Formula:DLMF:25.12:E11

<< Formula:DLMF:25.12:E10

formula in Polylogarithms

Formula:DLMF:25.12:E12 >>

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

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 ${\displaystyle{\displaystyle{\displaystyle z=1}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\mathrm{Li}_{s}}}}$ : polylogarithm : http://dlmf.nist.gov/25.12#E10
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2
${\displaystyle{\displaystyle{\displaystyle\mathrm{ph}}}}$ : phase : http://dlmf.nist.gov/1.9#E7
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4

Bibliography

Equation (11), Section 25.12 of DLMF.

URL links

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<< Formula:DLMF:25.12:E10

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