Generalization of Cauchy-residue formula

Abstract

This article concentrates on the application of Cauchy residue theorem in different fields and the links between the work of Cauchy, who developed his theory of residue in one variable, and other contributing mathematicians, such as Laurent and Sokhostkii. Laurent introduced Laurent series and developed Laurent theorem based on the work of Cauchy. Sokhostki studied the properties of residue. We will overview the work of these three mathematicians and mention the application of Cauchy residue theorem in integration and series summation. When calculating the integral of a real function, we transform the problem into calculating the integral of a complex function along a chosen contour, and then use the residue theorem to get the answer. When computing the sum of a series, we transform the problem into summing the residues of a function on specific points, and then use the residue theorem to transform the latter problem into computing the integral of the function along a contour.