Special Functions: A Graduate Text

by Richard Beals , Roderick Wong

• ISBN13: 9780521197977
• ISBN10: 052119797X
• Format: Hardcover
• Copyright: 2010-09-27
• Publisher: Cambridge University Press

Summary

The subject of special functions is often presented as a collection of disparate results, which are rarely organised in a coherent way. This book answers the need for a different approach to the subject. The authors' main goals are to emphasise general unifying principles coherently 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. Providing ready access to theory and formulas, this book serves as an ideal graduate-level textbook as well as a convenient reference.

Table of Contents

Orientation	p. 1
Power series solutions	p. 2
The gamma and beta functions	p. 5
Three questions	p. 6
Elliptic functions	p. 10
Exercises	p. 11
Summary	p. 14
Remarks	p. 16
Gamma, beta, zeta	p. 18
The gamma and beta functions	p. 19
Euler's product and reflection formulas	p. 22
Forulas of Legendre and Gauss	p. 26
Two characterizations of the gamma function	p. 28
Asymptotics of the gamma function	p. 29
The psi function and the incomplete gamma function	p. 33
The Selberg integral	p. 36
The zeta function	p. 40
Exercises	p. 43
Summary	p. 50
Remarks	p. 56
Second-order differential equations	p. 57
Transformations, symmetry	p. 58
Existence and uniqueness	p. 61
Wronskians, Green's functions, comparison	p. 63
Polynomials as eigenfunctions	p. 66
Maxima, minima, estimates	p. 72
Some equations of mathematical physics	p. 74
Equations and transformations	p. 78
Exercises	p. 81
Summary	p. 84
Remarks	p. 92
Orthogonal polynomials	p. 93
General orthogonal polynomials	p. 93
Classical polynomials: general properties, I	p. 98
Classical polynomials: general properties, II	p. 102
Hermite polynomials	p. 107
Laguerre polynomials	p. 113
Jacobi polynomials	p. 116
Legendre and Chebyshev polynomials	p. 120
Expansion theorems	p. 125
Functions of second kind	p. 131
Exercises	p. 134
Summary	p. 138
Remarks	p. 151
Discrete orthogonal polynomials	p. 154
Discrete weights and difference operators	p. 154
The discrete Rodrigues formula	p. 160
Charlier polynomials	p. 164
Krawtchouk polynomials	p. 167
Meixner polynomials	p. 170
Chebyshev-Hahn polynomials	p. 173
Exercises	p. 177
Summary	p. 179
Remarks	p. 188
Confluent hypergeometric functions	p. 189
Kummer functions	p. 190
Kummer functions of the second kind	p. 193
Solutions when c is an integer	p. 196
Special cases	p. 198
Contiguous functions	p. 199
Parabolic cylinder functions	p. 202
Whittaker functions	p. 205
Exercises	p. 209
Summary	p. 211
Remarks	p. 220
Cylinder functions	p. 221
Bessel functions	p. 222
Zeros of real cylinder functions	p. 226
Integral representations	p. 230
Hankel functions	p. 233
Modified Bessel functions	p. 237
Addition theorems	p. 239
Fourier transform and Hankel transform	p. 241
Integrals of Bessel functions	p. 242
Airy functions	p. 244
Exercises	p. 248
Summary	p. 253
Remarks	p. 262
Hypergeometric functions	p. 264
Hypergeometric series	p. 265
Solutions of the hypergeometric equation	p. 267
Linear relations of solutions	p. 270
Solutions when c is an integer	p. 274
Contiguous functions	p. 276
Quadratic transformations	p. 278
Transformations and special values	p. 282
Exercises	p. 286
Summary	p. 290
Remarks	p. 298
Spherical functions	p. 300
Harmonic polynomials; surface harmonics	p. 301
Legendre functions	p. 307
Relations among the Legendre functions	p. 311
Series expansions and asymptotics	p. 315
Associated Legendre functions	p. 318
Relations among associated functions	p. 321
Exercises	p. 323
Summary	p. 326
Remarks	p. 334
Asymptotics	p. 335
Hermite and parabolic cylinder functions	p. 336
Confluent hypergeometric functions	p. 339
Hypergeometric functions, Jacobi polynomials	p. 343
Legendre functions	p. 346
Steepest descents and stationary phase	p. 348
Exercises	p. 352
Summary	p. 364
Remarks	p. 369
Elliptic functions	p. 371
Integration	p. 372
Elliptic integrals	p. 375
Jacobi elliptic functions	p. 380
Theta functions	p. 384
Jacobi theta functions and integration	p. 389
Weierstrass elliptic functions	p. 394
Exercises	p. 398
Summary	p. 404
Remarks	p. 416
Complex analysis	p. 419
Fourier analysis	p. 425
Notation	p. 431
References	p. 433
Author index	p. 449
Index	p. 453
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