Polylogarithm
From specialfunctionswiki
The polylogarithm $\mathrm{Li}_s$ is defined by the formula $$\mathrm{Li}_s(z) = \sum_{k=1}^{\infty} \dfrac{z^k}{k^s} = z + \dfrac{z^2}{2^s} + \dfrac{z^3}{3^s} + \ldots$$
Various polylogarithms plotted on $[-2,1]$.
Contents
Properties
Theorem
The following formula holds: $$\Phi(z,n,1)=\dfrac{\mathrm{Li}_n(z)}{z},$$ where $\Phi$ denotes the Lerch transcendent and $\mathrm{Li_n}$ denotes the polylogarithm.
