{
  "id": "1211.3963",
  "version": "v2",
  "published": "2012-11-16T17:38:48.000Z",
  "updated": "2012-12-04T10:09:02.000Z",
  "title": "Series Expansion of Generalized Fresnel Integrals",
  "authors": [
    "Richard J. Mathar"
  ],
  "comment": "Remark 3 added. Corrected denominator in eq. (3.17) and enumeration of tables. Expanded references",
  "categories": [
    "math.CA"
  ],
  "abstract": "The two Fresnel Integrals are real and imaginary part of the integral over complex-valued exp(ix^2) as a function of the upper limit. They are special cases of the integrals over x^m*exp(i*x^n) for integer powers m and n, which are essentially Incomplete Gamma Functions. We generalize one step further and focus on evaluation of the integrals with kernel p(x)*exp[i*phi(x)] and polynomials p and phi. Series reversion of phi seems not to help much, but repeated partial integration leads to a first order differential equation for an auxiliary oscillating function which allows to fuse the integrals and their complementary integrals.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-12-04T10:09:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33B20",
      "28-04",
      "65D20"
    ],
    "keywords": [
      "generalized fresnel integrals",
      "series expansion",
      "first order differential equation",
      "essentially incomplete gamma functions",
      "series reversion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.3963M"
    }
  }
}