Elegant Theory of Complex Analysis

Abstract

A complex number is an element in a number system containing both real numbers and the imaginary unit , satisfying ^2 = 1. Since their discovery, complex numbers have been a powerful means of mathematical calculation. Complex analysis is a part of mathematical analysis that investigates complex numbers and their analyticity, holomorphicity, etc. Many renowned mathematical giants once had their own research in complex analysis, such as Cauchy, Gauss, Euler, etc. On the grounds that it deals with functions of complex numbers, complex analysis is a helpful area in the whole mathematics field. There are plenty of applications of complex analysis in both the mathematical field and the physics field. In this paper, the history of complex numbers and complex analysis is presented. Also, some contents of complex variables are shown, including the basic properties of complex numbers, the derivative and integral of functions of complex numbers, and several critical theorems in the area of complex analysis.