| 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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