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I was wondering about evaluating the following definite integral analytically: \begin{equation} \int_{-\infty}^{\infty}\frac{1}{\sqrt{k-p}\sqrt{k+p}}\,\mathrm dp \end{equation}

Does someone know how to approach this?

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Do you know the trick $(a - b)(a + b) = a^2 - b^2$? –  TMM Jun 6 '12 at 19:05
I would like to go around the singularity and possibly apply the Residue theorem to calculate it. Is that possible? –  Micheal Jun 6 '12 at 19:09
I'm not sure how you're going to go around the singularity when your integral seems to go right through it... –  Ben Millwood Jun 6 '12 at 19:14
And your integrand is non-real for some $p$ ... and anyway not absolutely convergent. So you need to provide some explanation for us to make any sense of it. –  GEdgar Jun 6 '12 at 19:22
1.Substitute p=z to work in the complex plane 2. Move one pole up to $k+i\gamma$ and after the integration add a limit of $\gamma$ going to zero and to the same thing for the other singularity. 3. Which results in two contour integrals in the complex plane one around the singularity in the upper plane plus an over the singularity in lower plane. –  Micheal Jun 6 '12 at 19:22
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closed as too localized by Lord_Farin, Martin, Amzoti, O.L., Pedro Tamaroff Jun 22 '13 at 23:18

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1 Answer

Wolfram Alpha checked that this improper integral does not converge.

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