(y)]-l

·f (z)jf (y-z) . F (a,S;r;-x) 2 1

0 < Re z < Re(a,S)

5.52 f (~+a-z) f (S-z)

0 f (l-z) x < 4

Re z < Re (S, ~+a) ~ ( 4) -lza-lzS a+ S (' lz) TI x- P a - S ~x

x > 4

00

1> (z) f z-l ¢(x) = x ¢(x)dx

0

5.53 z-l\)l(z) -1 < 1 -y-log (x -1) x

Re z > 0 0 x > 1

5.54 z -1\)1 (l+z) -y-log (I-x) x < 1

Re z > 0 0 x > 1

5.55 z-l[Y+\)I(l+z)] -log(l-x) x < 1

Re z > -1 0 x > 1

5.56 \)I' (z) -1 < 1 (x-I) log x x

Re z > 0 0 x > 1

5.57 csc (rrz) \)I (l+z) -1 -1 1T (l+I/x) [y+log(l+I/x)]

-1 < Re z < 0

5.58 csc(rrz) [Y+\)I(l+z) -1 -1 rr (1+I/x) log(I+I/x)

-1 < Re z < 1

5.59 \)I' (~+~z) -\)I' (~z) -1 2(x+l) log x x < 1

Re z > 0 0 x > 1

5.60 \)I (z+a) -\)I (z+S) (xS-Xa)/(l_X) x < 1

Re z > -Re(a,S) 0 x > 1

00

z-l <!l(z) = J x ¢(x)dx ¢ (x) 0

5.61 1)!(z+a)-ljJ(z+S) (x B+k _x a+k )/(l_x) x < 1

-h<Re(z+a) <l-h (xa_XB+XB+k_Xa+h)/(l_x)

-k<Re(z+S)<l+k x > 1

h,k = 0,1,2,'"

5.62 r(z) 1)!(z) -x e log x

Re z > 0

5.63 B(z,V) 1)!(z+v) v-I (I-x) [1)!(v)-log(l-x)]

Re(z,v) > 0 x<l

0 x > 1

5.64 B(z,V)1)!(z) v-I -1 (I-x) [1)!(v)-log(x -1)]

Re(z,v) > 0 x < 1

0 x > 1

5.65 B(z,v-z)1)!(v-z) -v (l+x ) [1)!(v)-log(l+x)]

-1 < Re z < Re v

5.66 B(z,v-J,z) 2(1+x2)-J,[(1+x2)~_x]2V-l

• [1)!(v-J,z)-1)!(v+J,z)] k 'log[ (l+x2) 2_X]

0 < Re z < 2 Re v

'" 1> (z) = J XZ-lcp (x) dx cp(x)

0

5.67 sin(~rrz)r(z)~(z) sin x log x -~rr cos x

0 < Re z < 1

5.68 co s ( ~rr z) r ( z) ~ ( z ) cos x log x + !z7T sin x

0 < Re z < 1

5.69 -z a f(z)r;(z) (eax_l) -1

Re z > 1

5.70 -z a r(z)r;(z-l) e ax (eax_l)-2

Re z > 2

5.71 a -zr (z) r; (Hz) -ax -log(l-e )

Re z > 0

5.72 (2a) -zr (z) 1; (z-1) \'csch 2 (ax)

Re z > 2

5.73 csc (rrz) r; (1-z) -rr -1 [y+~ (x+l) 1

-1 < Re z < 0

5.74 CSC(1fZ) r;(1-z) -1 -rr [~(l+x)-log xl

0 < Re z < 1

202 II. Inverse He11in Transforms

00

z-l <!> (z) = f x ¢(x)dx ¢ (xl

0

5.75 CSC(1TZ) 1T-1[~(1+1/x)-log(1+1/x)]

-1 [1;(1+z)-z ]

-1 < Re z < 0

5.76 -1 r(z) [1; (z) - (z-l) ] e-x [1_x-1+(ex _1)-1]

Re z > 0

5.77 f(z)f(n+1-z) (-1) n-1~n (1+x)

1; (n+1-z) 0 < Re z < n, n=l, 2,3, •••

5.78 -z a f(z)f(v-z) r (v) 1;(v,l+ax)

• 1; (v-z) 0 < Re z < Re v-I

5.79 -z a f(z) 1;(2z) J,[8

3 (0 I aX1T -2) -1]

Re z > J,

5.80 -1 ·-1 -1 x < 1 z 1; (z) n, (n+1) <x<n

Re z > 1-Re a n = 1,2,3,'"

0 x > 1

00

= L H(l - n) n=l x

'" z-l 1>(z) = J x ¢ (x) dx ¢(x) 0

-1 L -a 5.81 n z I; (z+a)

1 <n<!. Re z > l-Re a = =x

5.82 -v [r (v) ]-1 L [log (L) ] v-I z I; (z)

1 <n<!. nx

Re z > 1,Re v > 0 = =x

5.83 -z c r(z)l;(z,a) (ecx_l)-le cx(l-a)

Re z > 1,Re a > 0

5.84 -z a r(z)r(v-z) r(v) I;(v,a+ax)

"I;(v-z,a) 0 < Re z < Re v-I, Re a > 0

5.85 I;(v,z/a) -1 v v-I a-I [r (v)] a [log(l/x)] (l-x)

Re z > o ,Re a > 0 x < 1

Re v > 1 0 x > 1

5.86 I; ("0, \z+\zz/a) (2a)v[f(v)]-1(log!.)V-l(x- a _xa )-1 x

Re z > -Re a, Re v>l x < 1

0 x > 1

00

z-l 1> (z) = f x ¢(x)dx ¢(x) 0

5.87 <;; (v ,l+z/a) a V[f(v)]-l(log!)V-l(x-a_l)-l x x < 1

Re z > -Re a,Re v > 1 0 x > 1

5.88 esc (1TZ) -1 1T [~(a+x)-~(b+x)]

• [<;; (1- z , b) - <;; (1- z , a) ] 0 < Re z < 1

2.6 Orthogonal Polynomials and Legendre Functions 205

2.6 Orthogonal Polynomials and Legendre Functions

6.1

6.2

6.3

6.4

6.5

00

4(z) = J XZ-l~(x)dx o

-n -1 Z Tn(l-z )

Re z > 0

Re z > 0

Re z > 0

-!.:.:-n k z 2 He 2n [(2z) 2]

Re z > 0

-n-2 k z He 2n+l [(2z) 2]

Re z > 0

y = log(l/x)

~ (x)

n -1 n-l ~ (-1) [(2n-l)!] (2y) He 2n (y 2)

x < 1

o x > 1

x < 1

o x > 1

x < 1

o x > 1

2n+~2 (n+l) -1 (y/1T) l;i(y+l)n

·Un+l[(l-y)/(l+y)] x < 1

o x > 1

co z-l \p(z) = f x <P(x) dx 0 0 x > 1

6.7 -n-l -a/z ~ (-2)n(2/1I)~nsin[2(ay)~] < 1 z e He 2n+l [(2a/z) 2] x

Re z > 0 0 x > 1

z-~(l-~/z)He [a(l-~/z)-~] -k k k 6.8 (211Y) 2[Hen (a+y2)+Hen (a-y 2)] n

Re z > 0 x < 1

0 x > 1

6.9 -n-1 n (l-a/z) (~a)n+1ynL (~ay)/n! < 1 z P x n

Re z > 0 0 x > 1

6.10 -v n (l-a/z) n-1 F (-n,n+1;1,v;~ay) 1 z P Y x <

2 2

Re z > 0, Re v > 0 0 x > 1

6.11 z -~p -1 -n .1-z ~ [(-2ay) ~l n (a/z) (n!) i (1Iy) He n [(2ay) 2]He n

Re z > 0 x < 1

0 x > 1

Y = log(l/x)

00

z-l <jl (z) = J x <j>(x) dx <j> (x)

0

6.12 z-l:;-l:;np [(2z)-l:;] -1 kn -~ ~n-h k 1 (n ! ) 2' 11 y' 'He (y') x < n n

Re z > 0 0 x > 1

6.13 (z+b) -n-lp (z+a) -1 n -by < 1 (n!) y e L (l:;by-l:;ay) x n z+b n

Re z > -b 0 x > 1

6.14 (z+b) -v p (~) -1 v-1 e-by F (-n,n+li1,vi~by-l:;ay) n z+b [f (v) 1 y 2 2

Re z > -b, Re v > 0 x < 1

0 x > 1

6.15 (Z2+a 2) -l:;n-l:; -1 n < 1 (n!) y Jo(ay) x

.p [z (Z2+a 2) -~] n 0 X > 1

Re z > 0

6.16 (z2_a 2)-l:;n-l:; -1 n 1 ( n ! ) y 10 ( ay) x <

op [Z(Z2_a 2)-l:;] 0 x > 1 n

Re z > a

y = log (l/x)

00

z-l cjl (z) = f x ¢(x)dx ¢(x)

