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: http://math.stackexchange.com/questions/34767/int-infty-infty-e-x2-dx-with-complex-analysis/34772#34772

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

entirefunction solved byintroducingone or more poles, as in the example I mentioned. – B.W. Sep 10 '13 at 3:07