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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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    you have issues with the complex square root, but both pieces of this integral, $\int_{-k}^k and \int _k^{\infty} $ are standard trig integrals, substitute $x=kcos(\theta)$ in first and $x = ktan(\theta)$ in 2nd, and if you can't get anything sensible out of it, it is because of the problems limned above.2012-06-06

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Wolfram Alpha checked that this improper integral does not converge.