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Application of the Hurwitz Zeta Function to the Evaluation of Certain Integrals

Published online by Cambridge University Press:  20 November 2018

Zhang Nan Yue  and
Kenneth S. Williams
Zhang Nan Yue
Affiliation:
Information Department The People's University of China Beijing 100872 People 5 Republic of China
Kenneth S. Williams
Affiliation:
Department of Mathematics and Statistics Carleton University Ottawa, Ontario KIS 5B6
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Abstract

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The Hurwitz zeta function ζ(s, a) is defined by the series

for 0 < a ≤ 1 and σ = Re(s) > 1, and can be continued analytically to the whole complex plane except for a simple pole at s = 1 with residue 1. The integral functions C(s, a) and S(s, a) are defined in terms of the Hurwitz zeta function as follows:

Using integral representations of C(s, a) and S(s, a), we evaluate explicitly a class of improper integrals. For example if 0 < a < 1 we show that

Keywords

11M35Hurwitz zeta functionintegral representationevaluation of improper integralsrecurrence relations
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 36 , Issue 3 , 01 September 1993 , pp. 373 - 384
Copyright
Copyright © Canadian Mathematical Society 1993

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