{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nThe *inverse tangent integral* is the function $\\operatorname{Ti}_2 \\colon \\R \\to \\R$ given by\n$$\\operatorname{Ti}_2(x) = \\int_0^x \\frac{\\arctan t}{t} \\, dt.$$\nHere $\\arctan$ is the [[arctangent|0CE768XNVP]], and the integrand has the value $1$ at $t = 0$.\n\n## Remarks\n\nFor $\\abs{x} \\le 1$, integrating the series of $\\arctan t / t$ term by term gives\n$$\\operatorname{Ti}_2(x) = \\sum_{k=0}^{\\infty} \\frac{(-1)^k x^{2k+1}}{(2k+1)^2}.$$\nThe function is odd, and $\\operatorname{Ti}_2(x) = \\operatorname{Im} \\operatorname{Li}_2(ix)$ for real $x$, where $\\operatorname{Li}_2$ is the [[dilogarithm|G0X0NWSDB7]].\n\nExample: $\\operatorname{Ti}_2(1) = 1 - \\frac19 + \\frac1{25} - \\cdots = G$, Catalan's constant.\n\nInversion check: for $x \u003e 0$, the derivative of $\\operatorname{Ti}_2(x) - \\operatorname{Ti}_2(1/x)$ is $\\frac{\\arctan x + \\arctan(1/x)}{x} = \\frac{\\pi}{2x}$, and the difference is $0$ at $x = 1$, so\n$$\\operatorname{Ti}_2(x) - \\operatorname{Ti}_2\\left(\\tfrac1x\\right) = \\frac{\\pi}{2} \\log x.$$\n\nNon-example: $\\int_0^x \\frac{\\arctan t}{t^2} \\, dt$ is not a variant of this function; it diverges at $t = 0$, because $\\arctan t / t^2 \\sim 1/t$ there.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "NGGC4TTYNR",
    "kind": "definition",
    "msc": [
      "33B30",
      "33E20"
    ],
    "name": "Inverse tangent integral",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Inverse_tangent_integral"
      }
    ],
    "tag": "Z3AYQ5M633",
    "uses": [
      "0CE768XNVP"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/Z3AYQ5M633/",
  "links": [
    {
      "field": "defines",
      "target": "NGGC4TTYNR"
    },
    {
      "field": "uses",
      "target": "0CE768XNVP"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/Z3AYQ5M633.md",
  "msc": [
    "33B30",
    "33E20"
  ],
  "name": "Inverse tangent integral",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Z3A/Z3AYQ5M633.md",
  "status": null,
  "tag": "Z3AYQ5M633",
  "uri": "m2://Z3AYQ5M633",
  "url": "https://tamnd.github.io/math2/wiki/inverse-tangent-integral-definition/",
  "writing": {
    "tag": "Z3AYQ5M633",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}