INTEGRALS AND SERIES Volume 3 More Special Functions

INTEGRALS AND SERIES

Volume 3

More Special Functions

A.P. Prudnikov Yu. A. Brychkov

Computing Center of the USSR Academy of Sciences, Moscow

O.I. Marichev Byelorussian State University, Minsk

Translated from the Russian by G.G. Gould

GORDON AND BREACH SCIENCE PUBLISHERS New York • Philadelphia • London • Paris • Montreux • Tokyo • Melbourne

CONTENTS

Preface 21 Chapter 1. INDEFINITE INTEGRALS 23

1.1. Introduction 23

1.2. The Generalized Zeta Function! £ (s, x), Bernoulli Polynomiak Ba (x), Euler Polynomial E„(x) and Polylogarithms L i v (*) 23

1.2.1. Integrals containing ; J (s, x), Bn (x) and En (x) 23

1.2.2. Integrals of the form \ xa Un (ax) dx 23

1.2.3. Integrals of the form W (*) Li s (*) dx 23

1.3. The Generalized Fresnel Integrals S (x, v) and C (x, v) 24 f l S(ax, v)1

1.3.1. Integrals of the form \ / (*) [ \ dx 24 J \C (ax, v) f

1.4. The Struve Functions H V ( J C ) and L v ( x ) 24

1.4.1. Integrals of the form \ x^ H v (ax) dx 24

1.4.2. Integrals of the form \ xXeixHv(x)dx 25

1.4.3. Integrals of the form \ xX H (ax) H v (bx) dx 25

1.4.4. Integrals of the form \ x^/„ (ax) Hv(bx) dx 26

1.4.5. Integrals of the form \ x^Lv(ax)dx 27

1.4.6. Integrals of the form V x% e±xLv(x) dx 27

1.5. The Anger Function J v( .*) and Weber Function Ev(.r) 28

1.5.1. Integrals of the form f xK ) v<aJi:>! dx 28

i \Ev(ax)j

1.6. The Lommel Functions «„ > v (x) and S^t v (x) 28

1.6.1. Integrals of the form [ xh { sV-'âx)\ dx 28

1.6.2. Integrals containing / v (x) H S(1J V (x) 29

1.7. The Kelvin Functions b e r v ( * ) , b e l v ( * ) , kerv (x) and kelv (x) . 29

3

CONTENTS

1.7.1. Integrals of the form [ X* I*"*™\ ix and f x * [**^"*\ dx 2 9

J \ b e > v ( a x ) [ j \keiv(ax)[ 1.7.2. Integrals containing products of Kelvin functions 30

1.8. The Airy Functions (x) and Bl (x) 30 C fAi(x)" | 30

1.8.1. Integrals of the form \ f (x) l . \ dx

1.8.2. Integrals containing products of Airy functions 32

1.9. The Integral Functions of Bessel Jlv(x), Neumann Ylv(x) and MacDonald

tf'v«*) 3 3

1.9.1. Integrals of the form \ xaJiv(x)dx 33

r. (Yi {X)\ 1.9.2. Integrals of the form \ * \Ki <.x)( dx 3 3

1.10. The Incomplete Elliptic Integrals F(x, k), E (x, *) and n (*, v, *) 34 1.10.1. Integrals with respect to the argument x 34

1.10.2. Integrals with respect to the modulus ft 35

1.11. The Complete Elliptic Integrals K (ft), E (*) and II ( -^- , v, ft) 36

1.11.1. Integrals of the form f ka (l-k')& l u \ i k 3 S

1.11.2. Various integrals containing K (ft), E (ft) and II ( — . V, ft) 37

1.12. The Legendre Functions P§ (x) and QjJ (x) 37

1.12.1. Integrals of the form f / (x) P» (x) dx, \ f(x)Q§(x)dx 37

1.12.2. Integrals containing products of Legendre functions 3$

1.13. The Whittaker Functions Mp a (*) and »Pi a (x) 39

1.13.1. Integrals of the form f xa e^^/^M^ a{ax) dx 39

1.13.2. Integrals of the form [ xa e±ax/2Wp a (ax) dx 39

1.13.3. Integrals containing products of Whittaker functions 48

1.14. The Confluent Hypergeometric Functions of Kummer tFt {a; b; x) and Tricomi <F(o, b; x) 42

1.14.1. Integrals containing , F , (a; b; x) 42 1.14.2. Integrals containing <F (a, b; x) 43

1.15. The Gauss Hypergeometric Function zFt (a, b; c; x) 44

1.15.1. Integrals of the form \ xa tFt (a, b; c; x) dx 44

1.15.2. Integrals of the form \ (1 -x ) 1 3 2 F , (a, b; c; x) dx 44

1.15.3. Integrals of the form V x a ( l -x )ß 2 F , (a, b; C; x) dx 45

1.16. The Generalized Hypergeometric Function pFq ((ap); (bq); x),the Meijer G-Function, the Mac Robert ^-Function and the Fox //-Function 46

1.16.1. Integrals containing pFq {(ap); (bq); x) . 46 1.16.2. Integrals containing the Meijer G-function . • . . . . ' . 46 1.16.3. Integrals containing the MacRobert £-function • • • *7 1.16.4. Integrals containing the Fox //-function • • • • • • • • • • •» • • . . 47 4

CONTENTS

1.17. The Elliptic Functions of Jacobi and Weierstrassi 47

1.17.1. Integrals of the form \ f (sn u) du 47

1.17.2. Integrals of the form \ / (cn u) du 48

1.17.3. Integrals of the form \ f (dn u) du 49

1.17.4. Integrals of the form \ f (sn u, cn u, dn «) du 49

1.17.5. Integrals containing the Weierstrass elliptic functions - - ' ^ '

Chapter 2. DEFINITE INTEGRALS 53


2.2. The Gamma Function r (x) 53 2.2.1. Integrals along the line i (v-i'oo, y + icn) 53

2.3. The Generalized Zeta Function t % (s, x) 54

2.3.1. Integrals of f (x) t (s, a-tbx) 54

2.4. The Polynomials of Bernoulli Bn (x) and Euler En (x) 55 2.4.1. Integrals of / (x) Bn (x) 55 2.4.2. Integrals containing products of Bernoulli polynomials 55 2.4.3. Integrals of / (x) En (x) 56 2.4.4. Integrals of Em (x) E„(x + a), Bm(rx) E„(x) 57

