Formula:DLMF:25.5:E2: Difference between revisions

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<br /><div align="center"><math>{\displaystyle
<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>


== Proof ==
== Proof ==


We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.
<div align="left">Integrate
<br /><br />
<div align="left">Integrate <br />
<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><br />
}</math>
by parts.</div>
by parts.</div>
<br />


== Symbols List ==
== Symbols List ==
Line 53: 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__
 
== Testsection ==
{{#property:P11}}{{#property:P9}}

Latest revision as of 08:33, 22 December 2019


Constraint(s)

Proof

Integrate

\RiemannZeta ⁢ @ ⁢ s = 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ x s - 1 e x - 1 ⁢ d x \RiemannZeta @ 𝑠 1 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 𝑥 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{\Gamma% \left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{{\mathrm{e}^{x}}-1}\mathrm{d}x}}} {\displaystyle \RiemannZeta@{s} = \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1}}{\expe^x-1} \diff{x} }

by parts.

Symbols List

ζ 𝜁 {\displaystyle{\displaystyle{\displaystyle\zeta}}}  : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
Γ Γ {\displaystyle{\displaystyle{\displaystyle\Gamma}}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
∫ {\displaystyle{\displaystyle{\displaystyle\int}}}  : integral : http://dlmf.nist.gov/1.4#iv
e e {\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
d n ⁢ x superscript d 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\mathrm{d}^{n}x}}}  : differential : http://dlmf.nist.gov/1.4#iv
ℜ ⁡ z 𝑧 {\displaystyle{\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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