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Re: Can Mathematica NIntegrate a Log-type singularity?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg92662] Re: [mg92642] Can Mathematica NIntegrate a Log-type singularity?
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Fri, 10 Oct 2008 04:32:07 -0400 (EDT)
  • References: <200810091036.GAA24061@smc.vnet.net>

On 9 Oct 2008, at 19:36, Aaron Fude wrote:

> Hi,
>
> I would like to evaluate the following:
>
> NIntegrate[alpha^2 Log[2 Cos[alpha/2]]^2, { alpha, -Pi, Pi}]
>
> How do I help Mathematica deal with the LogSquared type singularity at
> either end of the interval. If I try it straight, Mathematica
> complains and gives a wrong answer.
>
> Please note, that Mathematica has absolutely no problem Integrating or
> NIntegrating the function
>
> Log[Cos[alpha/2]]^2
>
> from -Pi to Pi, each time giving the correct answer, but the multiple
> of alpha^2, throws it off.
>
> Many thanks in advance,
>
> Aaron.
>
> PS: By the way, I'm pretty sure that that integral must be some
> rational number times Pi^5.
>

Which version of Mathematica? Here with version 6.03 I get:

a = Chop[NIntegrate[alpha^2*Log[2*Cos[alpha/2]]^2, {alpha, -Pi, Pi},
        WorkingPrecision -> 30]]
  37.402405918201066509890604560864352814270789949531176\
  7786391`30.

and then:


RootApproximant[a/Pi^5]
11/90


Andrzej Kozlowski



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