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I am working out some computations in Yang-Mills theory and, in order to find the potential, I have to compute the following integral $$ V(r)=\int_0^\infty dp\frac{p(a+bp^2)}{cp^4+dp^2+e}\sin(pr). $$ I have thought to some techinques using residues with Cauchy theorem and Jordan's Lemma. I would appreciate any insight about.


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Fixed it, thanks. – Jon Sep 25 '12 at 7:49
up vote 3 down vote accepted

Since the integrand is even, you can extend the integral to $(-\infty,\infty)$ and divide by $2$. Then you can split the sine up into two complex exponentials. For each of them, you can close the contour with a semicircle at infinity in one of the half-planes, and then apply the residue theorem. The denominator is biquadratic in $p$, so you can solve for $p^2$ and take both square roots of both solutions to find the four poles.

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Thanks a lot. This seems the right track. – Jon Sep 25 '12 at 7:50

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