Solving Real Integrals Using the Residue Theorem

Abstract

Previously, it is difficult or even impossible to tackle some real integrals that may not be calculating by classic methods in calculus. However, this paper would summarize methods using the Residue theorem to calculate some types of these “impossible integrals”. However, before one calculates the integral using the residue theorem, one will introduce two important conclusions, one is about when the function decays faster than , which involve integral inequality and the method finding the upper bound. The second one would give reader a general conclusion of integration. At the same time, the readers have to understand the proof of the Cauchy’s Residue Theorem, and other theorems such as Cauchy’s theorem, Cauchy’s Integral formula. One should notice that the importance of branch cut and branch points, since the integrand one face could be multifunction, so one may try to turn them into single valued function. Also, the integral inequality plays a very important role in the first conclusion.