Formula:DLMF:25.12:E12

<< Formula:DLMF:25.12:E11

formula in Polylogarithms

Formula:DLMF:25.12:E13 >>

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

Constraint(s)

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

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\mathrm{Li}_{s}}}}$ : polylogarithm : http://dlmf.nist.gov/25.12#E10
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4

Bibliography

Equation (12), Section 25.12 of DLMF.

URL links

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<< Formula:DLMF:25.12:E11

formula in Polylogarithms

Formula:DLMF:25.12:E13 >>