Vistas of Special Functions Ii.
Title:
Vistas of Special Functions Ii.
Author:
Chakraborty, Kalyan.
ISBN:
9789814273985
Personal Author:
Physical Description:
1 online resource (228 pages)
Contents:
Contents -- Preface -- 1. The theory of Bernoulli and allied polynomials -- 2. The theory of the gamma and related functions -- 2.1 Gamma function -- 2.2 The Euler digamma function -- 3. The theory of the Hurwitz-Lerch zeta-functions -- 3.1 Introduction -- 3.2 Integral representations -- 3.3 A formula of Ramanujan -- 3.4 Some definite integrals -- 3.5 The functional equation -- 4. The theory of Bernoulli polynomilas via zeta-functions -- 5. The theory of the gamma and related functions via zeta-functions -- 5.1 Derivatives of the Hurwitz zeta-function -- 5.2 Asymptotic formulas for the Hurwitz and related zetafunctions in the second variable -- 5.3 An application of the Euler digamma function -- 5.4 The first circle -- 6. The theory of Bessel functions and the Epstein zeta-functions -- 6.1 Introduction and the theory of Bessel functions -- 6.2 The theory of Epstein zeta-functions -- 6.3 Lattice zeta-functions -- 6.4 Bessel series expansions for Epstein zeta-functions -- 7. Fourier series and Fourier transforms -- 7.1 Fourier series -- 7.2 Integral transforms -- 7.3 Fourier transform -- 7.4 Mellin transform -- 8. Around Dirichlet's L-functions -- 8.1 The theory of periodic Dirichlet series -- 8.2 The Dirichlet class number formula -- 8.3 Proof of the theorems -- Appendix A Complex functions -- A.1 Function series -- A.2 Residue theorem and its applications -- Appendix B Summation formulas and convergence theorems -- B.1 Summation formula and its applications -- B.2 Application to the Riemann zeta-function -- Bibliography -- Index.
Abstract:
This book (Vista II), is a sequel to Vistas of Special Functions (World Scientific, 2007), in which the authors made a unification of several formulas scattered around the relevant literature under the guiding principle of viewing them as manifestations of the functional equations of associated zeta-functions. In Vista II, which maintains the spirit of the theory of special functions through zeta-functions, the authors base their theory on a theorem which gives some arithmetical Fourier series as intermediate modular relations - avatars of the functional equations. Vista II gives an organic and elucidating presentation of the situations where special functions can be effectively used. Vista II will provide the reader ample opportunity to find suitable formulas and the means to apply them to practical problems for actual research. It can even be used during tutorials for paper writing. Sample Chapter(s). Chapter 1: The theory of Bernoulli and allied polynomials (779 KB). Contents: The Theory of Bernoulli and Allied Polynomials; The Theory of the Gamma and Related Functions; The Theory of the Lipschitz-Lerch Transcendent; Elucidation of Zeta-Identities; Hypergeometric Functions and Zeta-Functions; The Theory of Bessel Functions and the Epstein Zeta-Functions; The Theory of Arithmetical Fourier Series and the Parseval Identities; Around the Dirichlet L-Functions and the Deninger R-Function. Readership: Graduate students and researchers in pure mathematics.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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