Some Fundamental Results from Complex Analysis

Abstract

This paper is going to introduce the most basic theory of analytic functions of one complex variable. It begins at the discussion of meaning of complex number and the historical development from the formula of cubic equation to the square root of negative number. In the middle section, which is divided in to four small parts. First part states the expression of complex number z, algebraic properties, and the relationship of each single complex number with whole complex plane. Second part concerns about several elementary functions of complex number. Next part relates to the derivative of complex number, such as the partial derivative and Cauchy-Riemann Equation to verdict that whether a function is analytic in the domain. Last part is the integral of complex functions, which concludes some important theorems with the proof, and examples. After the section of introduction of complex number, it comes to the final section which is talking about an identity involving complex number and the prove of it.