Evaluate the integral of x^a / (x^2 + 1)^2 dx from x = 0 to infinity
We can define the function z^a in the complex plane unambiguously, by defining polar coordinates z = r exp(i theta) and then defining z^a = r^a exp(i a theta). Now defining the polar angle theta will involve setting the angle back by 2 pi somewhere, otherwise a point in the real axis could be assigned a polar angle of zero and any multiple of 2 pi, so the polar angle would not be unambiguously defined. The line (or curve) in the complex plane where the angle jumps back by 2 pi is called the branch cut (analogy: when you travel around the globe crossing time zones you will at some point have to reset your clock by 24 hours, as each point on the globe has an unambiguously defined local time and date).
When doing contour integration, you must make sure that the branch cut you choose does not intersect the contour. This is because the integrand has a discontinuity across the branch cut and thus cannot be meromorphic. So, you would not be able to apply the residue theorem. How should we then choose the branch cut in this problem? We can see that the contour can be taken to be a half circle in the upper half plane, as the integral over the negative real axis will be proportional to the integral over the positive real axis. We can combine the two integrals and then solve for the desired integral. So, the branch cut can be ...
The integral of x^a / (x^2 + 1)^2 dx from x = 0 to infinity is evaluated using contour integration techniques. All steps are explained in detail.