{
  "id": "2111.09623",
  "version": "v2",
  "published": "2021-11-18T11:02:21.000Z",
  "updated": "2021-12-06T09:39:06.000Z",
  "title": "The asymptotic expansion of a Mathieu-exponential series",
  "authors": [
    "R B Paris"
  ],
  "comment": "12 pages, 1 figure",
  "categories": [
    "math.CA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2021-12-06T09:39:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30E15",
      "33E20",
      "33C15",
      "34E05",
      "41A60"
    ],
    "keywords": [
      "asymptotic expansion",
      "mathieu-exponential series",
      "functional series",
      "approach employed consists",
      "contour integral"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}