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MathGroup Archive 2007

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Re: Problem with Integrate

  • To: mathgroup at smc.vnet.net
  • Subject: [mg77711] Re: Problem with Integrate
  • From: chuck009 <dmilioto at comcast.com>
  • Date: Fri, 15 Jun 2007 04:38:29 -0400 (EDT)

Hey did, I'm not especially sharp in Complex Analysis but this is what I beleve:  I think nothing is wrong with Integrate:  When using antiderivatives, you need to split-up the contour over regions in which the function and it's anti-derivative are analytical.  When you Integrate:

Exp[2 i t]/(Exp[i t]+1/4) you get:

-iExp[i t]+1/4 i Log[1+4 Exp[it]]

As t goes around one complete circle, the contour is crossing the Log branch cut causing analyticity to be lost.  NIntegrate works because it's not using antiderivatives.  To use antiderivatives involving the Log function, need to split up the integral into regions  and choose branches of the Log function which are analytical over each of the regions separately.  I would not expect the Integrate function by itself to detect or handle branch cuts in this fashion.  Why it appears to give the correct answer when you supply a specific value of T I'm not sure but I do not feel you're using Integrate correctly for this particular problem.


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