Formula:DLMF:25.11:E16

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formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E17 >>

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{1-s}{\frac{h}{k}}=\frac{2\Gamma\left(s\right)}{(2\pi k)^{s}}\*\sum_{r=1}^{k}\cos\left(\frac{\pi s}{2}-\frac{2\pi rh}{k}\right)\HurwitzZeta@{s}{\frac{r}{k}}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle s\neq 0,1}}}$ &
${\displaystyle{\displaystyle{\displaystyle h,k}}}$ integers, ${\displaystyle{\displaystyle{\displaystyle 1\leq h\leq k}}}$ &
${\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}$

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
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\mathrm{cos}}}}$ : cosine function : http://dlmf.nist.gov/4.14#E2
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (16), Section 25.11 of DLMF.

URL links

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formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E17 >>