2.5. The Polylogarithm Liv(x) 57

2.5.1. Integrals of general form 5 7 2.5.2. Integrals of A (x) Li„ {—ex) 59 2.5.3. Integrals of A (x) Li2 ( — ex) . 59

2.6. The Generalized Fresnel Integrals S(x, v) and C (jr, v) 60

2.6.1. Integrals of general form 60

2.6.2. Integrals of A (x) I \ 65

v l l 68

2.6.3. Integrals of xa ePx±r { S {CX' V ) \ \C(cx, v ) |

„ l sin bx\ IS (ex, v) l 2.6.4. Integrals of xa { ) { \ 68

[cos bx | \C(cx, v) ) i , > • r a / I n U + z) "1 IS (ex, v)) Integrals of xa { } { ) 69

Un x-z j \ C (ex, v) / 2.6.5.

jS(bx,n)-\ „ . . X, n )1 ( S (ex, v) 1 2.6.6. Integrals rf x " „ t } { „ > 69

\C(bx, i t )J \C(ex, v ) (

2.6.7. Integrals containing Ei (— bx") l ' > . . . . . . . 70 I.C ( « , v) |

2.6.8. Integrals of xa ( W > , 71 \c i (6x) / \ c<c* , v ) |

2.6.9. Integrals of j " ; \ t ) 71 \erfc <&*'•) / \C(cx, v) )

, „ „ iS(bx)} (S(ex, v ) l 2.6.10. Integrals ofi xa i • • > i > . . . . . . : 72

\c(bx){\C(cx, v ) / „ I y (H, bx) 1 | S (ex, v) 1

2.6.11. Integrals ofi xa l } i S 73 \ r ( n , bx)) \C(ex, v ) /

2.7. The Struve Functions H v ( * ) and Lv(jr) . . . . . . . . . . 73.

5

CONTENTS

2.7.1. Integrals of general form 73 2.7.2. Integrals of xaHv(cx) 80

R / H V < " > 1 2.7.3. Integrals of xa (z±xf < L ( w ) > 80

n fHv<e*>\ 2.7.4. Integrals of xa (z* ± x'f < L ( M ) > 81

2.7.5. Integrals of *"«-•?* <^Lv(cx)J 84

J"Hv(c*rt 2.7.6. Integrals of xae~Px < L ( c x ) > 84

2.7.7. Integrals containing the trigonometric functions and Hv(cx) 85 (Hv(cx)\

2.7.8. Integrals containing the logarithmic function and \ L (CX) ( "®

2.7.9. Integrals containing the inverse trigonometric functions and Hv (cx) 86

2.7.10. Integrals of xa H^bx^1) Hy (cx) 86 ( H V ( C J C ) - |

2.7.11. Integrals containing Ei (-bx1) < L ,£ X . > 86

fH v (cx) \ 2.7.12. Integrals of xa erfc (SA:'-) < L (CJC) > 87

fHv«:x)) 2.7.13. Integrals containing D^ (bx) < L ,fxy > 87

2.7.14. Integrals containing / ^ (bx±r) Hv(cx) 88 2.7.15. Integrals containing Y^ (<p (*)) Hv(cx) 89

2.7.16. Integrals containing products of special functions by Yf(ox) -Hv(cx) 90

2.7.17. Integrals containing products of elementary functions by / , V(CJ:)-LV(CMC) 91

2.7.18. Integrals containing products of special functions by l±y(cx) -Lv(cx) 93

[Hv(cx>\ 2.7.19. Integrals containing K^ ( 94

2.8. The functions of Anger Jy (x) and Weber Ev (*) 97

2.8.1. . Integrals of general form • 97

2.8.2. Integrals of A (*) \ E v ( « ) [ 99

„ f Jv<")1 2.8.3. Integrals of x a « - ' « B < E {cx) > 101

„ f sin 6x1 f J v ( c J : ) \ 2.8.4. Integrals of *» ( ^ J j ^ ^ f 101

f J„(cx)l 2.8.5. Integrals containing J„{bx) and < > 102

^ Ey \CX) )

2.9. The Lommel Functions *(i_ v (x) and 5„> v (* ) 102 2.9.1. Integrals of general form • 102

2.9.2. Integrals of A (x) { *•*• v ( " ) 106

2.9.3. Integrals of xae-Pxr I St>-*{cX)\ 1 0 8

\ S M t V ( « ) / '

6

CONTENTS

2.9.4. Integrals containing trigonometric or inverse trigonometric functions and < •*' v } ' "

\Su > v (c*) /

2.9.5. Integrals of * « / x ( t a ± l ) /„'•*• "™\ 110

2.9.6. Integrals of xaY% (bx) s^ v (ex) 111-

2.9.7. Integrals of *<***, W) { s ^ S } 1 U

2.9.8. Integrals with respect to the index containing Sp,, n[x (e) 112

<berv(x)~\ (kerv(x)1 2.10. The Kelvin Functions { b e i ^ J } and [ ^ W) " « 2.10.1. Integrals of general form 112 2.10.2. Integrals containing algebraic functions and Kelvin functions 117

2.10.3. Integrals containing e~Px and Kelvin functions 117 2.10.4. Integrals containing trigonometric or logarithmic functions and Kelvin functions . . . . 119 2.10.5. Integrals containing produets of two Kelvin functions 120

2.10.6. Integrals of xaE\ (-bx") | b e [ ^ ( « j } 1 2 1

2.10.7. Integrals of xVJ^bx) { ^ g j } 121

2.10.8. Integrals containing K„ (bxn) and Kelvin functions 122

2.11. The Airy Functions Ai (x) and Bi (x) 123 2.11.1. Integrals of general form 123

f Ai (cx)~\ 2.11.2. Integrals of A (x) l } 125

l Bi (ex) I 2.11.3. Integrals of xae-P*r{ ' (CX)\ 126

2.11.4. Integrals of xa < " " * i Ai (ex) 126 (̂ cos bxr )

2.11.5. Integrals of xaJv(bxn Ai (ex) 126 2.11.6. Integrals of Ai (ax+b) Ai (cx+d) 126

2.12. The Integral Functions of Bessel Jiv(x), Neumann Ylv(x) and MacDonald Klv (x) 126 2.12.1. Integrals of general form 127

t Yiv (ex) 1 2.12.2. Integrals of A (x) Jiv (ex) and A (x) l Ki\cx) j 134

2.12.3. Integrals of *« e~Px±nJiv (ex) and xa e~P*±n | ^ £*>j I 3 6

„ (sinbx) , (smbx\ ( Yiv (ex)) 2.12.4. Integrals of *« [ ^ Jtv (ex) and x* [ ^ ( * £ < „ > ) I 3 7

