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  "body": "\n## Definition\n\nThe *dilogarithm* is the function $\\operatorname{Li}_2 \\colon \\C \\setminus (1, \\infty) \\to \\C$ given by\n$$\\operatorname{Li}_2(z) = -\\int_0^z \\frac{\\log(1 - u)}{u}\\, du.$$\nHere $\\log$ is the principal branch of the [[complex logarithm|02QGGD5G87]].\nThe path of integration lies in $\\C \\setminus [1, \\infty)$, except possibly for its endpoint $z = 1$.\nThe function is holomorphic on $\\C \\setminus [1, \\infty)$ and continuous at $z = 1$.\n\n## Remarks\n\nFor $\\abs{z} \\le 1$ it agrees with the series $\\sum_{k \\ge 1} \\frac{z^k}{k^2}$, so it is the [[polylogarithm|96CHDZKGQ0]] $\\operatorname{Li}_2$.\nThe integral does not depend on the path, because $\\C \\setminus [1, \\infty)$ is simply connected and the integrand is holomorphic there (the singularity at $u = 0$ is removable).\n\nExample: $\\operatorname{Li}_2(1) = \\sum_{k \\ge 1} k^{-2} = \\frac{\\pi^2}{6}$, and $\\operatorname{Li}_2(-1) = \\sum_{k \\ge 1} \\frac{(-1)^k}{k^2} = -\\frac{\\pi^2}{12}$.\nEuler's reflection formula $\\operatorname{Li}_2(z) + \\operatorname{Li}_2(1 - z) = \\frac{\\pi^2}{6} - \\log z \\log(1 - z)$ at $z = \\frac12$ gives $\\operatorname{Li}_2(\\frac12) = \\frac{\\pi^2}{12} - \\frac{(\\ln 2)^2}{2}$.\n\nNon-example of a valid argument: for real $x \u003e 1$ the point $x$ lies on the cut $[1, \\infty)$.\nThe limits of $\\operatorname{Li}_2$ from above and below the cut differ by $2\\pi i \\ln x$, so $\\operatorname{Li}_2$ has no continuous extension there.\n",
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