0

6.17 z -n-2\1C\l (l-a/z) rr~21-n-4\1[f(\I)f(~\I+n)]-lan+2\1 n

Re z > 0 .yn+2\1-1L\I-~(~ay) n x < 1

0 x > 1

6.18 Z -IlC\l (l-a/z) [nB(n,2\1)f(Il)]-ly ll -1 n

Re z > 0, Re 11 > 0 F (-n,n+2\1;~+\I,Il;~ay) x < 1 2 2

0 X > 1

6.19 (z+b) -IlC\l (~) -1 ll-l -by n z+b [nB(n,2\1)f(Il)] y e

Re z > 0, Re (11, \I) > 0 F (-n,n+2\1;~+\I,Il;~by-~ay) 2 2

X < 1

0 x > 1

6.20 z -\l-~nc \I (z -~) 2~n[n!f(\I)]-ly\l+~n-l n

l< Re z > 0 'Hen [(2y) 2] x < 1

0 x > 1

6.21 z-n-1Ln (Z) (n!)-l(l+y)npn(~~i) x < 1

Re z > 0 0 x > 1

Y = log(l/x)

2.6 Orthogonal Polynomials and Legendre Functions

6.22

6.23

6.24

6.25

6.26

00

~(z) = f xz-1~(x)dx o

Re z > 0

Re z > 0, Re v > 0

Re z > O,Re(v+n) > -1

v -1 -1 ·Ln [az (l-z) ]

Re z > 0, Re v > -1

On (z/a)

Re z > 0

y = log(l/x)

Hx)

o

-1 v-I (a.+1)n[n!r(v)] y

• F (-nia.+1,viay) 1 2

o

o

o

o

209

x < 1

x > 1

x < 1

x > 1

x > 1

x > 1

x < 1

x > 1

00

<l> (z) f z-l 0 - [aY-(l+a 2y 2)'J } x < 1

0 x > 1

k _ 3,.... -k 6.28 P v (z) -2 21T 2sin (1TV) y 'K +, (y) x < 1

v 'i

Re z > -1,-1 < Re v < 0 0 x > 1

k 6.29 qv (z) (1,1T/y) 2Iv+1, (y) x < 1

Re z > 1,Re v > -1 0 x > 1

6.30 (Z2_a 2)-1,11 (2a/1T)1,[f(I1-V)f(I1+V+l)]-1

.p~(z/a) .yl1-~ (ay) \)+~

x < 1

Re z > -a,Re(l1+v) > -1 0 x > 1

Re()1-v) > 0

6.31 (Z2_a 2)-1,)1 k 11-!::: (1,1Ta)'y 21 +, (ay)

v 'i x < 1

-i 1T)1 )1 0 > 1 'e q (z/a) x v

Re z > a Re ()1+v) > -1

Y = log(l/x)

6.32

6.33

6.34

6.35

6.36

co

~(z) = f Xz-l¢(x)dX o

[ (z+a) (z-a) ]-\].!

Re z > -a

[(2+a) / (z-a) ]-\\1

Re z > a

Re z > 0,-1 < Re v < 0

p [(2ab) -1 (a 2 +b 2 _z 2 )] v

-(a+b) < Re z < a+b

-1 < Re v < 0

-(a-b) < Re z < a-b

y = log(l/x)

¢ (x)

Ox> 1

-\ < Re v < \

-1 \[r(2+2v)] r(l+v+\1)

-1 'y M 1 + (2ay)

1-1, ~ v

o Re v > -\

x < 1

x > 1

l< (ab) 2sin ('lTv) [J +, (by) Y " ' (ay) v '2 -y-'2

x < 1

o x x > 1

a > b, Re v > -1

212

6.37

6.38

6.39

6.40

6.41

00

~(z) = f xz-l~(x)dx o

Re z > 0, Re v > -1

Re z > a+b

-1 < Re v < 0

Re z > a+b, Re v > -~

k qn[(z/a) 2]

Re z > a

n = 1,2,3,'"

(2 z+a) -~ (2 z-a) ~)l

)l k 'pv [(~+z/a) 2]

Re z > ~a

Re(V+)l) < 1,Re(v-)l) > -1

y = log(l/x)


~ (x)

k 1T(ab) 2J +' (ay) J +' (by) x < 1 v :-z v ~

o x > 1

(ab)~tan(1Tv) [I +' (ay)I +' (by) v '2 v '2

o x > 1

k 1T(ab) 21 +, (ay)I +' (by) x < 1 v ~ V P2

o x > 1

-k 3 -1-~ ~a 2 r (l+~n) [r (~2+n) ]y

kay 'e 2 M, , +' (ay)

-~,~ ~

o

x < 1

x > 1

Ox> 1

00

z-l ¢(z) = J x ¢ (xl dx ¢(x) 0

6.42 (2z-a)~~p~[(~+z/a)~1 3 .... 2 - 3/211 -~ -1 v 2 a [r(~~-~v-~)r(-~v-~~)l

> ~a -kj.l- s/It

Re z 'y 2 Wy,~_~, ~v+~(ay) x < 1

Re (v-~) > -1 0 x > 1

Re (v+~l < 0

(2z-a) ~v ~ [(2z+a) y,

~v+~ -~ -1 6.43 Pv 2z-a 1 2 a [r(-~~-~v)r(l-~)l

-},:;v- 3 ....... 2 Re z > a 'y 2 M (ay) x < 1 ~V,-~l1

Re (v+~) < 0 0 x > 1

6.44 (z2+a2)-~V-~ [r(v+~+l)l-lyVJ~(ay) x < 1

.p-~[z(z2+a2)-~1 v

0 x > 1

Re z > 0 Re(~+v) > -1

6.45 (z2_a2)-~V-~ [r(v+~+l)J-lyVI (ay) x < 1 ~

.p-~[z(z2-a2)-~J 0 x > 1 v

Re z > a Re (v+~) > -1

6.46 (z2_a2)-~V-~e-iTI~ -1 v 1 [r(v-~+l)J y K~(ay) x <

'q~[z(z2_a2) -~] 0 v

x > 1

Re z > a Re (v±~) > -1

Y = log (l/x)

00

(z) = J x Z- 1q,(x)dx q, (x) 0

6.47 Z CSC(1TZ)PZ(COS 8) -1 x(x2+2ax 8+a2)-~ a -1T cos

-1 < Re Z < 0 -1T < 8 < 1T

6.48 a Zcsc(1TZ)p (cosh CI.) -1 x(x 2 +2axcosh CI.+a 2) -~ -1T Z

-1 < Re Z < 0

6.49 a Zr(l+z-v)r(-v-z) 1T-~r(~-V) (2a sin 8)-vx l-v

.pv(cos z 8) • (x2+2axcos 8+a2)V-~

-l+Re v < Re Z < -Re v -1T < 8 < 1T

6.50 a Zr(l+z-v)r(-v-z) 1T -~r(~-v) (2a sinh CI.) -v

.pv(cosh z CI.) • (x 2+2ax cosh CI.+a2) v-~

-l+Re v < Re z < -Re v

6.51 r (l+v+z) a z -1 -x -2 k a xe J v [x(a -1) 2]

-v 'P z (a),Re z > -l-Re v -1 < a < 1

6.52 r (l+v+z) (1,;2-1) ~z+~ -2 -~ x exp[-(l-1,; ) x]Iv(x)

-v -l-Re 'pz (1,;), Re z > v 1,; > 1

00

z-l <!>(z) = f x q,(x)dx q, (xl 0

6.53 2z[r(1-~z)J-l~-~z ~ k k ~ 2JV [~'(~;;:+~) 2XJJV[~ 2(~r;;-~) 'xl

-v -2 • r (v+~z) p -~z (;;:) Re v < Re z < 2

6.54 2zr(~z)r(V+~z)~-~z 41 [~~(~;;:-~)~xlK [~~(~;;:+~)~xl v v -v (0,-2 v) 'p-~z (;;:) Re z > Re

6.55 2zr(~z)r(v+~z) 4K [~~(~;;:-~)~xlK [~~(~;;:+~)~xl v v -!jz i'ITV -v

.~ e q~z-l (;;:) Re z > (0 , ± 2 Re v)

-i 7fll 11 a) (~7f)~[r(~-ll)l-l(sinh a) 11 6.56 e q (cosh z-~

Re < -~-Re ' (coshy -cosha)-ll-~ < e -a z 11 x

Re 11 < ~ 0 x > -a e

6.57 r (~+z) r (-ll-Z) (~7f~)-~(;;:2-1)-\x~

z p ~ (;;:)

-2 _1< -1 .~ 'q 1 {(l-;;: ) 2[l+X(~;;:) l}

-ll-~

-!:2 < Re z < -Re 11

6.58 r (z-~) r (l-ll-Z) 7f(~7f~) ~sec(7fll) (;;:2-1) -\x-~

.~ze-i7fllqll (;;:) -2 -1< -1 • P 1 {(l-;;: ) 2[l+X(~;;:) l} -z -ll-~

~ < Re z < l±Re 11

2.7 Bessel Functions and Related Functions

7.1

7.2

7.3

7.4

7.5

1> (zl z-l I

= f x O,Re v > -~

-v z [cos(az+b)Jv(az)

+sin(az+b)Yv(az)]

Re z > O,Re v > -~

J 2 (azl+y2(az) v v

Re z > 0

Re z > 0

y = log (l/x)