2.12.5. integralsof x« { ^ ( ^ ! J .«„(«> and *« { , n , ^ , J { ^ , „ , } . . . .38

(y i v ( cx ) i 2.12.6. Integrals containing Ei (-6,v) ./i'v (ex) 0r El (-6*) < f-i (cx){ 1 3 8

„ ( si (bx) 1 _ ( si (bx) \ I Yiv (cx)~i 2.12.7. Integrals of. *« ^ ^ j J*v(«) and x« | d ^ J ( ^ (e j t ) J 139

ferf(ft^) 1 ieti(bxr)\ (Yiv(ex)\ 2.12.8. Integrals containing { ^ ^ j 7IV («) or { ^ ^ {Kiy (ex)) "°

„ fS(&*)l „ {S(bx)\ (Yiv(cx)\ 2.12.9. Integrals of- *« { £ ( f a ) J " v <«> and * • ( £ ( f c t >) ( * £ ( « , , } • • • « «

7

CONTENTS

(\(U,bx) 1 „ f V ( | i . bx)} lYiv(cx)\ , ;„ 2.12.10. Integrals of J « ' ' > Jlvicx) and * * < „ ' } < * - , , . , > 1 4 2

\riu,bx)f v I T(n , bx) ) \Kiv(cx)f

Kiv ,CXA 143

2.12.12. Integrals containing products of Bessel functions by Jiw (ex) or

/ y ' v ( " ) \ \lt \«v<«>/ '

2.13. The I.aguerre Function Lv(x) 14 5

2.13.1. Integrals of general form , 145 2.13.2. Integrals of A (X) e-Px L v (ex) 151

2.13.3. Integrals of A f W ^ W 154

2.13.4. Integrals of « « « - • • l \ Lv (ex) 155

t In (xn+zn) \ 2.13.5. Integrals of xPe-'* i ' } L v (CJC> 156

Unix»-«"!) 2.13.6. Integrals of x*et WEi (bxk) Lv(cx) 159

2.13.7. Integrals of xae-cx i S ' " \ Lv(cx) 157 (.ci (bx?) I

2.13.8. Integrals of x<*e-°x ! " * ' \ L v (ex) 158 lerfc(6xO)

fS(6*' ' ) l 2.13.9. Integrals of A - M i } £ v (ex) 15*

lC(bx?)I

2.13.10. Integrals of x a e - < » / V (M" ^ ' l i v (ex) , 159

2.13.11. Integrals of xae-"Jll (bx?) L v (ex) 169

2.13.12. Integrals of x 0 « - » / ^ { j ' r j } ^ v < " ) • • . . . . . . 161

2.13.13. Integrals of x ae9 ) <*)£.,£ (6x0 ^ v ( « ) 162 2.13.14. Integrals of xP-ef W / / m (bx?) Lv (ex) 162 2.13.15. Integrals of xaef> (-")Z-n (bxk) Lv (ex) 163

2.14. The Bateman Function ftv (x) 163


2.14.2. Integrals of A (x) e<f <*>ftv (ex) 171

„ I sin bx > 2.14.3. Integrals of xae~Px { > kv (ex) 173

l cos bx ) 2.14.4. Integrals of x°-e-Px { " * \ kv (ex) 174

U n | * - z | ) 2.14.5. Integrals of x a e - P x E i (±bx)kv(cx) 174

2.14.6. Integrals of xae~Px / S ' {bX) \ kv (ex) 175 \ci(bx)i

( erf (bxr) \ ) kv (ex) 176

eric(bxr)f

2.14.8. Integrals of x*e-Px l \ ky (ex) : 177 lC(bx))

2.14.9. Integrals of xPe-P' \V \ kv (ex) 177 \T(u.,bx)\

2.14.10. Integrals containing /c„(foc) kw(cx) 178

2.15. The Incomplete Elliptic Integrals F(x, * ) , E (x, k), n (x, v, k) 179 ( F (x. k) \

2.15.1. Integrals of / (x) { \ . . 179 \E (x, k) f

8

CONTENTS

2.15.2. Integrals with respect to the modulus k , 1 7 9

2.16. The Complete Elliptic Integrals K (*) , E (x) 180 2.16.1. Integrals of general form 180

2.16.2. Integrals of xa < > 182 I E ( C A T ) /

2.16.3. Integrals of {x±a)a(b±x)$K (ex) 183

2.16.4. Integrals of (x'±a')a{b'±x')P l \ 183 (_ £ \CX) }

H „ ( K (ex) 1 2.16.5. Integrals ofi Ara (x-±a!)P(b2 ± x')V l } 185

l E (ex) \ 2.16.6. Integrals of A (x) K (ex) 187

( K (9(x))\ 2.16.7. Integrals of A (x) l \ 188

2.16.8. Integrals ofi A (x) eV <*> I } 189 l E (CAT) )

f s i n e p u n (K(CAT)) 2.16.9. Integrals of A (x) { , , } { „ . , } 190

\cos(j> (x) | \E (ex) ) 2.16.10. Integrals containing In A (x) K(cx) 191 2.16.11. Integrals containing Ei (q> (Ar)) orerfc ( (AT)) and K (CAT) 192

fK(CAr)1 2.16.13. Integrals containing Kv (<f (Ar)) ( \ 193

1, fc. \cx) f 2.16.14. Integrals of A (Ar) H„ (ip (Ar)) K (ex) 193 2.16.15. Integrals of A (x) S ( i t v ((p (Ar)) K (ex) 194

