Formula:DLMF:25.2:E9: Difference between revisions
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<br /><div align="center"><math>{\displaystyle | <br /><div align="center"><math>{\displaystyle | ||
\RiemannZeta@{s} | \RiemannZeta@{s} = \sum_{k=1}^N \frac{1}{k^s} + \frac{N^{1-s}}{s-1} - \frac{1}{2}N^{-s} + \sum_{k=1}^n \binom{s+2k-2}{2k-1} \frac{\BernoulliB{2k}}{2k} N^{1-s-2k} - \binom{s+2n}{2n+1} \int_N^\infty \frac{\PeriodicBernoulliB{2n+1}@{x}}{x^{s+2n+1}} \diff{x}</math></div> | ||
= \frac{1}{\ | |||
\frac{\ | |||
\diff{x | |||
== Constraint(s) == | == Constraint(s) == |
Revision as of 22:18, 9 May 2017
Constraint(s)
Proof
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \RiemannZeta@{s} = \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1}}{\expe^x-1} \diff{x} }}
Symbols List
& : logical and
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \zeta}}
: Riemann zeta function : http://dlmf.nist.gov/25.2#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 (9), Section 25.2 of DLMF.
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