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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
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