Formula:DLMF:25.11:E28

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formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E29 >>

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\sum_{k=1}^{n}\frac{\Gamma\left(s+2k-1\right)}{\Gamma\left(s\right)}\frac{B_{2k}}{(2k)!}a^{-2k-s+1}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{\expe^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}x^{2k-1}\right)x^{s-1}\expe^{-ax}\diffd x}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle\realpart{s}>-(2n+1)}}}$ &
${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$

Proof

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Argue as in

${\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{2}+\frac{1}{s-1}+\sum_{m=1}^{n}\frac{\BernoulliB{2m}}{(2m)!}\frac{\Gamma\left(s+2m-1\right)}{\Gamma\left(s\right)}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{{\mathrm{e}^{x}}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{m=1}^{n}\frac{\BernoulliB{2m}}{(2m)!}x^{2m-1}\right)\frac{x^{s-1}}{{\mathrm{e}^{x}}}\mathrm{d}x}}}$.

Symbols List

& : logical and
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \zeta}}  : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Sigma}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle\Gamma}}}$ : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle B_{n}}}  : Bernoulli polynomial : http://dlmf.nist.gov/24.2#i
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int}}  : integral : http://dlmf.nist.gov/1.4#iv
${\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}$ : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \mathrm{d}^nx}}  : differential : http://dlmf.nist.gov/1.4#iv
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Re {z}}}  : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (28), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E27

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E29 >>