 1

x < 1

o x > 1

x < 1

o x > 1

-2 -1 2 2 2~ Y exp(y,a /y)Kv(y,a /y) x < 1

o x > 1

o x < e-a

e-a < x < e a

o a x > e

2.7 Bessel Functions and Related Functions 217

7.6

7.7

7.8

7.9

I (z) = J ~(x)x-l (x)dx

o

Re z > 0

cos (az 2) J (az 2 ) o

Re z > 0

-Re b < Re z < Re b

Re\i<~

y = log y(l/x)

~(x)

(~a)-~[cosaJ (a)+sinaY (a)l x < 1 o 0

o x > 1

(~a)-~[cosaJ (a)-sinaY (a)l x < 1 o 0

o x > 1

'" z-l (jl(z) = J x ¢(x)dx ¢(x) 0

7.10 (b2_Z2) -lzv 0 x < e -a

.J [a(b 2_z 2) lz] v

(lzb/1T) lz(ab) -v (a2_y2) 'iV-l,j

Re > -~ .J [b (a 2 _y2) lz] -a <

a v v-!;z e < x e

0 > e a

x

7.11 e-azJ [a(b 2_z 2 )lz] 0

0 x < -2a

e

1T-1 (2ay-y2)-lzcos[b(2ay-y2)lz]

x > -2a

e

7.12 z -)1 J (az -lz) v -1 )1+'-v-1 v

(lza) [r (1+v)r (lzv+)1) ] y 2

Re z > 0 , Re (v+2)1) > 0 F ( ; )1+lzv, l+v; -l,ja 2y) x < 1 a 2

0 X > 1

7.13 z)1J v (a/z) v -1 v-)1-1

(lza) [r (1+v) r (v-)1)] y

Re z > O,Re(v-)1) > 0 F ( i l+v, ~V-~lJ I ~+~'J-!zll i _l~ 6a 2y2) a 3

x < 1

0 x > 1

Y = log

00

z-l (z) = f x q,(x)dx q, (x)

0

7.14 -1< Z 2 [sin (a/z) J v (a/z)

-~ -~ 41T 2y 2COS(lTV) [cos (J,1Tv)ker 2V (a)

-cos(a/z)Yv(a/z)] -sin(y,lTv)kei 2v (a)] x < 1

Re z > 0,-\ < Re v < Y, 0 x > 1

a = 2 (2ay) Y,

z - y,[cos (a/z) J v (a/z) _:t- -k

7.15 -41T 2y 2COS (lTv) [sin (y,1Tv)ker2V (a)

+sin(a/z)Yv(a/z)] +cos(y,1Tv)kei 2V(a)] x < 1

Re z > O,-~ < Re v < Y, 0 x > 1

a = 2 (2ay) Y,

7.16 z-le -a/zJ (b/z) h: k J v (Ay 2) Iv (By 2) x < 1

v

Re z > o ,Re v > -1 0 x > 1

A = 2lz [(a 2+b 2 ) y,±a)

B

7.17 z -le a/zJv (b/z) k k

Iv (Ay 2) J v (By 2) x < 1

Re z > O,Re v > -1 0 x > 1

A = 2Y,[ (a 2 +b 2 ) y,±a]

B

y = log (I/x)

7.18

7.19

7.20

7.21

7.22

00

~(z) = J x z- 1 ¢(x)dx ¢(x) o

kZ (a/b) 2 r (~-~z)

·J~z (2ab)

±2Re v < Re z < 1

kZ (a/b) 2 r (~-~z)

Y, (2ab) ':2Z

±2Re v < Re z < 1

-1 [r(l+v-~z)l r(~-~z)

kZ • r (v+~z) (a/b) 2 J, (2ab)

--'2Z

-2Re v < Re z < 1

kZ • r (v+~z) (a/b) 2 Y, (2ab) -;:.zz

-2 Re v < Re z < 1

[J (2ab) -J (2ab) 1 -z z

3/. ~ 2sec(~v)Yv(U)Jv(v)-Jv(u)Yv(v)l

U = b[(a2+x2)~±al v

2 ~~J (v) Y (u) v v

-1 < Re z < 1

00

z-l Hz) = f x q,(x)dx q,(x)

0

7.23 b z se c (l:PIZ) 211 -1 sin[a(x+b 2/x)]

• [Jz (2ab)+J_ z (2ab)] -1 < Re Z < 1

7.24 (2a/b) ~zr (~z) 2aV(a2+x2)-~VJ [b(a2+x2)~] v

'Jv_~z(ab) 0 < Re z < 'l.-2+Re v

7.25 (2a/b) ~zr(~z) 2aV(a2+x2)-~Vy [b(a2+x2)~] v

.y v-~ z

0 < Re z < ~2+Re v

7.26 (2a/b) ~zr (~z) 2a-V(a2_x2)~VJ [b(a2-x2)~] v

'Jv+~z (ab) x < a

Re z > 0, Re v > -1 0 x > a

-z -~ -k ~ 7.27 (~a) r (~+z) 11 2(1/x-l) 2COS [all-x) 2]

·Jz (a) x < 1

Re z > -~ 0 x > 1

7.28 (~a)zr(z) !:::v k (l-x) 2 J v [a (l-x) 2] x < 1

'Jv+ z (a) 0 x > 1

Re z > 0

'" <jl(z) = J xZ-lq,(x)dx q,(x)

o

7.29 (2a/b) ~zr (~z)

• [JV+~z (ab) cos (~7fz) o < Re Z < '!.-2-Re v

7.30 (2a/b) ~zr (~z)

o < Re z < ~2-V

7.31 J;- (a)+Y;- (a) ~z ~z

-2 I -11 47f Ko(a x-x )

Principal value

7.32 o x < 1

-2 -1 47f sin(27fv)K2V [a(x-x )]

x > 1

-~ < Re z < ~

7.33 -1 -1 -27f J 2V [a(x+x )]

- ~2 < Re z < ~2

y = log y (l/x)

7.34

7.35

7.36

7.37

co

$(z) = f xz-1¢(xldx o

Re \) > -~

Re v > -!z

y = log (l/x)

¢(x)

-1 -1 2TI Y2v[a(x+x )]

_TI- l (a+bx) v (b+ax)-v

'J2V{[a2+b2+ab(X+X-l)1~}

0 -TI

X < e

-1 -TI YzTI J 2V (2asiny)e < x < e TI

0 X > TI e

0 X < -TI e

-1 YzTI J 2v [2a cos (Yzx) 1

-TI < e x < TI e

0 X > TI e

00

z-l w(z) = f x <P(x)dx <P(x) 0

7.39 -az 0 -2a e 10 (az) x < e

11-1 (2ay_y z)-1:2 x > e -2a

7.40 -1 o (z) 1 -1 z I x < e

Re > 0 -1 arccos (-y) -1 z 11 e < x < e

0 x > e

7.41 I 2n (az) 0 x < -a e

n = 0,1,2,·· • 1I-l(aZ-yZ)-~2n(y/a) -a a e < x < e

0 x > e a

7.42 -v 0 -a z I v+2n (az) x < e

Re v > -1:2, n=0,1,2,'" -1 -v -1 1:211 (1:2a) (2n)!r(v) [r(2n+2v)]

'(a Z_yz)v-1:2cv (y/a) -a a 2n e <x<e

0 x > a e

7.43 Iv (z) -1I-12 Vs{n(1Iv) (yZ_l)-1:2

0 ~ !.:: -2v -1 Re z > • [(y-l') 2+ (y+l) 2] x < e

1I-l(1-y2)-1:2cos[v arccos (-y) ]

-1 e < x < e

y = log 0 x > e

00

z-l <jJ (z) = J x 4> (x)dx 4> (x)

0

7.44 -1 z Iv (z)

v -1 2 (ltv) sin (ltv)

k 12 -2v -1 Re z > 0 • [ (y-l) 2+ (y+l) ] x < e

-1 (ltv) sin[v arccos (-y)]

-1 < < e x e

0 x > e

7.45 -v -a z Iv (az) 0 x < e

Re v > -\ IT-\(2a)-v[r(\+v)]-1(a2_y 2)v-\

-a e < x < e

a

0 > a x e

7.46 -v cosh (az) Iv (az) 0 < a z x e

Re v > -!z -k -v -1 v-~ \IT 2(2a) [r (\+v)] (2ay-y 2)

-a e < x < e

a

0 x > a

e

7.47 I {\a[z+(z2-b 2 )\]} 0 < -a x e v

'I {\a[z~(z2-b2)\]} -1 -k 2 k IT (a 2_y2) 2J [b (a 2_y ) 2] V 2v

Re > -!z -a < < a v e x e

0 > a x e

y = log

226

7.48

7.49

7.50

7.51

7.52

<X>

W(z) = J XZ- 1 ¢(x)dX o

Re v > -l:i

Re z > 0, Re v > 0

Z-llr (a/z) v

Re z > O,Re(ll+v > 0

Re z > O,Re v > -l:i

Re z > O,Re v > -1

a > b

y = log (l/x)


