# Polylogarithms

## Dilogarithms

$\displaystyle {\displaystyle \Dilogarithm@{z} = \sum_{n=1}^\infty \frac{z^n}{n^2} }$

Constraint(s): $\displaystyle {\displaystyle |z| \leq 1}$

$\displaystyle {\displaystyle \Dilogarithm@{z} = -\int_0^z t^{-1} \ln@{1-t} \diff{t} }$

Constraint(s): $\displaystyle {\displaystyle z \in \Complex \setminus (1,\infty)}$

$\displaystyle {\displaystyle \Dilogarithm@{z} + \Dilogarithm@{\frac{z}{z-1}} = - \frac{1}{2} (\ln@{1-z})^2 }$

Constraint(s): $\displaystyle {\displaystyle z \in \Complex \setminus [1,\infty)}$

$\displaystyle {\displaystyle \Dilogarithm@{z} + \Dilogarithm@{\frac{1}{z}} = - \frac{1}{6} \cpi^2 -\frac{1}{2} (\ln@{-z})^2 }$

Constraint(s): $\displaystyle {\displaystyle z \in \Complex \setminus [0,\infty)}$

$\displaystyle {\displaystyle \Dilogarithm@{z^m} = m \sum_{k=0}^{m-1} \Dilogarithm@{z \expe^{2 \cpi \iunit k/m}} }$

Constraint(s): $\displaystyle {\displaystyle m = 1,2,3,\dots}$ &
$\displaystyle {\displaystyle \{z\in\Complex:|z|<1\}\setminus[0,1)}$

$\displaystyle {\displaystyle \Dilogarithm@{x} + \Dilogarithm@{1-x} = \frac{1}{6} \cpi^2 - (\ln x) \ln@{1-x} }$

Constraint(s): $\displaystyle {\displaystyle 0 < x < 1}$

$\displaystyle {\displaystyle \Dilogarithm@{\expe^{\iunit\theta}} = \sum_{n=1}^\infty \frac{\cos@{n \theta}}{n^2} + \iunit \sum_{n=1}^\infty \frac{\sin@{n \theta}}{n^2} }$

Constraint(s): $\displaystyle {\displaystyle 0 \leq \theta \leq 2\cpi}$

$\displaystyle {\displaystyle \sum_{n \hiderel{=} 1}^\infty \frac{\cos@{n \theta}}{n^2} = \frac{\cpi^2}{6} - \frac{\cpi \theta}{2} + \frac{\theta^2}{4} }$

Constraint(s): $\displaystyle {\displaystyle 0 \leq \theta \leq 2\cpi}$

$\displaystyle {\displaystyle \sum_{n \hiderel{=} 1}^\infty \frac{\sin@{n \theta}}{n^2} = - \int_0^\theta \ln@{2 \sin@{\tfrac{1}{2} x}} \diff{x} }$

Constraint(s): $\displaystyle {\displaystyle 0 \leq \theta \leq 2\cpi}$

## Polylogarithms

$\displaystyle {\displaystyle \Polylogarithm{s}@{z} = \sum_{n=1}^\infty \frac{z^n}{n^s} }$

Constraint(s): real or complex $\displaystyle {\displaystyle s}$ and $\displaystyle {\displaystyle z}$

### Integral Representation

$\displaystyle {\displaystyle \Polylogarithm{s}@{z} = \frac{z}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1}}{\expe^x - z} \diff{x} }$

Constraint(s): $\displaystyle {\displaystyle \realpart{s} > 0}$ and $\displaystyle {\displaystyle \abs{\ph@{1-z}} < \cpi}$ , or $\displaystyle {\displaystyle \realpart{s} > 1}$ and $\displaystyle {\displaystyle z = 1}$

$\displaystyle {\displaystyle \Polylogarithm{s}@{z} = \EulerGamma@{1-s} \left( \ln@@{\frac{1}{z}} \right)^{s-1} + \sum_{n=0}^\infty \RiemannZeta@{s-n} \frac{(\ln@@{z})^n}{n!} }$

Constraint(s): $\displaystyle {\displaystyle s \neq 1,2,3,\dots}$ &
$\displaystyle {\displaystyle |\ln@@{z}| < 2\cpi}$

$\displaystyle {\displaystyle \Polylogarithm{s}@{\expe^{2 \cpi \iunit a}} + \expe^{\cpi \iunit s} \Polylogarithm{s}@{\expe^{-2 \cpi \iunit a}} = \frac{(2 \cpi)^s \expe^{\cpi \iunit s/2}}{\EulerGamma@{s}} \HurwitzZeta@{1-s}{a} }$

Constraint(s): $\displaystyle {\displaystyle \realpart{s} > 0}$ , $\displaystyle {\displaystyle \imagpart{a} > 0}$ or $\displaystyle {\displaystyle \realpart{s} > 1}$ , $\displaystyle {\displaystyle \imagpart{a} = 0}$

## Fermi-Dirac and Bose-Einstein Integrals

$\displaystyle {\displaystyle F_s(x) = \frac{1}{\EulerGamma@{s+1}} \int_0^\infty \frac{t^s}{\expe^{t-x}+1} \diff{t} }$

Constraint(s): $\displaystyle {\displaystyle s > -1}$

This formula has the name: Fermi-Dirac integral

$\displaystyle {\displaystyle G_s(x) = \frac{1}{\EulerGamma@{s+1}} \int_0^\infty \frac{t^s}{\expe^{t-x}-1} \diff{t} }$

Constraint(s): $\displaystyle {\displaystyle s > -1}$ , $\displaystyle {\displaystyle x < 0}$ , or $\displaystyle {\displaystyle s > 0}$ , $\displaystyle {\displaystyle x \leq 0}$

This formula has the name: Bose-Einstein integral

$\displaystyle {\displaystyle F_s(x) = -\Polylogarithm{s+1}@{-\expe^x} }$
$\displaystyle {\displaystyle G_s(x) = \Polylogarithm{s+1}@{\expe^x} }$