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Ⅰ．Sine,cosine and logarithmie integral 1

Text-books and other tables 5

？(Cix) and ？(aix) 9

Ⅱ．Factorial function 9

ψ(x) 18

ψ＇(x) 18

Text-books and other tables 21

The incomplete factorial function 22

Ⅲ．Error integral and related functions 23

1．Error integral φ(x) 24

2．dn φ(x)/dxn 24

3．The functions of the parabolic cylinder 32

？ et2 dt 32

4．Fresnel＇s integrals 35

5．The after-effect function 38

Text-books and other tables 40

Ⅳ．Theta-functions 41

Modular function 43

Other tables 43

log q 49

Ⅴ．Elliptic Integrals 52

A．Incomplete Integrals 52

F(α,φ) 52

E(α,φ) 54

B．Complete Integrals 73

Integral of the third kind 81

Taxt-books and other tables 83

C．Inductance of coils 86

Ⅵ．Elliptic functions 90

Jacobian functions 90

Weicrstrass＇s functions 98

φ＇u,φu,ξu,σu for g2=0,g2=1 101

Text-books and other tables 106

Ⅶ．Legendre functions 107

Legendre＇s associated functions 110

Text-books and other tables 117

dPn/dθ 124

Ⅷ．Bessel funetions 126

1．Definitions 128

2．Asymptotic representations 137

3．Zeros 143

4．Elementary functional equations 144

5．Differential formulae 145

6．Integral formulae 145

7．Differential equations 146

8．Integral representations 147

a) Real argument． 154

Jn+0,5(x),J-n-ο,i(x) 154

Jο(x),J1(x) 156

Jο(x)=0;J1(x)=0 162

Text-books and other tables 164

J±0,25(x),J±0,75(x) 164

1st and 2nd zero of Jρ(x) 152

Jρ(xn)=0;p=0,1,2,．．．5;n=1,2,．．．9 152

Jρ(n);n=1,2,．．．29 174

An(x)=n!/(x∶2)n Jn(x);n=0,1,2,．．．8 189

Nο(x),N1(x) 190

Roots of equations containing Bessel functions 204

9．The function ？ρ(z) of H．F．Weber and Lommel 210

10．The Struve function Qρ(x) 211

b) Imaginary argument． 224

Jο(ix),J1(ix) 224

Jn(ix);n=0,1,．．．11 224

J-ρ(x+iy)=0 230

Jn(ix)Jn(x)-iJn(x)J＇n(ix)=0,n=0,1,2 234

J n/3(i x);n=±1,±2 235

H(1)ο (ix),H(1)1 (ix) 242

c) Complex argument,especially x？i． 242

H(z)ξ (riz)=0 242

H(1)p (z)=0;H？(e)=0 243

Jο(x？i)=uο+ivο 244

？iJ1(x？i)=u1+iv1 244

H(1)ο (x？i)=Uο+iVο 250

？iH？(x？i)=U1+iV1 256

J(r？i)=biB 260

H(1)？(r？i)=hiη 259

Jο(r？i)∶J1(r？i) 264

H(1)ο(r？i)∶H(1)1(r？i) 265

Ⅸ．The Riemann Zeta-Function 269

Rernoulli＇s numbers 272

Zeros 0,5+iαn 274

Text-books and other tables 274

Ⅹ．Confiuent hypergeometric functions 275

Text-books and other tables 275

Ⅺ．Mathieu functions 283

Text-books and other tables 287

Some often used eonstants 295

Useful books for the computer 296

Index of tables of the elementary transcendentals 299

Supplementary Bibliography 302

General Index 305