Cauchy Residue Theorem Integral

**mathshelpee** · August 9th 2011, 01:16 PM

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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**Drexel28** · August 9th 2011, 05:06 PM

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?

**chisigma** · August 10th 2011, 04:45 AM

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...

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

$\chi$ $\sigma$

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