Formula:DLMF:25.11:E9

<< Formula:DLMF:25.11:E8

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E10 >>

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{1-s}{a}=\frac{2\Gamma\left(s\right)}{(2\pi)^{s}}\*\sum_{n=1}^{\infty}\frac{1}{n^{s}}\cos\left(\tfrac{1}{2}\pi s-2n\pi a\right)}}}$

Constraint(s)

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

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
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : 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 (9), Section 25.11 of DLMF.

URL links

We ask users to provide relevant URL links in this space.

<< Formula:DLMF:25.11:E8

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E10 >>