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MathGroup Archive 1999

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Tough Integral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg20284] Tough Integral
  • From: Ranko Bojanic <bojanic at math.ohio-state.edu>
  • Date: Mon, 11 Oct 1999 02:19:56 -0400
  • Organization: Ohio State University
  • Sender: owner-wri-mathgroup at wolfram.com

In [mg20262] Tough Integral, Lawrence Walker wrote:

> Mathematica is able to calculate the following integral.
> In[1]:=
> Integrate[x^3/(E^x-1),{x,0,Infinity}]
> Out[1]=
> Pi^4/15
> For the life of me, I cannot solve this by hand.  Does any
> know or has any ideas.

It is well known that for Re[s]>1
Integral of x^(s-1) / (Exp[x]-1)dx from 0 to Infinity  = Gamma[s]
Zeta[s]
To do this by hand, write
        1/(Exp[x]-1) = Exp[-x][1 + Exp[-x] + Exp[-2x]+.....]
and use integration by parts. Mathematica gives the same result.

Ranko



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