Cauchy Theorem and Cauchy Residue Theorem

Abstract

Cauchy theorem is widely used in solving analytic function problems in complex variables. It is an important theorem on path integrals of holomorphic functions in the complex plane. In this paper, the main work is about the application of the Cauchy theorem on the integrals which have singularities. The definition of the Cauchy theorem is given and proved by using the green theorem. By using the conclusion of Cauchy theorem, the integrals that have singularities can be solved, and the Cauchy residue theorem is proved. By using the Cauchy theorem, problems such as definite integrals, indefinite integrals, and differential equations can be solved. Some problems in other disciplines can also be solved such as the physics and computer. The Cauchy integral theorem states that if there are two distinct routes between two points and the function is holomorphic on both routes, then the integrals of the two pathways are equivalent.