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MathGroup Archive 2005

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Re: Bug in Integrate in Version 5.1?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg56814] Re: [mg56737] Bug in Integrate in Version 5.1?
  • From: yehuda ben-shimol <bsyehuda at gmail.com>
  • Date: Fri, 6 May 2005 03:01:28 -0400 (EDT)
  • References: <200505051001.GAA21888@smc.vnet.net>
  • Reply-to: yehuda ben-shimol <bsyehuda at gmail.com>
  • Sender: owner-wri-mathgroup at wolfram.com

There is more peculiar result there:
Try the indefinite integral 
a=Integrate[-3 (x^2 )Log[1 - Exp[-x]], x]
to get
-3*(x^4/12 + (1/3)*x^3*Log[1 - E^(-x)] + 
   (1/3)*((-x^3)*Log[1 - E^x] - 
     3*x^2*PolyLog[2, E^x] + 6*x*PolyLog[3, E^x] - 
     6*PolyLog[4, E^x]))
this expression does not have value in both x=0 and x-> infinity but
it does converge to a limit for both
So
Limit[a,x->Infinity]-Limit[a,x->0] does give the exact resut (i.e., Pi^4/15)
I cannot figure out why using the definite integral does not return
the true value.
yehuda

On 5/5/05, A.Reischl at gmail.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
> 
>


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