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arXiv · 2111.09623

The asymptotic expansion of a Mathieu-exponential series

R B Paris
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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.

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R B Paris. 2021-11-18. The asymptotic expansion of a Mathieu-exponential series. https://arxiv.org/abs/2111.09623

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