Deductions, Applications and Expansion of the Cauchy’s Residue Theorem

Abstract

Cauchy’s integral formula is one of the most important discovery in the development history of the complex variable integration. This article focuses on the methods of the integration in mathematics. The author aims to write about how to deduce the residue theorem from Cauchy’s integral formula and how to use the Cauchy’s integral theorem and residue theorem in the integral questions. The following parts of the article include using method of limits and Euler’s formula to deduce roughly the Cauchy’s Integral formula. In addition, the author also uses the differentiation to transfer the integral function. In part three, the author calculates three different types of examples and considers that residue theorem and Cauchy’s integral formula are very useful in the contour integral with singularities. The article is in order to show that the practicability of these theorem and formula. These formulas and theorems are used in a wide range of the subjects and areas.