===== Polylogarithm =====
==== Function ====
>Context: s

| @#FFBB00: definiendum | @#FFBB00: $ \mathrm{Li}_s: {\mathbb C} \to {\mathbb C}$ |
| @#FFBB00: definiendum | @#FFBB00: $ \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 ==
$\zeta(s)=\lim_{z\to{1}}\mathrm{Li}_s(z)$

=== Reference ===
Wikipedia: 
[[http://en.wikipedia.org/wiki/Polylogarithm|Polylogarithm]],
[[http://en.wikipedia.org/wiki/Complete_Fermi%E2%80%93Dirac_integral|Complete Fermi–Dirac integral]]

MathWorld: 
[[http://mathworld.wolfram.com/Polylogarithm.html|Polylogarithm]]

-----
=== Subset of ===
[[Holomorphic function]]