Formula:DLMF:25.12:E13

<< Formula:DLMF:25.12:E12

formula in Polylogarithms

Formula:DLMF:25.12:E14 >>

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

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}}$, ${\displaystyle{\displaystyle{\displaystyle\Im{a}>0}}}$ or ${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$, ${\displaystyle{\displaystyle{\displaystyle\Im{a}=0}}}$

Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Symbols List

${\displaystyle{\displaystyle{\displaystyle\mathrm{Li}_{s}}}}$ : polylogarithm : http://dlmf.nist.gov/25.12#E10
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}$ : imaginary unit : http://dlmf.nist.gov/1.9.i
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2
${\displaystyle{\displaystyle{\displaystyle\Im{z}}}}$ : imaginary part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (13), Section 25.12 of DLMF.

URL links

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<< Formula:DLMF:25.12:E12

formula in Polylogarithms

Formula:DLMF:25.12:E14 >>