i K(v(x)) K(cxn 2.16.16. Integrals of A (x) l ) 194

\ E (<f (Ar)) E (ex) /

2.17. The Legendre Function of the Ist Kind Pv (x), P|J (*) 194

2.17.1. Integrals of (zm±xm)PpV (Cx\ 194

2.17.2. Integrals of Ar« (z*±x')&P§ (ex) 195

2.17.3. Integrals of (Ar±a)a (b ±x )$P§ «p (x)) 197

2.17.4. Integrals of (xm ± am)^ (b*±x2)p'p>} (ex) 198

2.17.5. Integrals ot (x ± a)a (b ± x)$ (d ± j.)Yp^ (ex) 199

2.17.6. Integrals of A (x) [ p ^ (CAT) + P ^ (-CAT)] ' 2 0 1

2.17.7. Integrals of A (x) ef <*>P# (X (x)) . . . 202

2.17.8. Integrals containing the hyperbolic functions and PJJ (<P (Ar)) 203

2.17.9. Integrals containing the trigonometric functions and P§ ( (x)) P§ (X (x)) . . . 206

2.17.13. Integrals containing Y\ ( (A-)) P§ (ex) 209 fHi(<p(Ar))1 „

2.17.16. Integrals containing l L £ ( {x)) > P}} (ex) 211

2.17.17. Integrals containing S x ^ ( (Ar)) P v (ex) 212

2.17.19. Integrals of A (x) P ! J (bx) pg (ex) 212

9

CONTENTS

2.17.20. Integrals of A (x) Ptf (U))PgOC(*)) 216

2.17.22. Integrals of xaJx ( (x)) 217

2.17.24. Integrals with respect to the index of products of Pgx+v^ by elementary functions 218

2.17.25. Integrals with respect to the index of products of Pfx-ni (e) by elementary functions 218

2.17.26. Integrals with respect to the index of products of Pcx-i/2 <c) by special functions 219

2.17.27. Integrals with respect to the index of products of Pfx+V (c) by special functions 221

2.17.28. Integrals with respect to the index of products of P irx-l/2 O îx-iß ^ by elementary functions 225

2.17.29. Integrals with respect to the index of products of Pix-m (*) Ptx-iß W by special

functions • • 225

2.17.30. Integrals with respect to the index containing Pirx-iß CO ^ f . i - i / 2 (c) ^ "

2.18. The Legendre Functions of the 2nd Kind <?v (.*), Q§ (x) 228

2.18.1. Integrals of (z<" ± x">ft Q^(cx) 228

2.18.2. Integrals of xa (z! ± x'ft QrJ- (ex) 229

2.18.3. Integrals of <* ± a)a (b ± xft Q$ < W Q§ (ex) 232

2.18.7. Integrals containing hyperbolic or trigonometric functions and

<?tf(<P<*)) 233

2.18.8. Integrals of A (x) Jk (<f (x)) Q§ (% (x)) 234

2.18.9. Integrals of A (x) Kj, (<P <*)) Q$ (ex) 235

2.18.10. Integrals of A (x) K (<p (x)) Qtf (ex) • 236

2.18.11. Integrals of A (x) Q$J <cp (x)) Q# (ex) 236

2.18.12. Integrals of A (x) J^ (cp (x)) Q\ (ex) Q^ (ex) 237

2.18.13. Integrals of A (x) P * ( P ? (cx)Q$ (ex) 239 I. cos y (x) ) "•

2.18.15. Integrals ofi A (x) J^ <<p (x)) P* (ex) Q§ (ex) 240

2.18.16. Integrals with respect to the index containing Q-j- ix —1/2 (c) 240

2.19. The Whittaker Functions Mp_ a (x) and Wp , CT (*) 2411


2.19.2. Integrals of xaWPt a (ex) 254

/ Alpp a (ex) \ 2.19.3. Integrals of, xae~Px < w (cx) > 255

{ M0 0 (ex) 1

w (cx) > 256 2.19.5. Integrals of A (x)e±cx'z <w <cx) > 258

, 1 , 6 . t n t e g r a U o f Ä ± ^ { - £ } { " r a £ ! } » t t

10

CONTENTS

f M 0 a (ex) \ 2.19.7. Integrals of x*e±cxl2 ln<p(x) < w ( M ) > 26S

f Af D 0 (ex) 1 2.19.8. Integrals of J ^ e ± cxl* Ei (— 6x) < w . . > 267

„ „ , ferf (6 VT±7) \ fMp, CT ( « ) \ 2.19.9. Integrals * * « ^ { ^ ( ^ — , J ^ „ ( c J t ) | 267

fAfD a(cx)} 2.19.10. Integrals of * a ^ * ß v (M*+z> ± 1 / 2 ) W 0 (cJC) f 2T0

l Af p CT (ex) | 2.19.11. Integrals containing e ± C J C 'S^V (6* ± r ) W ' ( c ; t ) f 2 7 1

fMpt „(ex)') 2.19.12. Integrals containing Iv(bxr) <w ,cx. > 276

<Yv(bxr) | (Mpia(cx) \ 2.19.13. Integrals containing j ^ ^ r , | | u r p > a ( « ) f 2 7 8

{Af „ „ (ex)! r

P " ( M ) > 283

(M0 „(ex) 1 2.19.15. Integrals of A (x) e±cx'2 P„ (ax ± r - f t ) < ^ (ffje) > 284

i f M p , CT*") 1 2.19.16. Integrals of ^ (x) ePxL* (a + bx) < w (cx) > 288

(Mn „(ex) \ 2.19.17. Integrals of xae~Px H„ (bV~) < , „ / ' > • • 289

{ w p, CT (ex) J

[M„ „(«) | 2.19.18. Integrals of A (x) e ± ™ / 2 C £ (ax±r_ b) J * ^ V 2 8 9

(Mn „(ex) ) 2.19.19. Integrals of A (x) e±c*/2 P<»' v > ( a x ± l - l ) <m

P , , > 294 l w p, CT ( " ' J

2.19.20. Integrals of A (x) , ± « / t / K ( (*)) < „ / ' ( M ) > 300

/ ^ n . v ^ x + z ) ! /Af p a ( c x ) ) 2.19.23. Integrals of A (x) ^ v ( M + f ) j j , ^ ^ ^ | 302

2.19.24. Integrals of W <*> J J M ^ v ( i j ± 1) W p „ ( ^ * T 1 ) 305 j fr J J RR

2.19.25. Integrals of xa cos ax Af v(bx)M a (ex) 308

2.19.26. Integrals of xa Jv (bx ± r ) J J Afp ._ CT . (6 .x) W p CT.(eftx) 308 i,k J 1 *' *

2,9.27. Integrals of rfJM*^ l / * » • ( " *> " * • < f l , „

2.19.28. Integrals with respect to the index containing M(>, <j (c) o r wp, a<c) 3 U

2.20. The Confluent Hypergeometric Functions of Kummer 1pi ( a ; b; x) and Tricomi V(a, b;x) .' 314

2.21. The Gauss Hypergeometric Function 2Ft ((*, b; c; x) 314 2.21.1. Integrals of A (x) , F , (a, b; c; (x)) 318

11

CONTENTS

[ S j n c ; c ± l / 2 1 2.21.3. Integrals of A (x) < , , . > 2 F , (a, i ; <:; t F , (a, t ; c; (x)) 2 F , (a, 6; c; X (x)) 327

