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Re: Can Mathematica NIntegrate a Log-type singularity?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg92675] Re: Can Mathematica NIntegrate a Log-type singularity?
  • From: Peter Pein <petsie at dordos.net>
  • Date: Fri, 10 Oct 2008 04:34:29 -0400 (EDT)
  • References: <gckmrr$ng7$1@smc.vnet.net>

This must be another version of Mathematica than
"6.0 for Linux x86 (64-bit) (March 13, 2008)"

I get:
In[1]:= NIntegrate[alpha^2*Log[2*Cos[alpha/2]]^2, {alpha, -Pi, Pi}]
Out[1]= 37.40240591819764
In[2]:= NIntegrate[alpha^2*Log[2*Cos[alpha/2]]^2, {alpha, -Pi, Pi},
   WorkingPrecision -> 30] == Pi^5*(11/90)
Out[2]= True

Aaron Fude schrieb:
> 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.
> 


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