{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $q \\in \\C$ with $\\abs{q} \u003c 1$.\nThe *quantum dilogarithm* is the function of $x \\in \\C$ given by the infinite [[q-Pochhammer symbol|Y1X8ZZQ7VM]]\n$$(x; q)_\\infty = \\prod_{n=0}^{\\infty} (1 - x q^n).$$\nFor $\\abs{x} \u003c 1$ it is related to the series\n$$\\operatorname{Li}_2(x; q) = \\sum_{n=1}^{\\infty} \\frac{x^n}{n (1 - q^n)}$$\nby $(x; q)_\\infty = \\exp\\big(-\\operatorname{Li}_2(x; q)\\big)$, and $\\operatorname{Li}_2(x; q)$ is also called the quantum dilogarithm.\n\n## Remarks\n\nThe product converges for every $x$, because $\\sum_n \\abs{x q^n} \u003c \\infty$, so $(x; q)_\\infty$ is an entire function of $x$; for $q \\neq 0$ its zeros are the points $x = q^{-n}$, $n \\ge 0$.\nThe relation with the series comes from $\\log(1 - x q^n) = -\\sum_{k \\ge 1} x^k q^{nk} / k$ and summing the geometric series in $n$.\n\nCheck of the classical limit: for fixed $x$ with $\\abs{x} \u003c 1$ and real $q \\to 1^-$, one has $(1 - q) / (1 - q^n) \\to 1/n$, so $(1 - q) \\operatorname{Li}_2(x; q) \\to \\sum_{n \\ge 1} x^n / n^2 = \\operatorname{Li}_2(x)$, the [[dilogarithm|G0X0NWSDB7]].\n\nNon-example: for $\\abs{q} = 1$ and $x \\neq 0$ the product $\\prod_{n \\ge 0} (1 - x q^n)$ does not converge, since its factors do not tend to $1$.\nFor $\\abs{q} = 1$ other versions are used, such as the non-compact quantum dilogarithm of Faddeev, given by an integral.\n\nEuler's expansion gives $\\frac{1}{(x; q)_\\infty} = \\sum_{n \\ge 0} \\frac{x^n}{(q; q)_n}$ for $\\abs{x} \u003c 1$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "6HMT17DVX7",
    "kind": "definition",
    "msc": [
      "33B30",
      "33D05"
    ],
    "name": "Quantum dilogarithm",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Quantum_dilogarithm"
      }
    ],
    "tag": "GY5EEKHA45",
    "uses": [
      "Y1X8ZZQ7VM",
      "G0X0NWSDB7"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/GY5EEKHA45/",
  "links": [
    {
      "field": "defines",
      "target": "6HMT17DVX7"
    },
    {
      "field": "uses",
      "target": "Y1X8ZZQ7VM"
    },
    {
      "field": "uses",
      "target": "G0X0NWSDB7"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/GY5EEKHA45.md",
  "msc": [
    "33B30",
    "33D05"
  ],
  "name": "Quantum dilogarithm",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GY5/GY5EEKHA45.md",
  "status": null,
  "tag": "GY5EEKHA45",
  "uri": "m2://GY5EEKHA45",
  "url": "https://tamnd.github.io/math2/wiki/quantum-dilogarithm-definition/",
  "writing": {
    "tag": "GY5EEKHA45",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}