{
  "id": "1003.2170",
  "version": "v3",
  "published": "2010-03-10T18:47:55.000Z",
  "updated": "2011-08-29T22:58:07.000Z",
  "title": "New definite integrals and a two-term dilogarithm identity",
  "authors": [
    "F. M. S. Lima"
  ],
  "comment": "10 pages, 1 figure. Accepted for publication: Indagat. Mathematicae (08/29/2011)",
  "journal": "Indagationes Mathematicae 23 (2012) 1-9",
  "doi": "10.1016/j.indag.2011.08.008",
  "categories": [
    "math.CA",
    "math.NT"
  ],
  "abstract": "Among the several proofs known for $\\sum_{n=1}^\\infty{1/n^2} = {\\pi^2/6}$, the one by Beukers, Calabi, and Kolk involves the evaluation of $\\,\\int_0^1 {\\int_0^1{1/(1-x^2 y^2) \\, dx} \\, dy}$. It starts by showing that this double integral is equivalent to $\\frac34 \\sum_{n=1}^\\infty{1/n^2}$, and then a non-trivial \\emph{trigonometric} change of variables is applied which transforms that integral into $\\,{\\int \\int}_T \\: 1 \\; du \\, dv$, where $T$ is a triangular domain whose area is simply ${\\pi^2/8}$. Here in this note, I introduce a hyperbolic version of this change of variables and, by applying it to the above integral, I find exact closed-form expressions for $\\int_0^\\infty{[\\sinh^{-1}{(\\cosh{u})}-u] d u}$, $\\,\\int_{\\alpha}^\\infty{[u-\\cosh^{-1}{(\\sinh{u})}] d u}$, and $\\,\\int_{\\,\\alpha/2}^\\infty{\\ln{(\\tanh{u})} \\: d u}$, where $\\alpha = \\sinh^{-1}(1)$. From the latter integral, I also derive a two-term dilogarithm identity.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-08-29T22:58:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M06",
      "40C10",
      "33B30"
    ],
    "keywords": [
      "two-term dilogarithm identity",
      "definite integrals",
      "exact closed-form expressions",
      "triangular domain",
      "hyperbolic version"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1003.2170L"
    }
  }
}