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Special values of the Lerch zeta function and the evaluation of certain integrals

Authors: Kenneth S. Williams and Nan Yue Zhang
Journal: Proc. Amer. Math. Soc. 119 (1993), 35-49
MSC: Primary 11M35; Secondary 11M06
MathSciNet review: 1172963
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Abstract: The Lerch zeta function $ \Phi (x,a,s)$ is defined by the series

$\displaystyle \Phi (x,a,s) = \sum\limits_{n = 0}^\infty {\frac{{{e^{2n\pi ix}}}} {{{{(n + a)}^s}}}} ,$

where $ x$ is real, $ 0 < a \leqslant 1$, and $ \sigma = \operatorname{Re} (s) > 1$ if $ x$ is an integer and $ \sigma > 0$ otherwise. In this paper we study the function $ J\left( {s,a} \right) = \Phi (\tfrac{1} {2},a,s)$. We use its integral representation

$\displaystyle J\left( {s,a} \right) = \frac{{{a^{ - s}}}} {2} + 2\int_0^\infty ... ...tan }^{ - 1}}\frac{y} {a}} \right)} \frac{{{e^{\pi y}}dy}} {{{e^{2\pi y}} - 1}}$

to obtain the values of certain definite integrals; for example, we show that

\begin{displaymath}\begin{gathered}\int_0^\infty {\frac{{\cosh x\log x}} {{\cosh... ...} \right)} \right\},\qquad 0 < a < 1. \hfill \\ \end{gathered} \end{displaymath}

References [Enhancements On Off] (What's this?)

  • [1] W. Gröbner and N. Hofreiter, Integraltafel, zweiter teil, Bestimmte integrale, Springer-Verlag, Berlin, 1966.
  • [2] E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469
  • [3] Zhang Nan Yue and Kenneth S. Williams, Application of the Hurwitz zeta function to the evaluation of certain integrals, Canad. Math. Bull. 36 (1993), no. 3, 373–384. MR 1245823,

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Keywords: Lerch zeta function, Hurwitz zeta function, integral representation, recurrence relations
Article copyright: © Copyright 1993 American Mathematical Society

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