# Cauchy Residue Theorem Integral

• Aug 9th 2011, 02:16 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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• Aug 9th 2011, 06:06 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?
• Aug 10th 2011, 05:45 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

$\chi$ $\sigma$