{
  "id": "1809.08794",
  "version": "v1",
  "published": "2018-09-24T08:16:59.000Z",
  "updated": "2018-09-24T08:16:59.000Z",
  "title": "Asymptotics of a Gauss hypergeometric function with large parameters, IV: A uniform expansion",
  "authors": [
    "R B Paris"
  ],
  "comment": "9 pages, 0 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "We consider the uniform asymptotic expansion for the Gauss hypergeometric function \\[F(a+\\epsilon\\lambda,m;c+\\lambda;x),\\qquad \\lambda\\to+\\infty\\] for $x<1$ and positive integer $m$ when the parameter $\\epsilon>1$ and the constants $a$ and $c$ are supposed finite. When $m=1$, we employ the standard procedure of the method of steepest descents modified to deal with the situation when a saddle point is near a simple pole. It is shown that it is possible to give a closed-form expression for the coefficients in the resulting uniform expansion. The expansion when $m\\geq 2$ is obtained by means of a recurrence relation. Numerical results illustrating the accuracy of the resulting expansion are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-24T08:16:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C05",
      "34E05",
      "41A60"
    ],
    "keywords": [
      "gauss hypergeometric function",
      "large parameters",
      "uniform asymptotic expansion",
      "standard procedure",
      "steepest descents"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}