Application and Analysis of The Residue Theorem

Abstract

This paper studies the derivation of the residue theorem and its application. The residue theorem is an important part of the theory of complex functions and plays a critical role in the development and application of complex functions. At the same time, the residue theorem advances the method of solving the value of definite integrals to a new stage. The issue of the Cauchy residue theorem has received considerable attention. Also, residue theorem lays the foundation for the development of integral theory, and it has been widely used in many scientific areas. The goal of this study is to investigate the properties and applications of the residue theorem. To this end, the strategies to deduce the residue theorem are summarized. Then this paper uses several examples that relate to the residue theorem to help people understand the residue theorem more deeply. The integrals under consideration include trigonometric integrals and integrals involving power functions.