Formula:DLMF:25.2:E9: Difference between revisions

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== Proof ==
== Proof ==
<div align="left">Follows from <br />
<div align="left">Follows from <br />
<math id="DLMF:25.2:E8">{\displaystyle \RiemannZeta@{s} = \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1}}{\expe^x-1} \diff{x} }</math><br />
<math id="DLMF:25.2:E8">{\displaystyle \Riemannzeta@{s} = \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1}}{\expe^x-1} \diff{x} }</math><br />
by repeated integration by parts.</div>
by repeated integration by parts.</div>



Revision as of 16:04, 2 January 2020


Constraint(s)



Proof

Follows from


by repeated integration by parts.

Symbols List

& : logical and
 : Riemann zeta function : http://dlmf.nist.gov/25.2#E1
 : sum : http://drmf.wmflabs.org/wiki/Definition:sum
 : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
 : Bernoulli polynomial : http://dlmf.nist.gov/24.2#i
 : integral : http://dlmf.nist.gov/1.4#iv
 : periodic Bernoulli functions : http://dlmf.nist.gov/24.2#iii
 : differential : http://dlmf.nist.gov/1.4#iv
 : real part : http://dlmf.nist.gov/1.9#E2

Bibliography

Equation (9), Section 25.2 of DLMF.

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