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Suppose we have a contour integral of an entire function along a line in the plane (or perhaps a line segment). I am looking for examples of such integrals that are computed by first altering the integrand so that poles are introduced.

An example that I have found is the proof of $\int_{-\infty}^\infty e^{-x^2}dx=\sqrt{\pi}$ from Remmert's book on complex analysis. A discussion of that can be found here:

Can anyone direct me to other examples that might be similar to this?

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A simple Google gives you, for example, this: link But any decent book in Complex Analysis, like Stein, Gamelin, Conway should have plenty of examples. – Alin Galatan Sep 10 '13 at 2:51
I agree that a "simple" Google search gives plenty of examples of residue integrals, but I am looking for something a bit more specific: an integral of an entire function solved by introducing one or more poles, as in the example I mentioned. – B.W. Sep 10 '13 at 3:07

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