# Asymptotic expansion of Mathieu power series and trigonometric Mathieu series

## TL;DR

This paper derives a comprehensive asymptotic expansion for a generalized Mathieu series involving complex powers and explores asymptotic behavior of trigonometric Mathieu series, extending previous results.

## Contribution

It introduces a full asymptotic expansion for generalized Mathieu series using Mellin transforms and polylogarithm functions, expanding on known results for alternating series.

## Key findings

- Asymptotic expansion expressed via polylogarithm and Hurwitz zeta functions
- Extension of known expansions to generalized Mathieu series with complex powers
- First-order asymptotic behavior of trigonometric Mathieu series analyzed

## Abstract

We consider a generalized Mathieu series where the summands of the classical Mathieu series are multiplied by powers of a complex number. The Mellin transform of this series can be expressed by the polylogarithm or the Hurwitz zeta function. From this we derive a full asymptotic expansion, generalizing known expansions for alternating Mathieu series. Another asymptotic regime for trigonometric Mathieu series is also considered, to first order, by applying known results on the asymptotic behavior of trigonometric series.