Difference between revisions of "Formula:DLMF:25.2:E9"

From DRMF
Jump to navigation Jump to search
 
(5 intermediate revisions by the same user not shown)
Line 6: Line 6:
 
</div>
 
</div>
  
<br /><div align="center"><math>
+
<br /><div align="center">{{#invoke:Wd|property|P1|}}</div>
\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>
 
  
 
== Constraint(s) ==
 
== Constraint(s) ==
 
+
{{#invoke:Wd|properties|P9|format=%p<br/>}}
<div align="left"><math>{\displaystyle \realpart{s} > -2n} </math>
 
<br /> <math>{\displaystyle n,N = 1,2,3,\dots} </math>
 
<br />
 
  
 
== 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>
  
 
== Symbols List ==
 
== Symbols List ==
 +
<ul>{{#invoke:Wd|properties|linked|raw|qualifier|P4|P1|format=<li>%q: %p</li>}}</ul>
 +
  
 
& : logical and<br />
 
& : logical and<br />

Latest revision as of 16:09, 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.

URL links

We ask users to provide relevant URL links in this space.