{
  "id": "1107.2007",
  "version": "v2",
  "published": "2011-07-11T12:09:57.000Z",
  "updated": "2011-07-14T12:19:01.000Z",
  "title": "Some asymptotics for the Bessel functions with an explicit error term",
  "authors": [
    "Ilia Krasikov"
  ],
  "comment": "Typos corrected",
  "categories": [
    "math.CA"
  ],
  "abstract": "We show how one can obtain an asymptotic expression for some special functions satisfying a second order differential equation with a very explicit error term starting from appropriate upper bounds. We will work out the details for the Bessel function $J_\\nu (x)$ and the Airy function $Ai(x)$ and find a sharp approximation for their zeros. We also answer the question raised by Olenko by showing that $$c_1 | \\nu^2-1/4\\,| < \\sup_{x \\ge 0} x^{3/2}|J_\\nu(x)-\\sqrt{\\frac{2}{\\pi x}} \\, \\cos (x-\\frac{\\pi \\nu}{2}-\\frac{\\pi}{4}\\,)| <c_2 |\\nu^2-1/4\\,|, $$ $ \\nu \\ge -1/2 \\, ,$ for some explicit numerical constants $c_1$ and $c_2.$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-07-14T12:19:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bessel function",
      "second order differential equation",
      "appropriate upper bounds",
      "asymptotic expression",
      "explicit numerical constants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1107.2007K"
    }
  }
}