Formula:DLMF:25.5:E2: Difference between revisions

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<br /><div align="center"><math>{\displaystyle
<br /><div align="center">{{#invoke:Math|render|P11}}</div>
  \RiemannZeta@{s}
  = \frac{1}{\EulerGamma@{s+1}}
    \int_0^\infty \frac{\expe^x x^s}{(\expe^x-1)^2} \diff{x}
}</math></div>


== Constraint(s) ==
== Constraint(s) ==


<div align="left"><math>{\displaystyle \realpart{s} > 1}</math></div><br />
<div align="left">{{#invoke:Math|render|P9}}</div><br />


== Proof ==
== Proof ==
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<div id="alignright"> [[Formula:DLMF:25.5:E3|Formula:DLMF:25.5:E3]] >> </div>
<div id="alignright"> [[Formula:DLMF:25.5:E3|Formula:DLMF:25.5:E3]] >> </div>
</div>
</div>
== Testsection ==
{{#invoke:Math|render|P9}}

Revision as of 20:01, 13 March 2017



Constraint(s)


Proof

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

Integrate

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} }}

by parts.


Symbols List

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 \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 \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 \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 (2), Section 25.5 of DLMF.

URL links

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