Vistas of special functions

Shigeru Kanemitsu, Haruo Tsukada9789812707741, 9789812708830, 9812707743

This is a unique book for studying special functions through zeta-functions. Many important formulas of special functions scattered throughout the literature are located in their proper positions and readers get enlightened access to them in this book. The areas covered include: Bernoulli polynomials, the gamma function (the beta and the digamma function), the zeta-functions (the Hurwitz, the Lerch, and the Epstein zeta-function), Bessel functions, an introduction to Fourier analysis, finite Fourier series, Dirichlet L-functions, the rudiments of complex functions and summation formulas. The Fourier series for the (first) periodic Bernoulli polynomial is effectively used, familiarizing the reader with the relationship between special functions and zeta-functions.

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Table of contents :
Contents……Page 12
Preface……Page 8
Abstract……Page 14
2.1 Gamma function……Page 42
2.2 The Euler digamma function……Page 55
3.1 Introduction……Page 64
3.2 Integral representations……Page 67
3.3 A formula of Ramanujan……Page 75
3.4 Some definite integrals……Page 78
3.5 The functional equation……Page 83
Abstract……Page 90
5.1 Derivatives of the Hurwitz zeta-function……Page 94
5.2 Asymptotic formulas for the Hurwitz and related zetafunctions in the second variable……Page 104
5.3 An application of the Euler digamma function……Page 106
5.4 The first circle……Page 110
6.1 Introduction and the theory of Bessel functions……Page 118
6.2 The theory of Epstein zeta-functions……Page 122
6.3 Lattice zeta-functions……Page 128
6.4 Bessel series expansions for Epstein zeta-functions……Page 138
7.1 Fourier series……Page 144
7.2 Integral transforms……Page 164
7.3 Fourier transform……Page 171
7.4 Mellin transform……Page 174
8.1 The theory of periodic Dirichlet series……Page 178
8.2 The Dirichlet class number formula……Page 187
8.3 Proof of the theorems……Page 189
A.1 Function series……Page 196
A.2 Residue theorem and its applications……Page 206
B.1 Summation formula and its applications……Page 210
B.2 Application to the Riemann zeta-function……Page 215
Bibliography……Page 220
Index……Page 226