¢ (x)

o x < e-a

o

-~ ~ k (l:iy/a) '{berv [(2ay) 21bei~ [(2ay) 'J

o

o

-!::: k (1TY) 'J 2v [ (8ay) ']

o

A = (a+b)~±{a-b)~ B

o

x > 1

x < 1

x > 1

x < 1

x > 1

x < 1

x > 1

7.53

7.54

7.55

7.56

00

¢(z) = J xz-l¢(X)dx o

z -Ie -a/z I (b/z) v

Re z > O,Re v > -1

a > b

Re z > 0, Re v > -~

-2 -2 -2 z exp(-z )Iv(Z )

Re z > 0

• [Iv_~ (a/z) -Iv+~ (a/z) 1

y = log (l/x)

¢ (x)

x < 1

o x > 1

x < 1

o x > 1

F (;1+v,1+2v;-~y2) X < 1 o 2

o x > 1

x < 1

o x > 1

Re z > 0

00

z-l <P(z) = J x CP(x)dx cp(x) 0

7.57 Ko(az) (y 2-a 2)-l:z -a x < e

Re > 0 0 -a z x > e

7.58 -1 log[y/a+(y2/a 2-1)l:z] -a z Ko (az) x < e

0 -a Re z > 0 x > e

7.59 Kv(az) l:z(y2_ a 2)-l:z [y/a+(y2/a 2_1)l:z]v

Re 0 +[y/a_(y2/a 2)l:z]v -a z > x < e

0 -a x > e

7.60 -1 z K v(az) J, -1 l:z v V {[y/a+ (y2/a 2_1) ]

Re z > 0 _[y/a_(y2/a 2_1)l:z]v}x -a < e

0 -a x > e

7.61 -v z K v(az) nl:z(2a)-v[r(l:z+v)]-1(y2_ a 2)v-l:z

Re > o ,Re -~ -a z v > x < e

0 x > -a e

7.62 z -ilK v(az) ~ 2 ~~-k ~-~ (l:zn/a) 2 (y2_a ) 2 2p 2 (y/a)

v-~

Re z > 0 -a x < e

0 -a x > e

y = log

'" ~ (z) J

z-l q,(x) = x q,(x)dx 0

eazK -h 7.63 o (az) (y2+2ay) 2 x < 1

Re z > 0 0 x > 1

7.64 -v eazK v (az)

k -v -1 2 V-k z 'JT 2(2a) [r (~+v) 1 (y +2ay) 2

Re z > 0, Re v > -}z x < 1

0 x > 1

7.65 z-)JeazK v (az) (~'JTja)~(y2+2ay)~)J-~

Re z > 0 ·p~-)J(l+yja) v-!:z x < 1

0 x > 1

7.66 z -IKO (az~) -~Ei(-~a2jy) x < 1

Re z > 0 0 x > 1

7.67 z~vK (az~) v -v-l _\a2jy x < 1 v a (2y) e

Re z > 0 0 x > 1

7.68 )J h

z Kv(az 2 ) a -ly -)J-~exp (-Yea 2 jy)

Re z > 0 'W~+)J,~v (\a2jy) x < 1

0 x > 1

y = log (ljx)

00

z-l <p(z) = f x q,(x)dx q,(x)

0

7.69 z -~K (az lz) lz('rry) -lzexp(_1/ea2jy)K, e/e a2 jy) \l ,,\l

Re z > 0 x < 1

0 x > 1

K [a(b 2+z 2) lz] _:!.:: k -a 7.70 (y2_a 2) "cos [bjy2_a 2) "] x<e

0

Re z > 11m bl 0 > e -a

x

eazK [a(b2+z2) lz] -~ k

7.71 (y2+2ay) "cos [b (y2+2ay) "] x < 1 0

Re z > 11m bl 0 x > 1

7.72 (b2+Z2) -lz\l (lzTIb)lz(ab)-\l(y2_a2)lz\l-~

'K [a (b 2+Z 2) lz] .J ,[b (y2_a 2) lz] < e -a

x \l "V-~

Re z > 11m b I, Re \l > -~ 0 > e -a

x

7.73 e az (b2+Z2) -lz\l (lzTIb)lz(ab)-\l(y2+2ay)lz\l-~

'K [a(b 2+z 2) lz] 'J ,[b (y2+2ay) lz] x < 1 \l V-Yz

Re z > 11m bl, Re \l > -lz 0 x > 1

7.74 K [a(b 2_z 2)lz] lz(a 2+y2)- lzexp[_b(a 2+y2)lz] 0

-Re b<Re Z<Re b, Re b>O

7.75

7.76

7.77

7.78

7.79

00

¢(z) = J Xz-l¢(x)dx o

-Re b<Re z<Re b

Re b > 0

(b 2 _Z 2 )-1-;,v

.K [a(b 2-z 2) 1-;,] v

-Re b<Re z<Re b

Re z > IRe bl

(Z2_b 2) -1-;,v

oK [a(z2-b2) 1-;,] v

Re z > IRebl

Re v > -~

y = log(l/x)

¢ (x)

-Re b < Re z < Re b, Re b >0

Re b > 0

o

-a x < e

-a x > e

(1-;,nb)1-;,(ab)-V(y2_a2)1-;,v-~

k a .r [b (y2_a 2 ) 2] x < e-v-Yz

o -a x > e

232

7.80

7.81

7.82

7.83

00

w(z) = ! xZ-l~(x)dx o

[ (z+b) / (z-b) ]l:;v

Re z > IRebl

Rez>O,a~b

Re z > 0

Re z >0, a~b

y = log n/x)


~ (x)

-v 2 2 -k V ub V -ub l:;a (y -a) 2 [(y+u) e + (y-u) e 1

o

-a x < e

-a x > e

x < e-(a+b)

l:;(ab)-\ I [(2ab)-1(a 2 +b 2 _ y 2)] v-~

e-(a+b) < x < e-(a-b)

Ox> e -a-b)

l:;1T(ab)\ I [(2ab)-1(y2_ a 2_b 2)] v-~

x < e-(a+b)

Ox> e-(a+b)

o x > 1

'" ~ (z) = f z-l x ¢ (x) dx

o

7.84

Rez>O,a2,b

7.85

Re z > 0

a 2: b

7.86

Re z > 0

7.87 -v l< 2 z [Kv (az 2) 1

Re z > 0

7.88

Re z > 11m bl

y = log(l/x)

<jJ (x)

o

o

o

o

o

233

x < 1

x > 1

x < 1

x > 1

x < 1

x > 1

x < 1

x > 1

-2a x < e

-2a x > e

234

7.89

7.90

7.91

7.92

00

¢(z) = J XZ-l~(x)dx o

Re Z > lIm b I

Re z > lIm b I

oK {a [(z2+b 2 ) J,+z] } \!

Re z > lIm b I

Re z > IRe bl

y = log(l/x)


~ (x)

x < 1

o x > 1

-k !,.; (y2_4a2 ) 2J2 \! [b (y2_4a 2) 2]

o

x < e -2a

-2a x > e

Re \! > -J,

x < 1

o x > 1

Re \! > -J,

o

Re \! > -J,

-2a x < e

-2a x > e

7.93

7.94

7.95

7.96

7.97

00

~(z) = J xz-l~(x)dx o

e2azIV{a[Z-(Z2-b2)~1 }

oK {a[z+(z2-b2)~1} v

Re z > IRe bl

K {a[z-(z2-a2)~1} v

oK {a[z+(z2-a2)~1} v

Re z>IImbl,-~<Re v<~

e2azKv{a[Z-(Z2-b2)~1}

oK {a[z+(z2-b2)~1} v

Re z>IIm bl,-~<Re v<~

z -~ea/zKv (a/z)

-~ < Re v < ~

Re z > O,-~ < Re v < ~

Re z > 0, -~ < Re V < ~

y = log(l/x)

~(x)

o x < 1

Re v > -~

2 cos (TTV) (y2_4a2)-~

oK [b (y2-4ay) ~l -2a x < e 2v

0 x > e -2a

2 cos (TTV) (y2+4ay)-~

oK [b (y2+4ay) ~l 2v x < 1

0 x > 1

x < 1

Ox> 1

-(TTy)~{sin(TTV)J2V[(8aY)~1

+cOS(TTV)Y2V[(8aY)~1} x < 1

o x > 1

00

<j>(z) = f x z - 1 <j>(x)dx <j>( x) 0

7.98 z -le a/ zK (biz) 2lT-1 sin(-rrv) [K (AY~)K (BY~) v v v

Re z > O,-l<Re v<l +K (AY~) I (BY~) 1 +I (AY~) K (BY~) v v v V

A = (a+b)~±(a-b)~ < 1 B x

0 x > 1

7.99 -1 e-a/zK v (biz)

~ k z -~lT{sin (lTv) [J (Ay ) J (By 2) v v

Re z > 0,-1 < Re k ~

v < 1 -Yv (Ay 2) Yv (By 2) 1

A (a+b) ~± (a-b) ~ k !.:: = +cos(lTv) [JV (Ay 2)Yv (By 2) B

k !:z +JV (By 2)Yv (Ay)]} x < 1

0 x > 1

7.100 z -2ve az 2 Kv (az 2 ) 2lT~[r(1+2v)]-1(8a)~VyV-l

Re z > 0, Re v > -~ 'exp(-Yrsy 2 Ia)M 3 1 (~8y2Ia) - ""'2 V , ~v

X < 1

0 x > 1

Y = log(l/x)

