Application of Residue Theorem in Calculating Improper Integral in Calculus

Abstract

In this paper, the residue theorem and its applications in the field of calculus are discussed. This study first emphasizes the importance of the residue theorem in the analysis of complex functions, and points out its wide applicability in various branches of mathematics. In the second part, this article explores some basic concepts in depth, including a section on singularities and higher-order singularities, precise definitions of residuals, and different methods used to calculate them. Building on this theoretical foundation, this paper transitions to practical applications, showing how to use the residual theorem to compute integrals, with particular emphasis on its ability to simplify calculations and provide efficient solutions to other thorny problems. Through a series of detailed examples, the validity of the theorem in solving difficult integrals solved by standard methods is emphasized. The conclusion reiterates the key role of the residue theorem in calculus and emphasizes its indispensable contribution to calculus calculation by providing accurate and simplified solutions.