The Residue Theorem and Its Generalization

Abstract

In multivariate analysis, the residue theorem is an effective tool to calculate the results of analytic functions along closed curves. It can also be used to compute the integrals of real functions. The residue theorem reveals the relationship between integrals and singular residue on closed paths. If the Laurent expansion for a given point is computed, the corresponding residue for calculating the circumference integral can directly be obtained. It’s a generalization of Cauchy’s integral theorem and Cauchy’s integral formula. In this paper, the definition of the Residue will be mentioned. Then the proof of Residue Theorem will be shown. Three examples of how to use the Residue Theorem to the closed curve integral takes the derivative. Finally, the generalization of residue theorem is mentioned. In mathematical analysis and practical life problems, some primitive functions that cannot be represented by elementary functions as integrands can be calculated using the residue theorem. Converting integrals into residues for calculation can make integral problems simpler.