Polylogarithm

Function

Context: s
 definiendum $\mathrm{Li}_s: {\mathbb C} \to {\mathbb C}$ definiendum $\mathrm{Li}_s(z) := \begin{cases} \sum_{n=0}^\infty\, n^{-s} z^n&\hspace{.5cm} \mathrm{if}\hspace{.5cm} |z|<1,\hspace{.5cm} \\\\ \text{analytic continuation}\hspace{.5cm} &\hspace{.5cm} \mathrm{else} \end{cases}$
todo “$\text{analytic continuation}$”

Theorems

Representations
$\mathrm{Li}_s(z) = z\dfrac{\int_0^\infty\frac{x^{s}}{e^x-z}\frac{{\mathrm d}x}{x}}{\int_0^\infty \frac{x^{s}}{e^x-0}\frac{{\mathrm d}x}{x}}=\frac{1}{\Gamma(s)}\int_0^\infty\frac{x^{s}}{e^x\,z^{-1}-1}\frac{{\mathrm d}x}{x}$

This relates to the Bose-Einstein distribution where $z$ is the Fugacity.

Relation to other functions

$\zeta(s)=\lim_{z\to{1}}\mathrm{Li}_s(z)$