Cauchy Residue Theorem Integral

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August 9th 2011, 01:16 PM
mathshelpee

Cauchy Residue Theorem Integral

Hey Guys,

Could you please clarify this problem to me?
When I separate the denominator, I get two values that I can use in the Cauchy Residue theorem.
Thus, should I only take one value (depending on the contour I use obviously), if so do I take the negative one or the positive one?

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August 9th 2011, 05:06 PM
Drexel28

Re: Cauchy Residue Theorem Integral

Quote:

Originally Posted by mathshelpee

Hey Guys,

Could you please clarify this problem to me?
When I separate the denominator, I get two values that I can use in the Cauchy Residue theorem.
Thus, should I only take one value (depending on the contour I use obviously), if so do I take the negative one or the positive one?

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What precisely do you mean? What contour would you use?
August 10th 2011, 04:45 AM
chisigma

Re: Cauchy Residue Theorem Integral

Quote:

Originally Posted by mathshelpee

Hey Guys,

Could you please clarify this problem to me?
When I separate the denominator, I get two values that I can use in the Cauchy Residue theorem.
Thus, should I only take one value (depending on the contour I use obviously), if so do I take the negative one or the positive one?

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The integral is...

$\int_{- \infty}^{+ \infty} \frac {e^{i x}}{(x-\pi)^{2}+a^{2}}$ (1)

The solution with the Cauchy residue theorem requires the choice os an integration path in the complex plane. In this case it is adequate the path of figure...

http://digilander.libero.it/luposabatini/MHF1100.bmp

The poles of the complex function in (1) are at $z=\pi \pm i\ a$ and inside the path is those with positive imaginary part...

Kind regards

$\chi$ $\sigma$