Table of Contents
ISRN Applied Mathematics
Volume 2014, Article ID 985782, 8 pages
http://dx.doi.org/10.1155/2014/985782
Research Article

Some Remarks on the Mathieu Series

Landesbank Baden-Württemberg (LBBW), Am Hauptbahnhof 2, 70173 Stuttgart, Germany

Received 11 December 2013; Accepted 29 January 2014; Published 13 March 2014

Academic Editors: G. Fikioris and C.-H. Lien

Copyright © 2014 Robert Frontczak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The object of this note is to present new expressions for the classical Mathieu series in terms of hyperbolic functions. The derivation is based on elementary arguments concerning the integral representation of the series. The results are used afterwards to prove, among others, a new relationship between the Mathieu series and its alternating companion. A recursion formula for the Mathieu series is also presented. As a byproduct, some closed-form evaluations of integrals involving hyperbolic functions are inferred.