Special functions of applied mathematics

Title page......Page 1
PREFACE......Page 8
ACKNOWLEDGMENTS......Page 14
1.1 Introduction......Page 16
1.2 The seventeenth century......Page 17
1.3 The eighteenth century......Page 18
1.4 The nineteenth century......Page 19
1.5 The twentieth century......Page 20
Notes......Page 21
2.1 Introduction......Page 22
2.2 Appell's symbol......Page 23
2.3 Vandermonde's theorem......Page 27
2.4 Hypergeometric series......Page 29
2.5 Wallis' formula for Pi......Page 35
2.6 Arithmetic, geometric, and logarithmic means......Page 36
2.7 Stirling's formula for n!......Page 38
Notes......Page 39
Exercises......Page 40
3.1 Introduction......Page 46
3.2 Definition and difference equation......Page 47
3.3 Analyticity......Page 49
3.4 Limit formulas......Page 50
3.5 Logarithmic convexity......Page 53
3.6 The reciprocal of the gamma function......Page 56
3.7 The duplication theorem......Page 59
3.8 Stirling's formula in the complex plane......Page 60
3.9 Euler's reflection formula......Page 63
3.10 Inequalities......Page 65
3.11 Euler measures......Page 67
Formulary......Page 69
Exercises......Page 70
4.1 Introduction......Page 74
4.2 The beta function of two variables......Page 75
4.3 The beta function of several variables......Page 76
4.4 Dirichlet measures......Page 79
Notes......Page 83
Exercises......Page 84
5.1 Introduction......Page 88
5.2 The averaging process......Page 90
5.3 Averages of derivatives......Page 93
5.4 Euler-Poisson differential equations......Page 95
5.5 Newton-Taylor series with remainder......Page 99
5.6 Associated functions......Page 103
5.7 Averages of power series......Page 106
5.8 Averages of e^x......Page 108
5.9 Averages of x^t......Page 112
5.10 Confluence......Page 117
5.11 Averages of Cauchy's integral formula......Page 120
5.12 Averages of e^(1/x)......Page 122
Formulary......Page 131
Exercises......Page 133
6.1 Introduction......Page 143
6.2 Representation as a polynomial......Page 144
6.3 Dirichlet averages with negative parameters......Page 150
6.4 The binomial theorem......Page 155
6.5 Linear transformation......Page 156
6.6 Generating functions......Page 157
6.7 Polynomials of Legendre, Gegenbauer, and Chebyshev......Page 163
6.8 Analytic continuation of the R function......Page 168
6.9 The first quadratic transformation......Page 174
6.10 The second quadratic transformation......Page 179
6.11 Gegenbauer's product formula......Page 186
Notes......Page 189
Formulary......Page 190
Exercises......Page 192
7.1 Taylor series and Jacobi series......Page 204
7.2 Biorthogonality......Page 207
7.3 The addition theorem for Gegenbauer polynomials......Page 210
7.4 Asymptotic properties......Page 212
7.5 Convergence of series......Page 215
7.6 Jacobi series of an analytic function......Page 218
7.7 Applications......Page 221
7.8 Rodrigues' formula and orthogonality......Page 224
7.9 Laguerre polynomials......Page 226
7.10 Hermite polynomials......Page 232
Exercises......Page 236
8.1 Evaluation of integrals......Page 246
8.2 The Schwarz-Christoffel mapping and elliptic functions......Page 248
8.3 Gauss's theorem and dependence on a small variable......Page 255
8.4 Existence theorem for associated functions......Page 260
8.5 Removal of integral parameters......Page 263
Exercises......Page 268
9.1 Introduction......Page 272
9.2 Symmetric standard functions......Page 276
9.3 Reduction to standard functions......Page 281
9.4 Applications......Page 285
9.5 Landen's transformation......Page 290
9.6 The duplication theorem......Page 293
9.7 The addition theorem......Page 296
9.8 A reduction theorem......Page 298
Exercises......Page 302
A Notation for sets......Page 306
B Integrals depending on a parameter......Page 307
Solutions to exercises......Page 314
References......Page 330
INDEX......Page 338