# Formula:DLMF:25.13:E2

$\displaystyle {\displaystyle \PeriodicZeta@{x}{s} = \frac{\EulerGamma@{1-s}}{(2 \cpi)^{1-s}} \* \left( \expe^{\cpi \iunit (1-s)/2} \HurwitzZeta@{1-s}{x} + \expe^{\cpi \iunit (s-1)/2} \HurwitzZeta@{1-s}{1-x} \right) }$

## Constraint(s)

$\displaystyle {\displaystyle 0 < x < 1}$ &
$\displaystyle {\displaystyle \realpart{s} > 1}$

## 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

& : logical and
$\displaystyle {\displaystyle F}$  : periodic zeta function : http://dlmf.nist.gov/25.13#E1
$\displaystyle {\displaystyle \Gamma}$  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
$\displaystyle {\displaystyle \pi}$  : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
$\displaystyle {\displaystyle \mathrm{e}}$  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
$\displaystyle {\displaystyle \mathrm{i}}$  : imaginary unit : http://dlmf.nist.gov/1.9.i
$\displaystyle {\displaystyle \zeta}$  : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
$\displaystyle {\displaystyle \Re {z}}$  : real part : http://dlmf.nist.gov/1.9#E2