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[资源] 剑桥2010年英文原版Special.Functions-.A.Graduate.Text

The subject of special functions is often presented as a collection of disparate results,
which are rarely organized in a coherent way. This book answers the need for a
different approach to the subject. The authors’ main goals are to emphasize general
unifying principles and to provide clear motivation, efficient proofs, and original
references for all of the principal results.
The book covers standard material, but also much more, including chapters on
discrete orthogonal polynomials and elliptic functions. The authors show how a very
large part of the subject traces back to two equations – the hypergeometric equation
and the confluent hypergeometric equation – and describe the various ways in which
these equations are canonical and special.
Each chapter closes with a summary that provides both a convenient guide to the
logical development and a useful compilation of the formulas. This book serves as an
ideal graduate-level textbook as well as a convenient reference.
Richard Beals is Professor Emeritus of Mathematics at Yale University.
Roderick Wong is Professor of Mathematics and Vice President for Research at
City University of Hong Kong.
1 Orientation 1
1.1 Power series solutions 2
1.2 The gamma and beta functions 5
1.3 Three questions 6
1.4 Elliptic functions 10
1.5 Exercises 11
1.6 Summary 14
1.7 Remarks 16
2 Gamma, beta, zeta 18
2.1 The gamma and beta functions 19
2.2 Euler’s product and reflection formulas 22
2.3 Formulas of Legendre and Gauss 26
2.4 Two characterizations of the gamma function 28
2.5 Asymptotics of the gamma function 29
2.6 The psi function and the incomplete gamma function 33
2.7 The Selberg integral 36
2.8 The zeta function 40
2.9 Exercises 43
2.10 Summary 50
2.11 Remarks 56
3 Second-order differential equations 57
3.1 Transformations, symmetry 58
3.2 Existence and uniqueness 61
3.3 Wronskians, Green’s functions, comparison 63
v
vi Contents
3.4 Polynomials as eigenfunctions 66
3.5 Maxima, minima, estimates 72
3.6 Some equations of mathematical physics 74
3.7 Equations and transformations 78
3.8 Exercises 81
3.9 Summary 84
3.10 Remarks 92
4 Orthogonal polynomials 93
4.1 General orthogonal polynomials 93
4.2 Classical polynomials: general properties, I 98
4.3 Classical polynomials: general properties, II 102
4.4 Hermite polynomials 107
4.5 Laguerre polynomials 113
4.6 Jacobi polynomials 116
4.7 Legendre and Chebyshev polynomials 120
4.8 Expansion theorems 125
4.9 Functions of second kind 131
4.10 Exercises 134
4.11 Summary 138
4.12 Remarks 151
5 Discrete orthogonal polynomials 154
5.1 Discrete weights and difference operators 154
5.2 The discrete Rodrigues formula 160
5.3 Charlier polynomials 164
5.4 Krawtchouk polynomials 167
5.5 Meixner polynomials 170
5.6 Chebyshev–Hahn polynomials 173
5.7 Exercises 177
5.8 Summary 179
5.9 Remarks 188
6 Confluent hypergeometric functions 189
6.1 Kummer functions 190
6.2 Kummer functions of the second kind 193
6.3 Solutions when c is an integer 196
6.4 Special cases 198
6.5 Contiguous functions 199
6.6 Parabolic cylinder functions 202
6.7 Whittaker functions 205
Contents vii
6.8 Exercises 209
6.9 Summary 211
6.10 Remarks 220
7 Cylinder functions 221
7.1 Bessel functions 222
7.2 Zeros of real cylinder functions 226
7.3 Integral representations 230
7.4 Hankel functions 233
7.5 Modified Bessel functions 237
7.6 Addition theorems 239
7.7 Fourier transform and Hankel transform 241
7.8 Integrals of Bessel functions 242
7.9 Airy functions 244
7.10 Exercises 248
7.11 Summary 253
7.12 Remarks 262
8 Hypergeometric functions 264
8.1 Hypergeometric series 265
8.2 Solutions of the hypergeometric equation 267
8.3 Linear relations of solutions 270
8.4 Solutions when c is an integer 274
8.5 Contiguous functions 276
8.6 Quadratic transformations 278
8.7 Transformations and special values 282
8.8 Exercises 286
8.9 Summary 290
8.10 Remarks 298
9 Spherical functions 300
9.1 Harmonic polynomials; surface harmonics 301
9.2 Legendre functions 307
9.3 Relations among the Legendre functions 311
9.4 Series expansions and asymptotics 315
9.5 Associated Legendre functions 318
9.6 Relations among associated functions 321
9.7 Exercises 323
9.8 Summary 326
9.9 Remarks 334
viii Contents
10 Asymptotics 335
10.1 Hermite and parabolic cylinder functions 336
10.2 Confluent hypergeometric functions 339
10.3 Hypergeometric functions, Jacobi polynomials 343
10.4 Legendre functions 346
10.5 Steepest descents and stationary phase 348
10.6 Exercises 352
10.7 Summary 364
10.8 Remarks 369
11 Elliptic functions 371
11.1 Integration 372
11.2 Elliptic integrals 375
11.3 Jacobi elliptic functions 380
11.4 Theta functions 384
11.5 Jacobi theta functions and integration 389
11.6 Weierstrass elliptic functions 394
11.7 Exercises 398
11.8 Summary 404
11.9 Remarks 416
Appendix A: Complex analysis 419
Appendix B: Fourier analysis 425
Notation 431
References 433
Author index 449
Index 453

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附件 1 : Beals,.Wong-.Special.Functions-.A.Graduate.Text,.CUP,.2010.pdf

2015-01-28 18:40:58, 2.54 M