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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