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