Re: Tough Integral

*To*: mathgroup at smc.vnet.net*Subject*: [mg20285] Re: Tough Integral*From*: "Bill Bertram" <wkb at ansto.gov.au>*Date*: Mon, 11 Oct 1999 02:19:56 -0400*Organization*: Australian Nuclear Science and Technology Organisation*References*: <7tp7fj$95g@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Lawrence Walker wrote in message <7tp7fj$95g at smc.vnet.net>... >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. > Yes, I have one. Rewrite the integral as Integrate[ E^(-x) x^3/(1-E^(-x)),{x,0,Infinity} Now expand the 1/(1-E^(-x)) factor as a power series, Sum[ E^(-k x),{k,0,Infinity}} and integrate the result term by term. You will end up with a sum of reciprical powers,( ie. a Zeta function) which gives you the desired numerical result. Cheers, Bill