Periodic Zeta Function

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Periodic Zeta Function

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Periodic Zeta Function

${\displaystyle{\displaystyle{\displaystyle F\left(x,s\right)=\sum_{n=1}^{\infty}\frac{\expe^{2\cpi\iunit nx}}{n^{s}}}}}$

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

${\displaystyle{\displaystyle{\displaystyle F\left(x,s\right)=\frac{\Gamma\left(1-s\right)}{(2\cpi)^{1-s}}\*\left(\expe^{\cpi\iunit(1-s)/2}\zeta\left(1-s,x\right)+\expe^{\cpi\iunit(s-1)/2}\zeta\left(1-s,1-x\right)\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle\zeta\left(1-s,x\right)=\frac{\Gamma\left(s\right)}{(2\cpi)^{s}}\left(\expe^{-\cpi\iunit s/2}F\left(x,s\right)+\expe^{\cpi\iunit s/2}F\left(-x,s\right)\right)}}}$

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

<< Polylogarithms

Periodic Zeta Function

Lerch's Transcendent >>