2.21.7. Integrals of .4 (x) K (ax) 2 F t (a, b; c; <p (x)) 329 2.21.8. Integrals containing Whittaker functions and 2F, (a, 6; c; cox) 329 2.21.9. Integrals containing products of two functions 2Ft (a, b; c; <f (x)) 329 2.21.10. Integrals with respect to the integrals containing 2 F t (a, b; c; x) 331

2.22. The Generalized Hypergeometric Function pFq ({ap); (bq); x) and Hypergeometric Functions of TwoVariables 333

2.22.1. Integrals of general form 333 2.22.2. Integrals of A (x) pFq ((ap); (bq); q> (x)) 334

2.22.3. Integrals of ef WpFq ((ap); (bQ); (p (x)) - 335 2.22.4. Integrals containing Bessel functions and pFq ((ap); (bq); ex) 337 2.22.5. Integrals containing various special functions and

pFq((ap); (&„); ex) 339 2.22.6. Integrals with respect to the parameters containing pFq ((ap); (bq); x) 340 2.22.7. Integrals of hypergeometric functions of two variables • • • •' 340

2.23. The MacRobert ^-Function E (p; ar;q; bs:x) 341 2.23.1. Integrals of A (x) E ((ap); (bq); <p (x)) 342 2.23.2. Integrals containing exponential or hyperbolic functions and'

E((ap); (bq); ex) 342 2.23.3. Integrals containing Bessel functions and E ((ap); (bq); ex) 343 2.23.4. Integrals containing Legendre functions and E ((ap); (bq); ex) 344 2.23.5. Integrals containing Whittaker functions and E ((ap); (bq); ex) 344 2.23.6. Integrals containing products of two /s-funetions 3 44

/ I (flp)\ 2.24. The Meijer G-Function Gj™ {* | r f 345 2.24.1. Integrals of general form 345

2.24.2. Integrals of A (x) G™ (<p (x) J °P J : . • • . 347

2.24.3. Integrals containing exponential, hyperbolic or trigonometric functions and

GwX°*l/k\Z) 35° 2.24.4. Integrals containing Bessel functions and G™j} ( © * ' / * . ) 351

2.24.5. Integrals containing orthogonal polynomials and amn ^(axl/k\("bP

)\ 351

2.24.6. Integrals containing Legendre functions and c!?" I tox'/* ,, 1 352 pq \ I (bq) /

2.24.7. Integrals containing Whittaker functions and G™" («>«"*. , 6 ) 3 5 2

2.24.8. Integrals containing the Gauss function 2F, (a, b; c; x) and G™£ ( <p (x) \ j . 3 5 3

2.24.9. Integrals containing sFf((cs); (äfi; - a x ) G™" ( wx'/* f ) 354

[ I [ap, Ap]-\

» , / / . 354 I [bq, Bq\ J

2.25.1. Integrals of general form 354 2.25.2. Integrals containing elementary functions and the //-function . . . 3 54 2.25.3. Integrals containing special functions and the //-function . . . i . . . . . . . . . . 355

12

CONTENTS

2.26. The Theta Functions 8 , (x, f ) 356

2.26.1. Integrals of f (x) 9, (x. q) 356

2.26.2. Integrals containing products of 6Aax, q) 357

2.26.3. Integrals with respect to q, containing 9.(*> <"?) 357

2.26.4. Integrals with respect to q, containing 6 . (x, aq) 358

2.27. The Mathieu Functions 359 2.27.1. Integrals containing elementary functions and c e n (x, q) or se n (x, q) 359 2.27.2. Integrals containing special functions and cen (*, q) or sen (x, q) 361 2.27.3. Integrals containing Ce„ (x, q) or Se„ (x, q) 3 64 2.27.4. Integrals containing fe„(.t, q) 0 r gen (x. q) 365 2.27.5. Integrals containing Fe„ (x, q) or Ge„ (x, q) 366 2.27.6. Integrals containing products of Mathieu functions 367

2.28. The functions v ( * ) , v (x, p), \x.(x, X), u (J?, m, « ) , X(x, a) 368 2.28.1. Integrals containing v (ex), v (ex, p) 368 2.28.2. Integrals containing p. (ex, A,), p- (ex, m, n) 369 2.28.3. Integrals containing A, (ex, a) 369

Chapter i. DEFINITE INTEGRALS OF PIECEWISE-CONTINUOUS FUNCTIONS . . I . . 370


3.2. Piecewise-Constant Functions •• 370 3.2.1. Bounded functions 370 3.2.2. Unbounded functions . . , . . 371

3.3. Sonic Piecewise-Continuous Functions 3 73 3.3.1. Power functions 373 3.3.2. Various functions 375

Chapter 4. MULTIPLE INTEGRALS 377


4.2. Double Integrals 3 77 4.2.1. Integrals containing the functions Hv(x), Jv(x), be r v (x ) , b e i v ( r ) , s ^ _ v (x) . . . 377 4.2.2. Integrals containing ,F, (a; b; x) . . . 378 4.2.3. Integrals containing ,F, (a, b; c; x) 379 4.2.4. Integrals containing pFq((ap); (bq); x) . 379

4.3. Multiple Integrals 3 8 » 4.3.1. Integrals containing pFq ((ap); (bq); x) 380 4.3.2. Integrals over a sphere 381 4.3.3. Various integrals 382

Chapter 5. FINITE SUMS 383

5.1. The Numbers and Polynomials of Bernoulli B„, Bn (x) and Euler En, En (x) . . . . 383 5.1.1. Sums containing Bn 383 5.1.2. Sums containing Bn (x) 384 5.1.3. Sums containing En ' 385 5.1.4. Sums containing En (.x) . . . . . . . . . . 385

13

CONTENTS

5.2. The Legendre Functions P ^ (x) and Q§ (JE) 38«

5.2.1. Sums of the form X « / ± I * W ^ < . j « v ^ U I 386

5.2.2. Sums containing products of Legendre functions 387

5.3. The Generalized Hypergeometric Function pFq ((dp); (bq); x) and the Meijeri C-Function 387

5.3.1. Sums of the form / . ah p + i f g (-k-tn, (ap); (6g); x) 387

5.3.2. Sums of the form J^ a f t p+,Fq (-k, v+ft, ( a p ) ; (6g); x) 389

5.3.3. Sums of the form 2âk p+lFq(-k, (ap) + k; (bq) + k\ x) 389

5.3.4. Sums of the form / , ah p+tFg (-k, (ap)-mk; (bq)-nk; x) 389

5.3.5. Sums of the form ^4aft pFq((ap) + k (cp); (bq) + k(dq); x) 390

5.3.6. Sums of the form J^dak pFq ((ap)-k (cp); (bq)-k(dq); x) 392

5.3.7. Various sums containing pFq((ap); (bq); x) 393 5.3.8. Sums containing the G-function 393 5.3.9. Sums containing the Neumann polynomials 0n(x) 395 5.3.10. Various sums 395

