On some integrals involving the Hurwitz zeta function. I
type: Item
instance of: scholarly article
title: On some integrals involving the Hurwitz zeta function. I
zbMATH Open document ID: 1019.33001
DOI: 10.1023/A:1015706300169
author: Olivier Espinosa
author: Victor H. Moll
published in: The Ramanujan Journal
publication date: 2002-08-21T00:00:00Z
review text: A series of integral formulae involving the Hurwitz zeta function are established. After reviewing some useful integral representations of the Hurwitz zeta function, as Hermite's and Yu's and Williams' ones, the authors do the same with a number of known definite integrals involving the Hurwitz zeta function, as Mikolas' identity for the integral of the product of two of them. They also mention the numerous physical applications as summarized in a book by the reviewer [\textit{E. Elizalde}, Ten physical applications of spectral zeta functions (1995; Zbl 0855.00002)]. After this introduction, the authors deal with the Fourier expansion of \(\zeta(z,q)\) and employ it to evaluate definite integrals of the form \(\int^1_0f(q) \zeta(z,q)dq\). A very useful theorem for the integral of the product of two Hurwitz zeta functions is proven in the next section, and classical relations for the Bernoulli polynomials are obtained as corollaries. Several explicit applications are then given, in particular to integrals of Bernoulli polynomials, \(\ln\Gamma (q)\) and \(\ln\sin q\), to an expression for Catalan's constant, to Clausen and related functions, and to Eisenstein series. Finally, to be remarked is the fact that most of the formulae involving elementary functions and \(\ln\Gamma(q)\) can be considered, with the help of equalities of the form \(\ln\Gamma (q)+\ln \Gamma (1-q)= \ln\pi-\ln \sin(\pi q)\), to be specail cases of the already mentioned theorem, which relates, in a linear way, all the above mentioned examples.
reviewed by: Emilio Elizalde
Mathematics Subject Classification ID: 33B15
Mathematics Subject Classification ID: 33E20
Mathematics Subject Classification ID: 11M35
Mathematics Subject Classification ID: 11B68
zbMATH DE Number: 1785613
zbMATH Keywords: Hurwitz zeta function
zbMATH Keywords: Bernoulli polynomials
zbMATH Keywords: Eisenstein series
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1785613