Integrating Complex Functions in Certain Situations by Two Methods

Abstract

This article will mainly discuss several ways of integrating complex functions. In real field, the basic idea of integrating a function is to find its anti-derivative. However, it is challenging to find an anti-derivative of a complex function that is not differentiable at some points. This is because that in complex field, the derivative of a differentiable complex function must be also differentiable. Thus, finding the anti-derivative of non-differentiable function in complex plain is unapproachable. Moreover, even though a differentiable complex function is given, computing the anti-derivative will be very complicated or time-consuming. As a result, exclusive methods are required to calculate the integral of complex functions. Fortunately, Cauchy Residue Theorem and Cauchy-Goursat Theorem provide comprehensive ways for computing the integrals. It is crucial to realize their importance and they will be discussed in the following context. Additionally, the specific steps of application will also be shown. This study demonstrates clearly that the Residue theorem is a power method to handle with a series of complex integrals.