Chapter 6. SERIES 396


6.2. The Generalized Zeta Function % (s, v) 396

6.2.1. Series of the form ^ aftr*£ (s ± k, v) 396

6.3. The numbers and polynomials of Bernoulli Bn, Bn (x) and Euler En, En (x) 397

6.3.1. Series of the form 2~iakBk 3 9 7

6.3.2. Series of the form ^ a f t ß Ä (x+ky) 397

6.3.3. Series of the form ^ a f t E f t 3 9 8

6.3.4. Series of the form " V o f t£ f t (x + ky) 398

6.4. The Functions of Struve Hv(x), Weber E v ( * ) a n d Anger J v ( * ) 3 98

6.4.1. Series containing Hv(x) 398

6.4.2. Series containing EV(JC) 400

6.4.3. Series containing J v ( x ) 4011

6.5. The Legendre Functions pV (x) and Q§ (x) 401

6.5.1. Series of the f o r m ^ o ^ P ^ S (z) 401

6.5.2. Series of the form / 1 a.uQ\n-kn (z) 403

6.5.3. Series of the form > Ä ah cos (kg+b) P^tkn (*) 404

E | s in ( J«+(>) l u + fcn

ak\cos(ka+b>) « v + toW < ° 5

H

CONTENTS

6.5.5. Series of the form £ , a f t ? £ ' + * £ (x) J > £ ' + * £ & ) 405

6.5.6. Series of the f o r m ^ ak <?£ '+*£ <*> «v. ' + ftSl^) • i06

6.5.7. Series of the f o r m ^ a ^ P ^ ^ <*) Q^tkal W 407

6.5.8. Series of the form ^ a f t cos (ka + b) P^+^[ W p v , ' + ftSj (f> 4 0 8

6.5.9. Series of the f o r m ^ a f t cos (fta+6) Q^'+JtoJ (*) <?v*+feS*, <*> 4 ° »

6.5.10. Series of the form ^ a f t cos (fta +6) />£ '+&£ (*> <# , '+&, , 'W 4 ° 9

6.6. The Kummer Confluent Hypergeometric Function tF, (a; b; x) • • • • • 409

6.6.1. Series of the form / 1ah tFt fe b^\ x) 409

6.6.2. Series of the form / ( au tF, (aft; bk; x)tF, (a'k; b'k; y) 411

6.6.3. Various series containing , P , (a; b; x) 412

6.7. The Gauss Hypergeometric Functioni tF, (a , b; c; x) 412

6.7.1. Series of the form / , ah 2/ r1 («L, bu\ Cu', x) 412

Z fak'bk\ fak'b'k\ ak*F'\ « M . 414 V CW x 1 \ck; y J

6.7.3. Various series containing 2^1 (a, b; c; x) 416

6.8. The Generalized Hypergeometric Function pFq ({dp); (bq); x) , . 417

6.8.1. Series of the form 7 { a.ktk pFa ((ap) ± k (tp); (bq) ± k(dq); x) 417

6.8.2. Series containing trigonometric functions and pFq ((ap); (bq); x) . • • • 420 6.8.3. Series containing special functions andpFq ((ap); (bq); x) 421 6.8.4. Series containing products of pFq ((ap); (bq); x) 422

6.9. Various Hypergeometric Functions . 422 6.9.1. Series containing 2 F 2 (a, b; c, d; x) 422 6.9.2. Series containing sFi(ai, a2, a3; &,, b2; xl** 423 6.9.3. Series containing various hypergeometriFfunctibnsi 425

6.10. The MacRobert ^-Function E (p; ar:q; bs'-z) 426

6.10.1. Series of the form ^ « f c E (("/>) ± m * l (bq) ± nk- *) 426

6.10.2. Series containing products of £-functions 427

(bq)) 4 2 7

(ap) ± k (cp) \

(bq) ±k(dq)J 4 2 7

6.11.2. Series containing trigonometric functions and the G-function 428

6.11. The Meijer G-function Oj™ \^z

6.11.1. Series of the form ^ a ^ O ^ l z

6.12. Various Series 429 6.12.1. Series containing the Neumann polynomials On (x) 429

15

CONTENTS

Chapter 7. THE HYPERGEOMETRIC FUNCTIONS: PROPERTIES, REPRESENTATIONS,

PARTICULAR VALUES 4 3 0


7.2. The Main Properties of the Hypergeometric Functions 430 7.2.1. The Uauss hypergeometric tunction; 2 F , (a, b; c; 2) 430 7.2.2. The confluent hypergeometric functions of Kummer , F , (a; b; z), Tricomi

V{a,b; z) and Whittaker Mp_ a (2), Wp_ a (z) 434

7.2.3. The generalized hypergeometric functiom pFq {(ap)\ (bq); z) 43 7 7.2.4. Hypergeometric functions of several variables 448

7.3. The Functions ,Fa (a; *) and 2F, (a, b; c; z) 453 7.3.1. Representations of ,F„ (a; z) and 2 F t (a, 6; c; z) 453 7.3.2. Particular values of 2F, (a, b; c; z) 468 7.3.3. Representations of 2 F t (a, b; c; - z ) 486 7.3.4. Particular values of 2 F t (a, b; c, - z ) 487 7.375. Values of 2 F, (a, b; c; 1) 489

7.3.6. Values of ,'F, (a, b; c; - 1) 489

7.3.7. Values of 2 F, fa, b; c; - M 491

7.3.8. Values of 2 F, ( - « , b; c; 2) 4 93 7.3.9. Values of 2 F, (a, i ; c; z0) for z0 ¥ = ± 1 . 2 * 1 493

7.4. The Function ,F,lait a2, a3; bu b2; z) 497 7.4.1. Representations of 3 F 2 (a,, a2, a3; b„ bt\ z) 497 7.4.2. Particular values of 3F2 (a„ a2, a3 ; fc„ ft2; 2) 500 7.4.3. Particular values of 3 F 2 (a„ a2, a3 ; 6j, f>2; - z ) 532 7.4.4. Values of 3 F 2 (a,, a2, a3; &,, fc2; 1) 533 7.4.5. Values of 3 F 2 (a„ a2, a3; b„ b2; - 1 ) 546 7.4.6.. Values of 3F2 (a„ a2, a3; fc,, f>2; z0) for z0 =*=±1 551

