Formula:DLMF:25.11:E8

<< Formula:DLMF:25.11:E7

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E9 >>

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{\tfrac{1}{2}a}=\HurwitzZeta@{s}{\tfrac{1}{2}a+\tfrac{1}{2}}+2^{s}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}}}}$

Constraint(s)

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

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(-1)}}}$ : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (8), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E7

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E9 >>