{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "GBXD38Y52R"
    }
  ],
  "body": "\n## Summary\n\nThe trigamma function $\\psi_1$ is the derivative of the [[digamma function|HWRRXB0YXS]], the second derivative of $\\log \\Gamma$.\nIt is the [[polygamma function|3J3QGKPMZQ]] of order $1$ and equals $\\sum_{k \\ge 0} (z+k)^{-2}$.\nIt is positive on the positive real axis, which expresses the log-convexity of the [[gamma function|0H5SYBSED9]].\n\n## Explanations\n\n### L1 human\n\nThe trigamma function measures how fast the slope of $\\log \\Gamma$ changes, that is, the bending of the graph of $\\log \\Gamma$.\nIt is a sum of squares of reciprocals: $\\frac{1}{z^2} + \\frac{1}{(z+1)^2} + \\frac{1}{(z+2)^2} + \\cdots$.\nFor $z = 1$ this is the famous Basel sum $1 + \\frac14 + \\frac19 + \\cdots = \\frac{\\pi^2}{6}$.\nSince every term is positive for $z \u003e 0$, the graph of $\\log \\Gamma$ always bends upward there.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "33B15"
    ],
    "name": "Trigamma function",
    "same_as": {
      "wikidata": "Q1244426",
      "wikipedia": "Trigamma function"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Trigamma_function"
      }
    ],
    "specializes": [
      "3J3QGKPMZQ"
    ],
    "tag": "5VGVPENA0F"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/5VGVPENA0F/",
  "links": [
    {
      "field": "specializes",
      "target": "3J3QGKPMZQ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/5VGVPENA0F.md",
  "msc": [
    "33B15"
  ],
  "name": "Trigamma function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/5VG/5VGVPENA0F.md",
  "status": null,
  "tag": "5VGVPENA0F",
  "uri": "m2://5VGVPENA0F",
  "url": "https://tamnd.github.io/math2/wiki/trigamma-function/",
  "writing": {
    "tag": "5VGVPENA0F",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}