Special functions: A graduate text

Preface......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
1 Orientation......Page 13
1.1 Power series solutions......Page 14
1.2 The gamma and beta functions......Page 17
1.3 Three questions......Page 18
1.4 Elliptic functions......Page 22
1.5 Exercises......Page 23
1.6.1 Power series solutions......Page 26
1.6.3 Three questions......Page 27
1.7 Remarks......Page 28
2 Gamma, beta, zeta......Page 30
2.1 The gamma and beta functions......Page 31
2.2 Euler's product and reflection formulas......Page 34
2.3 Formulas of Legendre and Gauss......Page 38
2.4 Two characterizations of the gamma function......Page 40
2.5 Asymptotics of the gamma function......Page 41
2.6 The psi function and the incomplete gamma function......Page 45
2.7 The Selberg integral......Page 48
2.8 The zeta function......Page 52
2.9 Exercises......Page 55
2.10.1 The gamma function......Page 62
2.10.2 Euler's product and reflection formulas......Page 63
2.10.5 Asymptotics of the gamma function......Page 64
2.10.6 The psi function and the incomplete gamma function......Page 65
2.10.7 The Selberg integral......Page 66
2.10.8 The zeta function......Page 67
2.11 Remarks......Page 68
3 Second-order differential equations......Page 69
3.1 Transformations, symmetry......Page 70
3.2 Existence and uniqueness......Page 73
3.3 Wronskians, Green's functions, comparison......Page 75
3.4 Polynomials as eigenfunctions......Page 78
3.5 Maxima, minima, estimates......Page 84
3.6 Some equations of mathematical physics......Page 86
3.7 Equations and transformations......Page 90
3.8 Exercises......Page 93
3.9.1 Transformations, symmetry......Page 96
3.9.3 Wronskians, Green's functions, comparison......Page 97
3.9.4 Polynomials as eigenfunctions......Page 99
3.9.5 Maxima, minima, estimates......Page 100
3.9.6 Some equations of mathematical physics......Page 101
3.9.7 Equations and transformations......Page 103
3.10 Remarks......Page 104
4.1 General orthogonal polynomials......Page 105
4.2 Classical polynomials: general properties, I......Page 110
4.3 Classical polynomials: general properties, II......Page 114
4.4 Hermite polynomials......Page 119
4.5 Laguerre polynomials......Page 125
4.6 Jacobi polynomials......Page 128
4.7 Legendre and Chebyshev polynomials......Page 132
4.8 Expansion theorems......Page 137
4.9 Functions of second kind......Page 143
4.10 Exercises......Page 146
4.11.1 General orthogonal polynomials......Page 150
4.11.2 Classical polynomials: general properties, I......Page 151
4.11.4 Hermite polynomials......Page 152
4.11.5 Laguerre polynomials......Page 155
4.11.6 Jacobi polynomials......Page 157
4.11.7 Legendre and Chebyshev polynomials......Page 159
4.11.8 Expansion theorems......Page 162
4.12 Remarks......Page 163
5.1 Discrete weights and difference operators......Page 166
5.2 The discrete Rodrigues formula......Page 172
5.3 Charlier polynomials......Page 176
5.4 Krawtchouk polynomials......Page 179
5.5 Meixner polynomials......Page 182
5.6 Chebyshev–Hahn polynomials......Page 185
5.7 Exercises......Page 189
5.8.1 Discrete weights and difference operators......Page 191
5.8.3 Charlier polynomials......Page 193
5.8.4 Krawtchouk polynomials......Page 195
5.8.5 Meixner polynomials......Page 197
5.8.6 Chebyshev–Hahn polynomials......Page 198
5.9 Remarks......Page 200
6 Confluent hypergeometric functions......Page 201
6.1 Kummer functions......Page 202
6.2 Kummer functions of the second kind......Page 205
6.3 Solutions when c is an integer......Page 208
6.4 Special cases......Page 210
6.5 Contiguous functions......Page 211
6.6 Parabolic cylinder functions......Page 214
6.7 Whittaker functions......Page 217
6.8 Exercises......Page 221
6.9.1 Kummer functions......Page 223
6.9.2 Kummer functions of the second kind......Page 225
6.9.4 Special cases......Page 226
6.9.5 Contiguous functions......Page 227
