Comparison of the Cauchy Integral Formula and Residue Theorem on the Calculation of Definite Integrals

Abstract

Integration is a useful mathematical tool which is applied in a wide range of studies and assists people to solve many problems. Despite methods of real analysis, complex integrations in complex analysis are more beneficial and more convenient than real integrations under certain circumstances. Cauchy’s Residue Theorem and Cauchy Integral Formula are two methods used to solve complex integrals. Cauchy’s Residue Theorem helps to solve complex integrals via the calculation involving singular points, whereas Cauchy Integral Formula is implemented to solve complex integrals by finding out poles enclosed by the contour. Herein, the comparison and analysis of the process of solving both real and complex integrations between Cauchy’s Residue Theorem and Cauchy Integral Formula are demonstrated through sample questions involving exponential integrations and polynomial integrations. This passage also includes definitions for Cauchy’s Residue Theorem, Cauchy Integral Formula, corollary of Cauchy Integral Formula, steps and solutions of sample contour integral questions.