Definition of Residue Theorem and its Basic Applications

Abstract

This paper explains the fundamentality of residue theorem in complex analysis integration termed contour integral. It also illustrates the importance of this theorem, followed by applying it in both mathematics and other fields where several typical examples coming from different fields are chosen. The paper introduces some basic definitions and theorems to support and finally prove residue theorem. Then two simple applications of residue theorem are presented. One is a contour integral of a fractional function with one singularity inside the contour. It can be solved by using partial fraction technique, then directly finding the residue at the singularity, and finally applying residue theorem to calculate for the result. The other one is a contour integral of a reciprocal of sine function with one singularity inside. This problem can be solved by finding the Laurent series of the integrand, thus finding the residue needed. However, the residue is the coefficient of negative-one-degree term of the singularity, and the value of the integral can be achieved by substituting the residue into the formula of the residue theorem.