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  "body": "\n## Definition\n\nLet $s$ be a complex number.\nFor $z \\in \\C$ with $\\abs{z} \u003c 1$, the *polylogarithm of order $s$* is the [[power series|0D6KXBHKC0]]\n$$\\operatorname{Li}_s(z) = \\sum_{k=1}^\\infty \\frac{z^k}{k^s},$$\ntogether with its [[analytic continuation|0M3YQJ81NB]] in $z$ to $\\C \\setminus [1, \\infty)$.\n\n## Remarks\n\nThe series has radius of convergence $1$ for every $s$, because $\\abs{k^{-s}}^{1/k} = k^{-\\operatorname{Re} s/k} \\to 1$.\n\nExample: for $s = 1$ the series is $-\\ln(1 - z)$, and for $s = 0$ and $s = -1$ it gives $\\operatorname{Li}_0(z) = \\frac{z}{1 - z}$ and $\\operatorname{Li}_{-1}(z) = \\frac{z}{(1 - z)^2}$.\nFor $\\operatorname{Re} s \u003e 1$ the series also converges at $z = 1$, and $\\operatorname{Li}_s(1) = \\zeta(s)$, the [[Riemann zeta function|1XWBY4N0VT]].\n\nNon-example of a valid argument: for $s = 1$ the series diverges at $z = 1$ (it is the harmonic series), and $-\\ln(1 - z)$ has a singularity there.\n\nTermwise differentiation gives $z \\frac{d}{dz} \\operatorname{Li}_s(z) = \\operatorname{Li}_{s-1}(z)$, so $\\operatorname{Li}_{s+1}(z) = \\int_0^z \\frac{\\operatorname{Li}_s(t)}{t}\\, dt$.\nFor $s = 2$ this is the [[dilogarithm|G0X0NWSDB7]].\n",
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