{
  "id": "2103.04284",
  "version": "v1",
  "published": "2021-03-07T07:07:04.000Z",
  "updated": "2021-03-07T07:07:04.000Z",
  "title": "On the asymptotic of Wright functions of the second kind",
  "authors": [
    "Richard Paris",
    "Armando Consiglio",
    "Francesco Mainardi"
  ],
  "comment": "13 pages, 7 coupled figures",
  "journal": "Fractional Calculus and Applied Analysis, Vol 24, No 1, pp. 54--72 (2021)",
  "doi": "10.1515/fca-2021-0003",
  "categories": [
    "math.CA"
  ],
  "abstract": "The asymptotic expansions of the Wright functions of the second kind, introduced by Mainardi [see Appendix F of his book {\\it Fractional Calculus and Waves in Linear Viscoelasticity}, (2010)], $$ F_\\sigma(x)=\\sum_{n=0}^\\infty \\frac{(-x)^n}{n! \\g(-n\\sigma)}~,\\quad M_\\sigma(x)=\\sum_{n=0}^\\infty \\frac{(-x)^n}{n! \\g(-n\\sigma+1-\\sigma)}\\quad(0<\\sigma<1)$$ for $x\\to\\pm\\infty$ are presented. The situation corresponding to the limit $\\sigma\\to1^-$ is considered, where $M_\\sigma(x)$ approaches the Dirac delta function $\\delta(x-1)$. Numerical results are given to demonstrate the accuracy of the expansions derived in the paper, together with graphical illustrations that reveal the transition to a Dirac delta function as $\\sigma\\to 1^-$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-07T07:07:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30B10",
      "30E15",
      "33C20",
      "34E05",
      "41A60"
    ],
    "keywords": [
      "second kind",
      "wright functions",
      "dirac delta function",
      "fractional calculus",
      "asymptotic expansions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}