Formula:DLMF:25.11:E7

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formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E8 >>

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\frac{1}{a^{s}}+\frac{1}{(1+a)^{s}}\left(\frac{1}{2}+\frac{1+a}{s-1}\right)+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}\frac{\BernoulliB{2k}}{2k}\frac{1}{(1+a)^{s+2k-1}}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{1}^{\infty}\frac{\PeriodicBernoulliB{2n+1}@{x}}{(x+a)^{s+2n+1}}\mathrm{d}x}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle a>0}}}$ &
${\displaystyle{\displaystyle{\displaystyle n=1,2,3,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle\Re{s}>-2n}}}$

Proof

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Take ${\displaystyle{\displaystyle{\displaystyle N=1}}}$ in

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\sum_{n=0}^{N}\frac{1}{(n+a)^{s}}+\frac{(N+a)^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right\rfloor}{(x+a)^{s+1}}\mathrm{d}x}}}$

and integrate by parts.

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
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle B_{n}}}}$ : Bernoulli polynomial : http://dlmf.nist.gov/24.2#i
${\displaystyle{\displaystyle{\displaystyle\int}}}$ : 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 (7), Section 25.11 of DLMF.

URL links

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