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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: [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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