{
  "id": "2007.11300",
  "version": "v1",
  "published": "2020-07-22T09:36:15.000Z",
  "updated": "2020-07-22T09:36:15.000Z",
  "title": "Bounds for an integral of the modified Bessel function of the first kind and expressions involving it",
  "authors": [
    "Robert E. Gaunt"
  ],
  "comment": "18 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "Simple upper and lower bounds are obtained for the integral $\\int_0^x\\mathrm{e}^{-\\gamma t}t^\\nu I_\\nu(t)\\,\\mathrm{d}t$, $x>0$, $\\nu>-\\frac{1}{2}$, $0<\\gamma<1$. Most of our bounds for this integral are tight as $x\\rightarrow\\infty$. We apply one of our inequalities to bound some expressions involving this integral. Two of these expressions appear in Stein's method for variance-gamma approximation, and our bounds will allow for a technical advancement to be made to the method.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-22T09:36:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C10",
      "26D15"
    ],
    "keywords": [
      "modified bessel function",
      "first kind",
      "simple upper",
      "steins method",
      "lower bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}