Formula:DLMF:25.11:E6

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formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E7 >>

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\frac{1}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)-s(s+1)\int_{0}^{\infty}\frac{\PeriodicBernoulliB{2}@{x}}{(x+a)^{s+2}}\mathrm{d}x}}}$

Constraint(s)

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

Proof

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Symbols List

& : logical and
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\widetilde{B}_{n}}}}$ : periodic Bernoulli functions : http://dlmf.nist.gov/24.2#iii
${\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}$ : differential : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\Re{z}}}}$ : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (6), Section 25.11 of DLMF.

URL links

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formula in Hurwitz Zeta Function

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