{
  "id": "1609.08365",
  "version": "v1",
  "published": "2016-09-27T11:55:09.000Z",
  "updated": "2016-09-27T11:55:09.000Z",
  "title": "Asymptotics of a Gauss hypergeometric function with large parameters, III: Application to the Legendre functions of large imaginary order and real degree",
  "authors": [
    "R. B. Paris"
  ],
  "comment": "14 pages, 4 figures",
  "categories": [
    "math.CA"
  ],
  "abstract": "We obtain the asymptotic expansion for the Gauss hypergeometric function \\[F(a-\\lambda,b+\\lambda;c+i\\alpha\\lambda;z)\\] for $\\lambda\\rightarrow+\\infty$ with $a$, $b$ and $c$ finite parameters by application of the method of steepest descents. The quantity $\\alpha$ is real, so that the denominatorial parameter is complex and $z$ is a finite complex variable restricted to lie in the sector $|\\arg (1-z)|<\\pi$. We concentrate on the particular case $a=0$, $b=c=1$, which is associated with the Legendre functions of real degree and imaginary order. The resulting expansions are of Poincar\\'e type and hold in restricted domains of the $z$-plane. An expansion is given at the coalescence of two saddle points. Numerical results illustrating the accuracy of the different expansions are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-27T11:55:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C05",
      "34E05",
      "41A60"
    ],
    "keywords": [
      "gauss hypergeometric function",
      "large imaginary order",
      "real degree",
      "legendre functions",
      "large parameters"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}