The asymptotic expansion of a Mathieu-exponential series

Abstract

We consider the asymptotic expansion of the functional series \[S_μ^\pm(a;λ)=\sum_{n=0}^\infty \frac{(\pm 1)^n e^{-λn}}{(n^2+a^2)^μ}\] for $\lambda >0$ and $\mu \ge 0$ as $|a|\to \infty$ in the sector $|\arg a|<\pi /2$. The approach employed consists of expressing $S_{\mu }^{\pm }(a;\lambda )$ as a contour integral combined with suitable deformation of the integration path. Numerical examples are provided to illustrate the accuracy of the various expansions obtained.

12 pages, 1 figure

References in corpus (3)

• Asymptotic expansions of several series and their application
• Asymptotic expoansions of mathieu-Bessel series. I
• Asymptotics of a Mathieu-Gaussian series