Formula:DLMF:25.5:E2: Difference between revisions
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<div align="center">{{#invoke:Math|render|P11}}</div> | |||
== Constraint(s) == | == Constraint(s) == | ||
<div align="left">{{#invoke:Math|render|P9}}</div | <div align="left">{{#invoke:Math|render|P9}}</div> | ||
== Proof == | == Proof == | ||
<div align="left">Integrate | |||
<div align="left">Integrate | |||
<math id="DLMF:25.5:E1">{\displaystyle | <math id="DLMF:25.5:E1">{\displaystyle | ||
\RiemannZeta@{s} | \RiemannZeta@{s} | ||
= \frac{1}{\EulerGamma@{s}} | = \frac{1}{\EulerGamma@{s}} | ||
\int_0^\infty \frac{x^{s-1}}{\expe^x-1} \diff{x} | \int_0^\infty \frac{x^{s-1}}{\expe^x-1} \diff{x} | ||
}</math | }</math> | ||
by parts.</div> | by parts.</div> | ||
== Symbols List == | == Symbols List == | ||
| Line 49: | Line 46: | ||
<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> | ||
__NOTOC__ | |||
Latest revision as of 08:33, 22 December 2019
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} }}
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.
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