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Re: Can Mathematica NIntegrate a Log-type singularity?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg92666] Re: [mg92642] Can Mathematica NIntegrate a Log-type singularity?
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Fri, 10 Oct 2008 04:32:50 -0400 (EDT)
  • Reply-to: hanlonr at cox.net

$Version

6.0 for Mac OS X PowerPC (32-bit) (June 19, 2007)

nint = NIntegrate[alpha^2 * Log[2 Cos[alpha/2]]^2, {alpha, -Pi, Pi}]

37.4024

nint/Pi^5

0.122222

% // Rationalize

11/90

The result is stable for changes in WorkingPrecision. The integral appears to be 11/90 * Pi^5


Bob Hanlon

---- Aaron Fude <aaronfude at gmail.com> 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.


--

Bob Hanlon



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