Re: Bug in Integrate in Version 5.1?

*To*: mathgroup at smc.vnet.net*Subject*: [mg56797] Re: Bug in Integrate in Version 5.1?*From*: Peter Pein <petsie at dordos.net>*Date*: Fri, 6 May 2005 03:00:12 -0400 (EDT)*References*: <d5crvb$lrs$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

A.Reischl at gmail.com wrote: > Hello, > > Integrate gives the following answer for this integral: > > a = Integrate[x^3 /(Exp[x] - 1), {x, 0, Infinity}] > N[a] > > Out[1]= Pi^4/15 > Out[2]= 6.49394 > > which I think is correct. > This integral, which should be the same ( by partial integration), > gives: > b = Integrate[-3 x^2 Log[1 - Exp[-x]], {x, 0, Infinity}] > N[b] > > Out[3]= (11*Pi^4)/60 > Out[4]= 17.8583 > > while numerical integration gives: > NIntegrate[-3x^2 Log[1 - Exp[-x]], {x, 0, Infinity}] > Out[5]= 6.49394 > > This is done with version 5.1. > > Version 4.2 gives > c=Integrate[-3*x^2*Log[1 - Exp[-x]], {x, 0, Infinity}] > N[c] > > Out[1]= Pi^4/15 > Out[2]= 6.49394 > > (Remarkably version 4.2. complaints: "Series::esss: Essential > singularity > encountered in ..." while calculating the correct result. ) > > So the result in version 5.1. looks wrong. > Or did I make a mistake? > > Cheers > Alexander > And to make things more confusing (in 5.1): ival = Integrate[-3*x^2*Log[1 - Exp[-x]], {x, x0, z}]; Simplify[Subtract @@ (Limit[ival, z -> #1] & ) /@ {Infinity, 0}] Pi^4/15 Mathematica fails to use it's capability to calculate the correct answer? And all of these give Pi^4/15 too: the "b" from above in other form: Integrate[-3*x^2*(Log[Exp[x] - 1] - x), {x, 0, Infinity}] after integrating by parts again: Integrate[6*x*PolyLog[2, Exp[-x]], {x, 0, Infinity}] and int. by parts again: -6*Integrate[PolyLog[2, Exp[-x]], {x, 0, Infinity}, x] -- Peter Pein Berlin