The asymptotic expansion of a Mathieu-exponential series

TL;DR

This paper derives the asymptotic expansion of a Mathieu-exponential series for large complex arguments using contour integral techniques, providing numerical validation of the expansions.

Contribution

It introduces a method to obtain asymptotic expansions of a specific Mathieu-exponential series via contour integral deformation.

Findings

Derived explicit asymptotic expansions for the series as |a|→∞.

Validated the expansions with numerical examples.

Provided a contour integral approach for similar series.

Abstract

We consider the asymptotic expansion of the functional series \[S_{\mu}^\pm(a;\lambda)=\sum_{n=0}^\infty \frac{(\pm 1)^n e^{-\lambda n}}{(n^2+a^2)^\mu}\] for $\lambda>0$ and $\mu\geq0$ 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.