Integration with residue calculus example with order 2, Complex Analysis
Let's compute the improper integral of x^2/(x^2+25)^2 from negative infinity to infinity using residue calculus (not required, but probably faster). I start by extending our real variable function to a function of a complex variable: f(z) = z^2/(z^2+25)^2. We find poles at +/- z=5i, of second order. I then introduce an upper semicircular contour in the complex plane, which will help us evaluate the integral over the real axis (the contribution from the semicircle arc vanishes). We compute the residue at the enclosed singularity z = 5i so that we can apply the residue theorem.
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