Polylogarithms

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Polylogarithms

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Dilogarithms

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

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

${\displaystyle{\displaystyle{\displaystyle\Dilogarithm@{z}=-\int_{0}^{z}t^{-1}\ln\left(1-t\right)\mathrm{d}t}}}$

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

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

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

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

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

${\displaystyle{\displaystyle{\displaystyle\Dilogarithm@{z^{m}}=m\sum_{k=0}^{m-1}\Dilogarithm@{z{\mathrm{e}^{2\pi\mathrm{i}k/m}}}}}}$

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

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

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

${\displaystyle{\displaystyle{\displaystyle\Dilogarithm@{{\mathrm{e}^{\mathrm{i}\theta}}}=\sum_{n=1}^{\infty}\frac{\cos\left(n\theta\right)}{n^{2}}+\mathrm{i}\sum_{n=1}^{\infty}\frac{\sin\left(n\theta\right)}{n^{2}}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\sum_{n=1}^{\infty}\frac{\cos\left(n\theta\right)}{n^{2}}=\frac{{\pi^{2}}}{6}-\frac{\pi\theta}{2}+\frac{\theta^{2}}{4}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\sum_{n=1}^{\infty}\frac{\sin\left(n\theta\right)}{n^{2}}=-\int_{0}^{\theta}\ln\left(2\sin\left(\tfrac{1}{2}x\right)\right)\mathrm{d}x}}}$

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

Polylogarithms

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

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

Integral Representation

${\displaystyle{\displaystyle{\displaystyle\Polylogarithm{s}@{z}=\frac{z}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{{\mathrm{e}^{x}}-z}\mathrm{d}x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}}$ and ${\displaystyle{\displaystyle{\displaystyle\left|\ph@{1-z}\right|<\pi}}}$, or ${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$ and ${\displaystyle{\displaystyle{\displaystyle z=1}}}$

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

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

${\displaystyle{\displaystyle{\displaystyle\Polylogarithm{s}@{{\mathrm{e}^{2\pi\mathrm{i}a}}}+{\mathrm{e}^{\pi\mathrm{i}s}}\Polylogarithm{s}@{{\mathrm{e}^{-2\pi\mathrm{i}a}}}=\frac{(2\pi)^{s}{\mathrm{e}^{\pi\mathrm{i}s/2}}}{\Gamma\left(s\right)}\HurwitzZeta@{1-s}{a}}}}$

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

Fermi-Dirac and Bose-Einstein Integrals

${\displaystyle{\displaystyle{\displaystyle F_{s}(x)=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{t^{s}}{{\mathrm{e}^{t-x}}+1}\mathrm{d}t}}}$

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

This formula has the name: Fermi-Dirac integral

${\displaystyle{\displaystyle{\displaystyle G_{s}(x)=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{t^{s}}{{\mathrm{e}^{t-x}}-1}\mathrm{d}t}}}$

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

This formula has the name: Bose-Einstein integral

${\displaystyle{\displaystyle{\displaystyle F_{s}(x)=-\Polylogarithm{s+1}@{-{\mathrm{e}^{x}}}}}}$
${\displaystyle{\displaystyle{\displaystyle G_{s}(x)=\Polylogarithm{s+1}@{{\mathrm{e}^{x}}}}}}$

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Polylogarithms

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