6.9.6 Parabolic cylinder functions......Page 228
6.9.7 Whittaker functions......Page 230
6.10 Remarks......Page 232
7 Cylinder functions......Page 233
7.1 Bessel functions......Page 234
7.2 Zeros of real cylinder functions......Page 238
7.3 Integral representations......Page 242
7.4 Hankel functions......Page 245
7.5 Modified Bessel functions......Page 249
7.6 Addition theorems......Page 251
7.7 Fourier transform and Hankel transform......Page 253
7.8 Integrals of Bessel functions......Page 254
7.9 Airy functions......Page 256
7.10 Exercises......Page 260
7.11.1 Bessel functions......Page 265
7.11.2 Zeros of real cylinder functions......Page 266
7.11.3 Integral representations......Page 267
7.11.4 Hankel functions......Page 268
7.11.5 Modified Bessel functions......Page 270
7.11.7 Fourier transform and Hankel transform......Page 271
7.11.8 Integrals of Bessel functions......Page 272
7.11.9 Airy functions......Page 273
7.12 Remarks......Page 274
8 Hypergeometric functions......Page 276
8.1 Hypergeometric series......Page 277
8.1.1 Examples of generalized hypergeometric functions......Page 278
8.2 Solutions of the hypergeometric equation......Page 279
8.3 Linear relations of solutions......Page 282
8.4 Solutions when c is an integer......Page 286
8.5 Contiguous functions......Page 288
8.6 Quadratic transformations......Page 290
8.7 Transformations and special values......Page 294
8.8 Exercises......Page 298
8.9.1 Hypergeometric series......Page 302
8.9.2 Solutions of the hypergeometric equation......Page 303
8.9.3 Linear relations of solutions......Page 304
8.9.5 Contiguous functions......Page 305
8.9.6 Quadratic transformations......Page 306
8.9.7 Transformations and special values......Page 308
8.10 Remarks......Page 310
9 Spherical functions......Page 312
9.1 Harmonic polynomials; surface harmonics......Page 313
9.2 Legendre functions......Page 319
9.3 Relations among the Legendre functions......Page 323
9.4 Series expansions and asymptotics......Page 327
9.5 Associated Legendre functions......Page 330
9.6 Relations among associated functions......Page 333
9.7 Exercises......Page 335
9.8.1 Harmonic polynomials; surface harmonics......Page 338
9.8.2 Legendre functions......Page 340
9.8.3 Relations among Legendre functions......Page 341
9.8.4 Series expansions and asymptotics......Page 342
9.8.5 Associated Legendre functions......Page 343
9.8.6 Relations among associated functions......Page 345
9.9 Remarks......Page 346
10 Asymptotics......Page 347
10.1 Hermite and parabolic cylinder functions......Page 348
10.2 Confluent hypergeometric functions......Page 351
10.3 Hypergeometric functions, Jacobi polynomials......Page 355
10.4 Legendre functions......Page 358
10.5 Steepest descents and stationary phase......Page 360
10.6 Exercises......Page 364
10.7.1 Hermite and parabolic cylinder functions......Page 376
10.7.2 Confluent hypergeometric functions......Page 377
10.7.3 Hypergeometric functions, Jacobi polynomials......Page 378
10.7.4 Legendre functions......Page 379
10.7.5 Steepest descents and stationary phase......Page 380
10.8 Remarks......Page 381
11 Elliptic functions......Page 383
11.1 Integration......Page 384
11.2 Elliptic integrals......Page 387
11.3 Jacobi elliptic functions......Page 392
11.4 Theta functions......Page 396
11.5 Jacobi theta functions and integration......Page 401
11.6 Weierstrass elliptic functions......Page 406
11.7 Exercises......Page 410
11.8.1 Integration......Page 416
11.8.2 Elliptic integrals......Page 417
11.8.3 Jacobi elliptic functions......Page 418
11.8.4 Theta functions......Page 422
11.8.5 Jacobi theta functions and integration......Page 424
11.8.6 Weierstrass elliptic functions......Page 426
Appendix A: Complex analysis......Page 428
Appendix B: Fourier analysis......Page 434
Notation......Page 443
References......Page 445
Author index......Page 461
Index......Page 465