Formula:DLMF:25.11:E7

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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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 \binom{s+2k-2}{2k-1} \frac{\BernoulliB{2k}}{2k} \frac{1}{(1+a)^{s+2k-1}} - \binom{s+2n}{2n+1} \int_1^\infty \frac{\PeriodicBernoulliB{2n+1}@{x}}{(x+a)^{s+2n+1}} \diff{x} }}

Constraint(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle s \neq 1}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle a > 0}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle n = 1,2,3,\dots}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \realpart{s} > -2n}}


Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

Take Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle N = 1}} in

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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-\floor{x}}{(x+a)^{s+1}} \diff{x} }}

and integrate by parts.


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
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \binom{n}{k}}}  : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
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
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \widetilde{B}_{n}}}  : periodic Bernoulli functions : http://dlmf.nist.gov/24.2#iii
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 (7), Section 25.11 of DLMF.

URL links

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