'" z-l ~(z) = f x ¢(x)dx ¢ (x)

0

7.101 (2a/b)~z[r(1-~z)1-1 0 x < a

'Kv+~z (ab) a-V(x2_a2)~VJ [b(x2-a2)~1 v

Re z < ~2-Re V x > a

Re v > -1

7.102 (2a/b) ~zr (~z) 2aV(a2+x2)-~vK [b(a2+x2)~1 v

Kv_~z (ab) Re z > 0

7.103 (2a/b)~zr(~z) 2a-V(a2+z2)~vK [b(a2+x2)~1 v

'Kv+~z (ab) Re z > 0

kz 2a -v (a 2_X 2) ~vI [b (a 2_X 2) ~l 7.104 (2a/b) 2 r(~z) v

'Iv+~z (ab) x < a

Re z > 0 0 x > a

7.105 bZcos(~~z)K (2ab) ~cos[a(x-b2/x)1 z

-1 < Re z < 1

7.106 bZsin(~~z)K (2ab) ~sin[a(x-b2/x)1 z

-1 < Re z < 1

'" z-l -2 Re v

7.109 r (v+~z) [r (l+v-~z) 1 -1 2')[~I v(a)Kv (13)

~z 13 = b [(a 2 +x2) ~±al • r (~-~z) (a/b) K, (2ab) ~z a

-Re v < Re z < 1

7.110 r(~z+v)r(J.,z-v) -k

2')[ "K)a)Kv (13)

[r(~+J.,z)1-1(a/b)~zK~z(2ab) 13 = b [(a 2+X2) ~±al a

Re z > ±2Re v

7.111 cos(~')[z)K: (a) ~')[y [a Ix-x-1 11 ~z 0

- ~2 < Re z < ~2 Principal value

, 7.112 Jz(a)Kz(a) ~J 0 [a (l/x-x) ~l x < 1

Re z > - ~ 0 x > 1

00

z-l -1 x < 1

0 x > 1

7.114 Kl;,z (a) Il;,z_;,(a) (;'rra)-;'(1-x 2)-;'cos[a(x-x- l )]

Re z > -1 x < 1

0 x > 1

7.115 Kv+;'z(a)Kv_;'z(a) -1 K2v [a(x+x )]

-k k 7.116 Iv+z (a) Kv_z (a) ;'J2v [a(x '-x')] x < 1

Re z > _3/4 ,Re v > -1z 0 x > 1

7.117 K;,V+;'z (a) I;,v-;'z (a) 0 x < 1

~2 , -1 -1 1 Re z < Re v > Jv[a(x-x )] x >

7.118 I;'v_;'z(a)K;'v+;'z(a) I -1 J [a x-x 11 v

+I;'v+;,z(a)K;'v_;'z(a) - ~2 < Re z < 3-2 , Re v > -1

7.119 K (a)K (b) v-z v-z ;'(a+bx) v (b+ax)-v

oK2V{[a2+b2+ab(x+x-l)1;'}

co z-l

~(z) = f x q,(x) dx 4>(x) 0

7.120 cosh (~1TZ) 0 x < -7T e

.IV_~iz(a)IV+~iz(al -1

~7T I 2v (2a sin y)e -7T < x < e 1T

Re v > -~ 0 x > e 7T

7.121 Iv_iz(a)Iv+iz(a) 0 x < e -7T

> -~ -1 cos (~x) ] -1T < e 7T Rev ~7T I 2v [2a e < x

0 x > e 7T

7.122 z -1 [H; (az) -Y (az)] 27T-110g[y(a+(1+y2/a2)~] x < 1 o 0

Re z > 0 0 x > 1

7.123 -1 -a z [Io (az) -1.0 (az) ] 1 x < e

Re z > 0 2 7T -larcsin (y fa) e -a < x < 1

0 x > 1

7.124 -v z ~ (az) -Yv (az) ] 27T-~(2a)-v[r(~+v)]-1

Re z > 0 2 2 v-~ • (y +a ) x < 1

0 x > 1

y = 10g(1/x)

co z-l (z) = f x <p(x) dx <j>(x)

0

7.125 z -v [I (az) -10 (az) 1 0 x < -a e v v

Re z > 0, Re v > -~ 2~-~(2a)-v[r(~+v)1-1

.(a2_y2)V-~ -a < x < 1 e

0 x > 1

7.126 Z~v[H (az~)-Y (az~)l -1 v -v-1 v v 7T cos(~v) (~a) y

Re z > 0, Rev<~ .exp (~a 2/y ) Erfc (~ay -~) x < 1

0 x > 1

7.127 (~a)-zr(~+z) 0 x < 1

• [~(a) -Y z (a) 1 ~ -~x~(x-1) -~exp [-a (x-1) ~l x > 1

7.128 (~) -zr(~+z) ~-~~(l-x)-~exp[-a(l-x)~l x < 1

• [I z (a) -loz (a) 1 0 x > 1

Re z > -~

7.129 -z (~a) r (Z)Hv+ z (a) (l-x)~vH [a(l-x)~l

v x < 1

Re z > 0, Rev>- 3~ 0 X > 1

7.130 (~a)-zr(z)lov+z(a) (l-x)~vL [a(l-x)~l v x < 1

Be z > 0, Rev > - ~2 0 X > 1

Y = log(l/x)

co

~(z) = f X z- 1 $(x)dx ¢(x) o

7.131 (2/a) z-l r (\+z)

o x > 1

Re z > -~

7.132 -z (2a) r (\+z) o x < 1

[I_Z (a) -Lz (a) 1

Re z > -~

7.133

. r (\+v+z)

o < Re z < \-Re v

7.134 -1 v 2 -k '1T sin ('1Tv) a (y+a 2 ) 2

Re z > 0 x < 1

o x > 1

7.135 -1 -1 '1T exp[-\a(x -x)] x < 1

o x > 1

y = log(l/x)

7.136

7.137

7.138

7.139

7.140

'" ~(z) = f x z- 1 ¢(x) o

Re z > 0, Re(Il±V) > -~

z-2 (2b) r (~z-~v)

(b/a)zr(z)r(~-z)

sz+v,z_v(ab)

O<Rez<~

-1 r (z) [r (~+z) 1

Re z > 0

-z a r(z)r(~-~Il-~v-z)

y = log(l/x)

¢ (xl

2,,-1 -k " 2 ... a "y"'exp (~a2/y)

·W (a/yl Il,V

o

(b 2+X2) -lK (ax) v

Rez>±ReIl

[r(~+v)1-12V-1

o

s [a(l+x) ~l Il,V

243

x < 1

x > 1

x > 1

2.8 Whittaker Functions and Special Cases

'" z-l <!> (z) = J x <P(x)dx <P(x)

0

8.1 Ei[-b(a+z)] -1 a < e -b -y x x

Re z > -a 0 > e -b x

8.2 -1 log (a/y) -a z Ei (-az) x < e

> 0 0 -a Re z x > e

8.3 eazEi (-az) -1 1 -(a+y) x <

0 x > 1

8.4 -1 az . z e El.(-az) -log (l+y/a) x < 1

Re z > 0 0 x > 1

8.5 -az e Ei (-az-bz) -(a+y) -1 x < e -b

0 x > e -b

8.6 e az [Ei (-az)] 2 2 (a+y) -llog(y/a) < e -a x

Re z > 0 0 > e -a x

y = log (l/x)

00

z-l <J>(z) = J x ¢ (x) dx ¢ (xl 0

8.7 e(a+b)z (y+a+b) -llog [ (abl -1 (y+a) (y+b) 1

·Ei (-az) Ei (-bz) x < 1

Re z > 0 0 x > 1

8.8 Ei (-az) Ei (-bz) -1 -1 y log [ (ab) (y-a) (y-b) 1

Re z > 0 x < e-(a+b)

0 x > e-(a+b)

8.9 E"i(az)Ei(-bz) y -llog 11-y2 /a 21 x < 1

Re z > 0 0 x > 1

8.10 z -le a/zEi (-a/z) -2Ko [2 (ay) ~l x < 1

Re z > 0 0 x > 1

8.11 z -le -a/zE"i (a/z) k 1TYo [2 (ay) 'J x < 1

Re z > 0 0 x > 1

8.12 az 2 e Ei (-az 2 )

_~ _~2/a k i(1Ta) e' Erf(~ia'y) x < 1

Re z > 0 0 x > 1

Y = log(l/x)

00

z-l 1>(z) = f x ¢(x)dx ¢(x) 0

8.13 z -J,Ei (-a/ z) -!.: .k 2 (lTy) 'ei [2 (ay) '] x < 1

Re z > 0 0 x > 1

8.14 z -J,E"i (a/z) -~ .k - k (lTY) '{Ei [-2 (ay) 2]+Ei [2 (ay) ']}

Re z > 0 x < 1

0 x > 1

8.15 z -J,ea/zEi (-a/z) -k k k (lTY) '{exp [2 (ay) 2]Ei [-2 (ay) 2]

