{
  "id": "2104.12912",
  "version": "v1",
  "published": "2021-04-26T23:47:16.000Z",
  "updated": "2021-04-26T23:47:16.000Z",
  "title": "Uniform asymptotic expansions for the Whittaker functions $M_{κ,μ}(z)$ and $W_{κ,μ}(z)$ with $μ$ large",
  "authors": [
    "T. M. Dunster"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Uniform asymptotic expansions are derived for Whittaker's confluent hypergeometric functions $M_{\\kappa,\\mu}(z)$ and $W_{\\kappa,\\mu}(z)$, as well as the numerically satisfactory companion function $W_{-\\kappa,\\mu}(ze^{-\\pi i})$. The expansions are uniformly valid for $\\mu \\rightarrow \\infty$, $0 \\leq \\kappa/\\mu \\leq 1-\\delta <1$, and for $0 \\leq \\arg(z) \\leq \\pi$. By using appropriate connection and analytic continuation formulas these expansions can be extended to all unbounded nonzero complex $z$. The approximations come from recent asymptotic expansions involving elementary functions and Airy functions, and explicit error bounds are either provided or available.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-26T23:47:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C15",
      "34E20",
      "34M60",
      "34E05"
    ],
    "keywords": [
      "uniform asymptotic expansions",
      "whittaker functions",
      "whittakers confluent hypergeometric functions",
      "numerically satisfactory companion function",
      "analytic continuation formulas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}