7.5. The Function tF3 («i, a2 , a3 , o 4 ; &,, 62, 63; a) 552 7.5.1. Representations of 4 F 3 (n„ a2, a3, a4; 6 t, 62, ba; z) 552 7.5.2. Particular values of 4 F 3 (a,, a2, a3> a4; bu b2, b3; z) 553 7.5.3. Values of 4 F 3 (a,, a» a„ at; bu b„ bt; 1) 554 7.5.4. Values of 4 F 3 (o„ o„ a„ at; blt b„ bt; -l) 560 7.5.5. Values of 4 F , (o„ a2, a3, a4; 6„ fc2, 63; z„) for z0 ¥= ± 1 563

7.6. The Function 5 /^ (0 , o6; ft, 64; «) 564 7.6.1. Particular values of 5 F 4 (a,, . . . a6; b, b4; ±z) . . 564 7.6.2. Values of 5 F 4 (a„ . . ., a5; * „ . . . , b,; 1) 564 7.6.3. Values of 6 F 4 ( a , a s; fc„ i>4: - 1 ) 567 7.6.4. Values of 5 F 4 ( a , a6; 6,, . . . , b,; z0) for z„ ^=±1 568.

7.7. The Function ,F 5 (a , , ..., a„; ft, 65;«) 568 7.7.1. Representations of , F S (a, a,; fc,, . . . , b s ; z) 568 7.7.2. Values of , F 5 (a, a,; h ft5; 1) 569 7.7.3. Values of , F 5 (a, a,; 6, fc5; - 1 ) 569

7.8. The Function ,F, (a, a7; 6j ft«; «) 570 7.8.1. Values of ,F, (a, a,; b, b,; 1) . . . . . . . . . . . . . . . . . . . . 570 7.8.2. Values of , F , (a, a7; fc, b,; - 1 ) 571

7.9. The Functions ,F, (a, a,; b, b,; z) and,/7, (<!,, ..., a,; b, 6„; z). . 571 7.9.1. Values of „F, (a, a,; fc,, . . . , b,; ±1) . . . . s . 571 7.9.2. Values of „F, (a, a,; *, b,; 1) . . . . . . ' . 571

7.10. The Function ? + i l » , ( ( o 0 + 1 ) ; ( 6 , ) ; *) • • 572

7.10.1. Representations of q + tFq U<*q + i)\ {bq); z) 572

7.10.2. Values of a + .Fq «afl+,1; 0>u); ± ' ) 5 73

7.11. The Functions of Kummer ,F, (a; b; z) and Tricomi V (a, b; z) 579

IG

CONTENTS

7.11.1. Representations of 0F 0 (z) a n d ^ f a ; b\ z). . . . . . . . . . . . . . ' . ' 579 7.11.2. Particular values of ,F , (<J; 6; z) . . 580 7.11.3. Representations of ,F , (a; 6; - z ) 583 7.11.4. Representations and particular values ot *P (a, b; z) 584

7.12. The Functions 2F2 (at, a2; &,, b2; «) and 9 f 4 ((aq); (bq); z) 585 7.12.1. Representations of 2F2 (a,, a2; 6,, fc2; z) 585 7.12.2. Particular values of 2F2 (a„ a2; 6„ fr2; z) 585 7.12.3. Representations of 3F 3 (a,, a2, a3; bu b2, b3; - z ) 593 7.12.4. Representations of qFq ((aq); (bq); z) 593

7.13. The Function „f, (6; z) 594 7.13.1. Representations and particular values of 0^1 (*: ± z ) 594

7.14. The Function ,/?2 ( a , ; » , , b2; z) 595 7.14.1. Representations of ,^,(11; 6,, b2; z) . 595 7.14.2. Particular values of ,F , (a; bu b2; z) 596 7.14.3. Representations of tF2(a; fc,, b2; —z) 608 7.14.4. Particular values of lF2 (a; bu b2; -z) 609

7.15. " The Function 2F, (a,, o2; b„ b2, b3; z). 609 7.15.1. Representations of ,F, (at, a2; bt, b2, b3; z) 609 7.15.2. Particular values of 2F3 (a„ a2: bu b2, b3; z) 610 7.15.3. Representations and particular values of 2 F 3 (a i , a2; bu b,, b3; -z) 611

7.16. Functions of the Form 0Fq ((bq); z), q = 2, 3 611 7.16.1. Particular values of „F2 (bt, b2; z) . . . 611 7.16.2. Representations and particular values of 0 F 3 (b„ b2, b3; z) 611 7.16.3. Representations and particular values of 0F3 (&,, b2, b3\ -z) 612 7.16.4. Representations of ,F 4 (b„ b2, b3, bt; z) sr\d0Fq((bq); z) 613

7.17. Functions of the form pFa (— n, (ap_t); z), p-2, 3 614 7.17.1. Representations of 2F0(-n, a; z) 614 7.17.2. Representations of 2Fa(-n, a; - z ) 614 7.17.3. Representations of 3F0 (—n, a„ a2; z) 614

7.18. Various Hypergeometric Functions . 615 7.18.1. Representations of ,Fq (a; (bq); z) 615 7.18.2. Representations of 3 F 8 (a,, a2, a3; bi, • • -. *«; z) 615 7.18.3. Representations of 4FX (—n, alt ait a3; b; z) 615

Chapter 8. THE MEIJER G-FUNCTION AND THE FOX »FUNCTION 616

8.1. Introduction >. 616

8.2. The Meiier G-Function O™" (z (Ap \ 617

8.2.1. Definition and Notation. 61 7 8.2.2. Basic Properties , 618

8.3. The Fox »Function HfJ} \ z p' P 1 . . . 626 pq L \[bq, Bg]_i

8.3.1. Definition and Notation 626 8.3.2. Basic Properties 627

8.4. Table of Mellin Transforms and Representations of Elementary and Special Functions in Terms of the Meijer G-Function and the Fox »Function 630