Re z > 0 !::: _ :k

+exp [-2 (ay) '] Ei [2 (ay) ']}x < 1

0 x > 1

8.16 z -J,e -a/zE"i (a/z) -k :k :k 2 (lTY) '{cos [2 (ay) 2] ei [2 (ay) 2]

Re z > 0 1: 1.:

+sin [2 (ay) ']Si [2 (ay) 2]} X < 1

0 x > 1

z -~i (-azJ,) _k

8.17 J, (lTY) 'Ei (-!,;a 2 /y) x < 1

Re z > 0 0 x > 1

8.18 exp (a 2 Z2) -!z -1 2 2 IT a exp (-!,;y / a ) x < e -2ab

'Erfc (b+az) 0 x > e -2ab

y = log(l/x)

00

<f>(z) = f x z - 1 (x)dx q, (x) 0

8.19 -1

exp (a 2 z 2 ) Erf(~/a) < 1 z x

·Erfc (az) 0 x > 1

Re z > 0

8.20 -1 Z2

z e Erfc(a+z) Erf (J,y) -Erfa x < e -2a

Re 0 0 -2a

z > x > e

8.21 Erfc [ (az) J,] -1 J, -1 -1

Tr a y (y-a) x < e -a

Re z > 0 0 x > e -a

8.22 z -J,Erf [(az) J,] 0 < e '""a x

Re 0 (TrY) -~ -a < x < 1 z > e

0 x > 1

-k k -~ -a 8.23 z "Erfc [(az) 2] (TrY) x < e

Re z > 0 0 > e -a

x

8.24 eazErfc[ (az) J,] -1 -l< -1

Tr (y/a) '(y+a) x < 1

Re z > 0 0 x > 1

Y = log (l/x)

00

<l>(z) = f z-l x CP(x) dx cp(x)

0

8.25 -~ a 2/z -~ z e Erfc(az) (7Ty) -~exp (-2ay~) x < 1

Re z > 0 0 x > 1

8.26 Erf(az-~) -1 ~ (7Ty) sin(2ay) x < 1

Re z > 0 0 x > 1

8.27 -v a 2/z z e a1-Vy~V-~ (2ay~)

-1 x < 1

-Erf (az -~) 0 x > 1

Re z > 0

8.28 z-ve a2 / z I-v ~v-~ ~ ~ a y [Iv _1 (2ay )-~v_1(2ay )1

'Erfc(az-~) x < 1

Re z > 0 0 x > 1

8.29 b-Zy(z,ab) e -bx

x < a

Re z > 0 0 x > a

8.30 -z 0 b r (z,ab) x < a

e -bx

x > a

y = log (l/x)

co

 (z) = f Xz - 1 $(x) $(X) 0

8.31 r(z)r(l-z,a) (l+x)-le -a(x+l)

Re z > 0

8.32 r (v ,az) v -1 -1 -v a [r(l-v)] y (y-a) x < e -a

Re z > 0, Rev < 1 0 x > -a e

8.33 eazr(v,az) v -1 -v -1 a [r (l-v) ] y (y+a) x < 1

Re z > 0, Rev < 1 0 x > 1

8.34 aZy(z,a) e -ax x < 1

Re z > 0 0 x > 1

8.35 -z e-a / x a r(-z,a) x < 1

0 x > 1

8.36 -v y(v,az) 0 -a z x < e

Rev > -1 v-1 -a < x < 1 y e

0 x > 1

8.37 -v v-1 -a z r(v,az) y x < e

Rev > -1 0 > e -a x

y = log (l/x)

co

<l> (z) f z-l <P(x) = x <P(x)dx

0

8.38 -v e aZ r (v, az) (y+a) v-I < 1 z x

Re z > 0 0 x > 1

8.39 -v -az 0

-a z e y(v,-az) x < e

v-I -a 1 (a-y) e < x <

0 x > 1

8.40 Y (v, a/z) kv 1.:\)-1 ~

< 1 a 2 y2 J v [2(ay) 2] x

Re z > 0, Re v > 0 0 x > 1

8.41 z V-l y (v, a/z) kV -kv h a 2 r(v)y 2 Iv [2 (ay) 2J x < 1

Re z > 0, Re v > 0 0 x > 1

8.42 zV-le a/z r (v,a/z) 2[r(1-V)]-1(y/a)-~vKv[2(ay)~]

Re z > 0, Re v < 1 x < 1

0 x > 1

8.43 z lle a/zy (v, a/z) [Vr(V_ll)]-l a Vy V-ll-l

Re z > 0, Re(v,ll) > 0 • jF 2 (l;v+l,v-ll;ay) x < 1

0 x > 1

Y = log(l/x)

00

(z) f z-l ¢ (x) = x ¢(x)dx 0

8.44 exp(l,;a2z 2)Dv (az) aV[f(_v)J-1y-V-1exp(_~2/a2)

Rev < 0 x < 1

0 x > 1

8.45 1<

Dv [2(az 2)] 2 lzv+lzalz [f (-lzv) J -1 (y-a) -l-lzv

!,n)-1.:: -a Re z > O,Re v < 0 • (y+a) 2 2 X < e

0 > e -a x

8.46 -k 1:: z 2DV [2 (az) 2] 2 lzv [f(lz_lzv)]-1(y_a)-lzv-lz(y+a)lzv

Re z > 0, Re v < 1 < e -a x

0 x > e -a

8.47 z lzv e lzaz -l-lzv -1 -1< 2 [f(-v)] (a+y) 2

'Dv [(2az) lz] !,; k -v-1 • [(a+y) 2_a 2] x < 1

Re z > 0, Re v < 0 0 x > 1

8.48 kv-l k

Z2 DV[(2az)2] 2- lzV [f(l_v)]-1[(a+y)lz_alz ]-v

Re z > 0, Re v < 1 x < 1

0 x > 1

8.49 zVexp(a/z) -1 -v-1 lz [f ( - 2 v) ] (2 y) exp [ - 2 (2 ay) ]

'D2v [2 (a/z) lz] x < 1

Re z > 0, Re v < 0 0 x > 1

Y = log(l/x)

252

8.50

8.51

8.52

8.53

00

¢(z) = f xz-l~(x)dx o

Re z > -b

Re z > -b

r(z)D (a) -z

Re z > 0

r(Z)D 2 [(2a)lz] -z

Re z > 0

y = log (l/x)


~ (x)

o

Re v < 0

o

Rev<O

2 -lz l+x (l-x) exp [-a (I-x) 1

o

x < 1

x > 1

-a x < e

-a x > e

x < 1

x > 1

8.54

8.55

8.56

8.57

8.58

00

4>(z) = f xz-l<p(x)dx o

-ll-~ ~az z e

'W (az) V,].I

Re z > 0, Re (].I-V) >-lz

z-].I-lzw (2az) V,].I

Re z > 0

Re (].I-V) > -lz

Re z > O,Re v > 1

-a. az z e W (az)

V,].I

Re z > 0, Re (a.-V) > 0

Re z > b

Re(].I-V) > -lz

y = 1og(1/x)

<p(x)

(y+a) v+].I-lz

o

V+].I-l< • (y+a) 2

o

x < 1

x > 1

-a x < e

-a x > e

kV v -a [(y+a) / (y-a) 1 2 P 1 (y/a) x<e

11-~

o -a x > e

F (lz-v+].I,lz-V-].Iia.-viy/a) 2 1

o

-1 abr (1+2].1) [r (lz+].I-v) 1

-v-k \j-k • (y-a) 2 (y+a) 2

o

-a x < e

-a x > e

-a x < e

-a x > e

8.59

8.60

8.61

8.62

'" f z-l ¢(z) = x ~(x)dx o

w (z)W (z) V,~ 2 V,~ 1

Re z > b

Re (~±V) > -~

Re z > 0

Re(~-v) > 0

Re z > 0

Re(~-v) > -~

'M (a/z) v,~

Re z > O,Re(v+~) > -~

y = log (l/x)

 e

x < 1

x > 1

a = ~(~-v-1), f3 = ~(1-v-3~)

x < 1

o x > 1

~ -1 v-~ a 2r (1+2~) [r (~+v+~) 1 y 2

x < 1

x > 1

'" ¢(z) = J XZ-l$(x)dx

o

8. 63 z-ve-'za/z

'W (a/z) V,]1

Re z > 0

Re (v±]1) > -~

8.64 zVe'za/z

oM (a/z) V,]1

Re z > 0, Re (]1-V) >-J,

8.65 z VeJ,a/z

'W (a/z) V,]1

Re z > 0, Re (v±]1) <J,

8.66 r (z+]1)W (a) -z,v

Re z > -Re ]1

y = log(l/x)

$ (xl

" -cos('ITV-'IT]1)J 2 [2(ay)']} x < 1 - ]1

Ox> 1

~ -1 -v-L a 2 r(l+2]1) [r(J,+]1-v)] y 2

o

o

-1 -1 'W [a(x -1) ] ]1,V

o

x < 1

x > 1

x < 1

x > 1

x < 1

x > 1

8.67

8.68

8.69

8.70

8.71

'" ~(Z) = ! XZ-l¢(x}dx ¢(x) o

f(y,+v+z}f(y,-v+z} f (l-].l+z)

