Formula:DLMF:25.13:E1

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formula in Periodic Zeta Function

Formula:DLMF:25.13:E2 >>

${\displaystyle{\displaystyle{\displaystyle\PeriodicZeta@{x}{s}=\sum_{n=1}^{\infty}\frac{{\mathrm{e}^{2\pi\mathrm{i}nx}}}{n^{s}}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle x}}}$ real &
${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$ if ${\displaystyle{\displaystyle{\displaystyle x}}}$ is an integer, ${\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}}$ otherwise

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle F}}}$ : periodic zeta function : http://dlmf.nist.gov/25.13#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (1), Section 25.13 of DLMF.

URL links

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<< Polylogarithms

formula in Periodic Zeta Function

Formula:DLMF:25.13:E2 >>