8.4.1. Formulae of general form 630 8.4.2. Power functions and algebraic functions 631 8.4.3. The exponential function 633 8.4.4. The hyperbolic functions 634 8.4.5. The trigonometric functions . . . . . . . . . . . . . . . . . . . . . -. . . . . . . . . . 635

rr

CONTENTS

8.4.6. The logarithmic functiom 637 8.4.7. The inverse trigonometric functions *>39 8.4.8. The inverse hyperbolic functions, 640 8.4.9. The polylogarithm Li„ (Ar) M I 8.4.10. The function * (xr, s, v) 641 8.4.11. The integral exponential function| Ei (x> 642 8.4.12. The integral sines Si (x), si (x) and cosine ci (x) 642 8.4.13. The integral hyperbolic sinei shi (x) and cosine chi (x), , , . . . . . 644 8.4.14. The error function erf (x), erfc (x) and erfi (x) 645 8.4.15. The Fresnel integralsi S (x) and C (x) 646 8.4.16. The incomplete gamma functions y (v, x) and r (v, X). 647 8.4.17. The generalized Fresnel integrals S (x, v) and c (*. v> 649 8.4.18. The parabolic cylinder function £>v(x) 650 8.4.19. The Bessel function Jy{x) 651

8.4.20. The Neumann function Yv(x) 654

8.4.21. The Hankel functions //«> (X) a n d H<?> (x) 662

8.4.22. The modified Bessel function / v (x) • . • 662

8.4.23. The MacDonald functiom K v (x) 665

8.4.24. The integral Bessel functions Jiv(x), Vt'v (x) and Kiv(x) 669

8.4.25. The Struve functions H v (x) and L v (x ) 670

8.4.26. The functions of Weber E v ( x ) , E ^ (x) and Anger J v ( x ) , J{"? (X) 671

8.4.27. The Lommel functions s ^ v (x) and S^ v (x) 672 8.4.28. The Kelvin functions' b e r ^ x ) , be i v (x)', ker v (x) and ke i v (Ar) 673

8.4.29. The Airy functions AI (Ar) and Bl (x) 676 8.4.30. The Legendre polynomials Pn (x) 679 8.4.31. The Chebyshev polynomials of the Ist kind Tn (x) 681 8.4.32. The Chebyshev polynomials of the 2nd kind Un (x) 683

8.4.33. The Laguerre polynomials £.*• (Ar) and Ln(x) 685

8.4.34. The Hermite polynomials Hn (x) 686

8.4.35. The Gegenbauer polynomials C^ (x) 686

8.4.36. The Jacobi polynomials P{f' a > (x) 690

8.4.37. The Laguerre function Lv (x) 691

8.4.38. The Bateman function kv (x) 691

8.4.39. The Lommel function Uv(.x, z) 692

8.4.40. The complete elliptic integrals K (Ar), E (x) and D (X) 692

8.4.41. The Legendre functions of the Ist kind P§ (x) and Pv M 697

8.4.42. The Legendre functions of the 2nd kind Q^ (AT) and Q v (Ar) 704

8.4.43. The Whittaker function Mp, o (x) 713 8.4.44. The Whittaker function w'p> „(AT) 713

8.4.45. The confluent hypergeometric function of Kummer %Pt (<*'• b; x). . . , 715 8.4.46. The confluent hypergeometric function of Tricomi V (a, b\ x) 715 8.4.47. The function oî 0; x) 717 8.4.48. The function iF2 (a; ft„ 62; x) 717 8.4.49. The Gauss hypergeometric function 2^1 (", b; c; x) 718 8.4.50. The function 3^2 ("i. a2, a3; b„ b2; x) 725 8.4.51. Various functions of hypergeometric type 727 8.4.52. Index of particular cases of the Meijer G-function and the Fox //-function 730

Appendix I. Some Properties of Integrals, Series, Products and Operations with them . . . . . . . 733


1.2. Convergence of Integrals and Operations with them 733 1.2.1. Integrals along unbounded curves 1 733

18

CONTENTS

1.2.2. Criteria for the convergence of integrals with infinite limits of non-negative functions . 734 1.2.3. Convergence tests for integrals with infinite limits of arbitrary functions 73 7 1.2.4. Integrals of unbounded functions along bounded curves 73 8 1.2.5. Convergence tests for integrals of non-negative unbounded functions 738 1.2.6. Uniform convergence of functions and integrals depending on a parameter 740 1.2.7. Operations with integrals depending on a parameter 741

1.3. Convergence of Series and Products and Operations with them 742 1.3.1. Basic notions 742 1.3.2. Convergence tests for positive series 743 1.3.3. Convergence tests for arbitrary series 74 9 1.3.4. Tests for the uniform convergence of series depending on a parameter 750 1.3.5. Operations with series 751 1.3.6. Power series • 752 1.3.7. Trigonometrie series 755 1.3.8. Asymptotic series 756 1.3.9. Infinite produets 757

Appendix II. Special Functions and their Properties 757

11.1. The binomial coefficients ( . ) 757

11.2. The Pochhammer symbol (a)^ 758 11.3. The gamma funetion r (z) 759 11.4. The psi funetion i|> (z) 760 11.5. The polylogarithm Li v (z ) 762 11.6. The generalized Fresnel integrals S (z, v) and C (z, v) 764 11.7. The generalized zeta funetion J (z, u) • 764 II.8 The Bernoulli polynomials Bn (z) and the Bernoulli number Bn 765 11.9. The Euler polynomials En (z) and the Euler numbers En 766 11.10. The Struve functions H v (z) and L v ( z ) 767

11.11. The functions of Weber Ev ' (z), E ^ (z) and Anger J v (z). J$ (z) 767

11.12. The Lommel functions s v (z) a n ( j S„ v (z) 768

11.13. The Kelvin functions berv (z), ba i v (z), ke r v ( z ) a n d ke i v (z) 769

11.14. The Airy functions (z). Bi (z) 770 11.15. The Bessel integral functions Jiv(z), V i v (z), Kt'v(z) 770

11.16. The incomplete elliptic integrals F (cp, k), E (tp, k), D (tp, *), n (tp, v, k), A0 (<p. ß. *) and the complete elliptic integrals K (k), E (*), D (*) 771

11.17. The Bateman funetion * v ( z ) 773

11.18. The Legendre functions ^ v U). P% (z). Qv <*). Q$ (2) 773 11.19. The MacRobert £-function E (p; ar:q; bs:z) 779 11.20. The Jacobi elliptic functions cn u, dn u, sn u 779 11.21. The Weierstrass elliptic funetion ? ( « ) , Z (u), a (u) 780 11.22. The theta functions Q Az, q), / = 0, 1, 2, 3, 4 780 11.23. The Mathieu functions. • • • 780 11.24. The polynomials of Neumann On (z) and Schläfli Sn(z) 787 11.25. The functions V (z), V (z, p), H (z, %), H (z. K P) • • • • 787

Bibliography 788

Index of notations for functions and constants 791

Index of mathematical Symbols 800

19

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INTEGRALS AND SERIES Volume 3 More Special Functions