'W (a) -z,v

Re Z > -y,±Re v

Re v < ~

-z -1 a [f (2v+l-2z) 1

• f (2z) M].l-Z, v-z (a)

o < Re Z < ~+Y,Re(v+].l}

-Z a f(2z)W].l_Z,v+z(a)

Re z > 0

-Z a f (y,-].l-v-2z)

of(2Z)W].l+z,v+z(a)

y = log(l/x)

-1 -1 'W [a(x -l} 1 ].l,V

-ax-Y, e

o

o

y,f (y,+].l+v) [f (l+2v) J- l

x < 1

x > 1

x < 1

x > 1

k -v-k -kax~ 1 • (l+x 2) 2e 2 M [a(l+x~»)

].l,V

k 1; v-1: -~ax 2 1,; y,(l+x2) 2e W [a(l+x 2})

].l,V

~ v-~ ~f(~-].l-v) (l+x 2)

~(z) co z-l = 6 x cp(x)dx

8.72 r (2z) [r (l+2v-2z) ]-1

-z 'a M (a) ll-z,V-Z

o < Re z < ~+Re(v+ll)

8.73 -z a r(2z)W + (al ll-Z,v z

Re z > 0

8.74

·W (a) ll+z,V+z

o < Re z < ~-~Re (ll+V)

8.75 -z a r (~+V-ll-2z)

[r(1-2z)]-lW (a)W + (a) z-v z II

Re z < ~~Re (V-ll)

8.76 r (v+z) r(l+v-z)

-Re V < Re z < l+Re V

y = log(l/x)

CP(x)

- ax~ I 'e ~ M [a(l+x~)] ll,V

k V-I kax~ ~r (~-ll-V) (l+x 2) ~e 2

o

l-x ·exp[~(a+b) (l+x)]

x < 1

x > 1

8.77

8.78

8.79

8.80

co

~(z) = f xz-l¢(x)dx ¢(x) o

r (v+z) r (l+v-z)

oM, (a)M L (b) ,,-z,v Z--:z,v

-Re v < Re z < l+Re v

r (v+z)

Re z > -Re v I a > b

r (1+v-z)

Re z < l+Re v, a > b

r(l+v-z)r(l-v-z)

Re z < liRe v

y = log (l/x)

(ab~r2 (1+2v) (1+x) -lI [2 (abx) ~l 2v l+x

x-l 'exp[~(a-b) (x+l)]

a > b

- ~ (ab) ~r (1+2 ) (1- ) lJ [2 (abx) ] v x 2v l-x

.exp[-~(a-b) (l+x)] x < 1 l-x '

Ox> 1

o x < 1

~ (ab) ~r (1+2v) (x-l) -lI [2 (abx) ]

2v x-l

o

x+l ·exp[-~(a+b) (x-l)]

x > 1

x < 1

x > 1

Appendix. List of Notations and Definitions

Abbreviations: £ n

Neumann I S number

£ 0 1 I £n = 2 I n = 1 I 2 I 3 I

y 0.57721·· • Euler's constant

a (a+l)··· (a+n-l)

a (a-I)'" (a-n+l) In!

r (a+n) . r (a) ,

-1 r (Ha) [n! r (Ha-n)]

1. Elementary functions

Trigonometric and inverse trigonometric functions:

sinx, cosx , sinx cosx tanx = COSX' cotx = sinx

1 secx = COSX' cscx 1

sinx ' arcsinx , arccosx ,

arctanx , arccotx.

Hyperbolic and inverse hyperbolic functions:

sinhx ~(e x -x -e ), coshx x -x

~ (e +e ) I -1 sinh x,

tanhx sinhx cothx coshx -1 COShX' sinhx '

tanh x,

sechx 1 cschx 1 COShX' sinhx'

2. Orthogonal polynomials

Legendre polynomials Pn(x).

259

-1 cosh x,

-1 coth x,

F (-n , n+l; 1; ~-~x) 2 1

260 Appendix

Gegenbauer's polynomials C~(x)

Chebycheff polynomials Tn(x), Un (x)

Tn (x) = cos (narccosx) F (-n,n;J,;J,-J,x) 2 I

(1-x2 l-J,sin[ (n+llarccosx] = C~(x)

Jacobi polynomials

J,n lim r ('JlC~ (xl v=o

p(a,S)(x) n

-1 [n!r(l+a)] r(l+a+n) F (-n,n+a+S+l;a+l;J,-J,x)

2 I

Laguerre polynomials

-1 [nIr (l+al] r (a+l+n) F (-n;a+l;x)

I I

Appendix

Hermite polynomials

n x 2 Hn(X) = (-1) e

He 2n (x) =

F (-n;:Y2 ;~X2) 1 1

3. The Gamma function and related functions -----"'j -t z-l r(z) = e t dt o

ljI-function

,I, (z) ~ log r I (z) ~ dz z = -rTZf

Beta function B(x,y)

B(x,y) = r(x)r(y) r(x+y)

4. Legendre functions

Re z > 0

(Definition according to Hobson)

~ -1 z+l y,~ Pv (z) = [r (l-~) 1 (z-l) F (-v ,v+l;'l-~ ;Y,-y,z)

2 1

• F (y,v+Y,~+Y"Y,V+Y,~+1;V+3~;z-2) 2 1

z is a point in the complex z-plane cut along the real

z-axis from to +1

261

262

P~(x)

Q~(x)

p~(z) q~(z)

P~(z) = Pv(z); Q~ (z)

5. Bessel functions ---z 2n

v '" (_l)n (2") Jv(z) = (~z) I Ir( 1) n=O n. v+n+

6. Modified Bessel functions ---

Appendix

Appendix 263

7. Anger-Weber functions

1 71 71- f cos(z sint-vt)dt

a

n = 0,1,2,"·; e: (z) = -H. (z) o 0

(7, (z) = (y,7fz) -y,{ [C (z)-S (z) ]cosz+[C (z)+S (z) ]sinz} = E J (z) 7z -"2

-!< ,J'_!<2(Z) = (y,7fz) 2{[C(z)+S(z)]cosz-[C(z)-S(z)]sinz} = EJ (z)

'2

8. Struve functions

9. Lommel functions

(y,z) v+2n+l L r(n+ 312)r(v+n+ 312} n=O

zll+l sll,V (z) = (Il-v+l) (Il+v+l)

11 ±v + -1, - 2, - 3, •••

S (z) Il,V

s (z) V,1l

S (z) = S (z) Il,V Il,-V

Special cases:

264

S (z) Lim' ll-l,v

f(V-v)

S (z) o,V

S-l,v (z) -1

-~wv csc(wv) [J (z)+J (z») v -v

Appendix

-1 ~wv csc (wv) [Jv (z) +J -v (z) -.::rv (z) -:r_v (z) 1

sl,v (z) l-~wvcsc(wv) [.1' (z)+.1' (z») v -v

-1< S_~,~(z) = z '[sinz Ci(z)-cosz si(z»)

-1< S 3 1 (z) = -z '[sinz si (z)+cosz Ci (z) 1

-;.0"2 ,~

Kelvin's functions

" 3 J(ze-~'"4l1) b ()"b"() v erv z -~ e~v z

Appendix

.1T

KV (Ze-14) = kerV(z) - i keiv(z)

bero (z) ber(z), beio(z) bei(z) ,

kero(z) ker(z), keio (z) kei(z)

Neumann polynomials

<1m ~ =2: 2m-n-l n(n-m-l)! (y,x) 1m!

m=O

Schlafli polynomials

<y,n =1: 2m-n (n-m-l) ! (~:iz) 1m! m=O

10. Gauss' hypergeometric series

F (a,b;c;z) 2 1

r (c) r(a)r(b) L

n=O

r (a+n) r (b+n) r (c+n)

11. Confluent hypergeometric functions

Kummer's functions

F (a;c;z) 1 1

F (a;c;z) 1 1

r (c) \' r (a+n) 'F'(aT n~O r (c+n)

e Z F (c-a;c;-z) 1 I

0 0 (x) -1

x

o

265

I z [ < 1

266 Appendix

Whittaker functions

r(-2~) r(2~) r(~ ~-k) Mk,~(Z) + r(~+~-k) ~,~(Z)

Wk,_~(z) = Wk,~ (z)

Parabolic cylinder function

-kZ 2 e • Hen (z) n = 0, 1, 2, •••

(2 ~Z-1)-~ K (' 2) " 1 '4 Z

'4

Error integrals

Erf(x) = 2'IT-~ f e- t2 dt o

Erfc(x) l-Erf (x)

21T -!zx F (!z i 3/2 i _x 2 ) 1 1

2 ( ~x) -~e-~x2 M (2) " 1 1 X -'4,'4

• 'IT 1 14

Erf(x~e ) C(x) + S(x) + i[C(x) - S(x)]

• 'IT 1 ;1.4

Erfc(x~e ) l-C(x) - S(x) + i[S(x) - C(x)]

Appendix 267

Fresnel's integrals

x C(x) = (2TI)-~ J t-~cost dt;S(x)

o

X (2TI)-~ J t-~sint dt

o

Exponential integrals

-Ei(-z) = -y-log z - L (non!)-l(_z)n n=l

-Ei (-x)

