There are three general types of definite integrals for which the complex contour integration methods are useful. Integrals of functions of cosines and sines over the interval [0, 2π] are transformed to the complex unit circle. The second and third type of integrals are over the intervals [-∞,∞] and [0, ∞], and are solved by including the interval on the real axis as part of the contour of integration in the complex plane.
This article works out an example of a complex contour integration method that I have not seen in my textbooks or in online articles I have read. I don’t know if this is a novel method or not. Maybe someone can point me to the relevant literature. The method transforms an infinite integral on the real line from –∞ to ∞ to a circle in the complex plane using Mobius transformations. Read more…Methods of Integration Using Mobius Transformations