{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "2XB240D6MP"
    },
    {
      "field": "uses",
      "target": "34JP0GZ3ED"
    },
    {
      "field": "uses",
      "target": "3XNJ04AMCW"
    },
    {
      "field": "uses",
      "target": "J9Y7DBZVJA"
    },
    {
      "field": "specializes",
      "target": "G0X0NWSDB7"
    }
  ],
  "body": "\n## Summary\n\nThe polylogarithm $\\operatorname{Li}_s(z)$ is the power series $\\sum_{k \\ge 1} \\frac{z^k}{k^s}$ and its [[analytic continuation|0M3YQJ81NB]].\nFor $s = 1$ it is $-\\ln(1 - z)$, which explains the name, and for $\\operatorname{Re} s \u003e 1$ its value at $z = 1$ is the [[Riemann zeta function|1XWBY4N0VT]] $\\zeta(s)$.\nThe case $s = 2$ is the [[dilogarithm|G0X0NWSDB7]].\nPolylogarithms appear in number theory, in algebraic K-theory, in quantum field theory and in the statistics of Fermi and Bose gases.\n\n## Explanations\n\n### L1 human\n\nThe series $z + \\frac{z^2}{2} + \\frac{z^3}{3} + \\cdots$ adds up to $-\\ln(1 - z)$.\nDividing the $k$-th term by $k$ once more, or $s$ times in all, gives the polylogarithm of order $s$.\nEach step is the same as integrating the previous function after dividing by $z$.\nFor $z = 1$ and $s = 2$ one gets $1 + \\frac14 + \\frac19 + \\cdots = \\frac{\\pi^2}{6}$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "33B30",
      "11G55"
    ],
    "name": "Polylogarithm",
    "same_as": {
      "wikidata": "Q1238449",
      "wikipedia": "Polylogarithm"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Polylogarithm"
      }
    ],
    "tag": "96CHDZKGQ0"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/96CHDZKGQ0/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/96CHDZKGQ0.md",
  "msc": [
    "33B30",
    "11G55"
  ],
  "name": "Polylogarithm",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/96C/96CHDZKGQ0.md",
  "status": null,
  "tag": "96CHDZKGQ0",
  "uri": "m2://96CHDZKGQ0",
  "url": "https://tamnd.github.io/math2/wiki/polylogarithm/",
  "writing": {
    "tag": "96CHDZKGQ0",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}