Ei (z) y+log z + L n=l

-1 n (nonl) z

x > 0

co

Ei(x)=~[Ei(-xeiTI)+Ei(-xe-iTI)]= -pov. f t-le-tdt, x> 0 -x

Ei(-ze±iTI) = ±iTI+Ei(z); Ei(ze~iTI) = ±iTI+Ei(-z)

Ei(-ze±~iTI)=Ci(z)+i[~TI-Si(z)];Ei(ze±~iTI)=Ci(z)±i[~TI+Si(z)]

Ei(-xe±~iTI)=Ci(x)±isi(x);Ei(xe±~iTI)=Ci(x)±i[TI+si(x)]

x > 0

Sine and cosine integral

Si (z)

Ci(z)

Ci(x)

L (-1)n[(2n+l)(2n+l)!]-lz2n+l n=O

y+log z + I (_1)n[2n(2n)!]-lz2n n=l

- f t-lcos t dt, si(x) x

x > 0

TI Si (x)- 2"

268

Ci (x) x

y+log x - J t-l(l-coS t)dt, o

Incomplete gamma function

y (v,x) xJ v-l -t -1 v

t e dt=v x F (v,v+l;-x), o 1 1

f(v,x) f(v)-y(v,x)

1Tl:iErfc(z); f(O,z) = -Ei(-z)

Appendix

x > 0

Re v > 0

~ - -z 1T"Erf(z); y(l,z) = l-e , f(l,z) -z

e

12. Particular cases of Whittaker's functions

M () k21Tl:iz!:iel:izErf(zl:i) -!:i,!:i z

Appendix

~,k+J,(Z)

W 1 1 (Z) -~,~

Wk,k-J,(Z)

13. Elliptic integrals and elliptic theta functions

Complete elliptic integrals

~7T K(k) = f (1-k2sin2x)-~dx

o

E (k) !j1T f (1-k 2sin2x)J,dx o

Theta functions

e (Z It) = (7ft)-J, 1 n=-co

2 L (_l)n exp[-7f2t(n+J,)2jsin[ (2n+l)7fz] n=O

269

270

e (z I t) 2

e (z I t) 3

8 (z I t) 4

(rrt) -;, L n 2 (-1) exp[- (z+n) /t] n=-oo

2 L exp[-rr 2 t(n+;,)2]cOs[ (2n+1)rrz] n=O

-I< (rrt) 2

CD

L n=-oo

2 exp[-(z+n) /t]

L En exp(-rr 2 tn 2 )cos(2rrnz) n=O

(rrt)-;' L exp[-(z+n+;,)2/t ] n=-oo

L n=O

Modified theta functions

CD

~ (zit) = (rrt)-;'{ L (_l)n exp[-(z+n+;,)2/t1 1 n=O

e (z I t) 2

e (z I t) 3

L (_l)n exp[-(z+n+;,)2/t ]} n=-l

CD

(rrt)-;'{ L (_l)n exp[-(z+n)2/t ] n=O

L (_l)n exp[-(z+n)2/t ]} n=-l

(rrt)-;'{ I exp[-(z+n)2/t ] n=O

L exp[-(z+n)2/t ]} n=-l

Appendix

Appendix

6 (zl t) (1ft)-~ l exp[-(z+n+~)2/t] n=O

2 l exp[-(z+n+~ It]} n=-l

14. Generalized hypergeometric functions

F (a a "'a'b b "'b ·z) P q l' 2' p' l' 2' q'

(a) "'(a) n I' In pnz L (b) ••• (b) n!

n=O 1 n q n

P'q = 0, 1, 2, •••

271

Izl<l if p = q+l, Izl<oo if p~q; divergent otherwise.

15. Meijer'~ ~-function

(21Ti) -1 f L

A(z) z B (z) x dz

where

m n A(z) = IT r(bk-z) II r(l-ak+z)

k=l k=l

q p B(z) II r(l-bk+z) II r(~-z)

k=m+l k=n+l

L is a path separating the poles of r(b -z) "', 1

(see Erdelyi et. al. Higher transcendental functions,

Vol. 1, Sec. 5.3; 1953, McGraw Hill).

272 Appendix

16. Miscellaneous functions

Riemann's

Z;; (z) =

Hurwitz's

z;; (z, a)

zeta function

2: n -z

n=l

zeta function

2: (n+a)-z n=O

Lerch's zeta function

Y(z,s,a) = I n=O

-s n (a+n) z

Unit step function

H(t) = I, t > 0; H(t) 0,

List of Functions

Symbol Name of the Function

C(x) Fresnel's integral

ci (x) Cosine integral

CV (x) n Gegenbauer's polynomial

Dv (z) Parabolic cylinder function

E(k) Complete elliptic integral

Ei (-x)

} Exponential integrals Ei(x)

Erf(z)

J Error integrals Erfc (z)

Re z > 1

Re z > 1

I z I < 1

t < 0

Listed under

11

11

2

11

13

11

11

Appendix

Symbol

a , ••• , ap b 1 ••• b )

1 ' , q

H(x)

H(1),(2)(z) v

K(k)

LC\(x) n

Lv (z)

~,lJ (z) } wk,lJ (z)

On (z)

Pn (x)

Name of the Function

Anger-Weber function

Hypergeometric functions

Meijer's G-function

unit step function

Hermite's polynomial

Hankel's functions

Struve's function

Modified Bessel function

Bessel function

Anger-Weber function

Complete elliptic integral

Modified Hankel function

Laguerre's function

Laguerre's polynomial

Struve's function

Whittaker's functions

Neumann polynomials

Legendre's polynomials

Listed under

7

273

10,11,12,14

15

16

2

5

8

6

5

7

13

6

11

2

8

11

9

2

274 Appendix

Symbol Name of the Function Listed under

pea,S) (x) Jacobi's polynomials 2 n

pil(z) v

pil(x) v

Legendre functions 4

q~(z)

Q~(x)

Sex) Fresnel's integral 11

Sn (z) Sch1af1i polynomials 9

si(x) } Sine integrals 11 Si (x)

s (z)

} il,V Lomme1's functions 9

S (z) il,V

Tn (x) } Chebycheff's polynomials 2 Un (x)

W (z) il,V Whittaker's function 11

Y(z,s,a) Lerch's zeta function 16

Yv (z) Neumann's function 5

B(x,y) Beta function 3

r (z) Gamma function 3

r (v, z) } Incomplete gamma functions 11 y(v,z)

Appendix

Symbol

lJ! (z)

1,; (z)

1; (z, a)

e (z I t) 1

e (z I t) 2

e (z I t) 3

e (z It)

" e (zl t) 1

e (z It) 2

e (z It) 3

8 (zl t) "

Name of the Function

Psi function

Riemann's zeta function

Hurwitz's zeta function

Elliptic theta functions

Listed under

3

16

16

13

Modified elliptic theta functions 13

275

G. Doetsch: Introduction to the Theory and Application of the Laplace Transformation

Translated by W. Nader 51 figures and a table of Laplace transforms VII, 326 pages. 1974 Cloth DM 68,-; US $27.80 ISBN 3-540-06407-9 Prices are subject to change without notice

In anglo-american literature there exist numerous books, devoted to the application of the Laplace transformation in technical domains such as electro technics, mechanics etc. Chiefly, they treat problems which, in mathematical language, are governed by ordinary and partial differential equations, in various physically dressed forms. The theoretical foundations of the Laplace transformation are presented usually only in a simplified manner, presuming special properties with respect to the transformed functions, which allow easy proofs. By contrast, the present book intends principally to develop those parts of the theory of the Laplace transformation, which are needed by mathematicians, physicists and engineers in their daily routine work, but in complete generality and with detailed, exact proofs. The applications to other mathematical domains and to technical problems are inserted, when the theory is adequately developed to present the tools necessary for their treatment.

Springer-Verlag Berlin . Heidelberg . New York MOnchen Johannesburg London Madrid New Delhi Paris Rio de Janeiro Sydney Tokyo Utrecht Wien

F. Oberhettinger and L. Badii:

Tables of Laplace Transforms

VII, 428 pages. 1973 OM 39,-; US $16.00 ISBN 3-540-06350-1

Th is material rep resents a collection of integrals of the Laplace- and inverse Laplace Transform type. The usefulness of this kind of information as a tool in various branches of Mathematics is firmly established. Previous publications include the contributions by A. Erdelyi and Roberts and Kaufmann (see References). Special consideration is given to results involving higher functions as integrand and it is believed that a substantial amount of them is presented here for the first time. Greek letters denote complex parameters within the given range of val i d ity. Latin letters denote (unless otherwise stated) real positive parameters and a possi ble extension to complex values by analytic continuation will often pose no serious problem.

F.Oberhettinger:

Tables of Bessel Transforms

I x, 289 pages. 1972 OM 32,-; US $13.10 ISBN 3-540-05997-0

This book contains a comprehensive collection of integrals for integral transforms which have cylindrical functions as kernels; the selection reflects the extensive experience of the author. This particular type of integral transforms is of great importance for appl ied mathematicians, physicists, and engineers.

Prices are subject to change without notice

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Tables of Mellin Transforms

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