Formula:DLMF:25.13:E3

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formula in Periodic Zeta Function

Formula:DLMF:25.14:E1 >>

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

Constraint(s)

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

Proof

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Symbols List

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

Bibliography

Equation (3), Section 25.13 of DLMF.

URL links

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