An Introduction to the Study of Holomorphic Functions in Complex Analysis

Abstract

This paper aims to study holomorphic functions in the complex plane and demonstrate their fundamental characteristics within the framework of complex analysis theory. It provides a definition of holomorphic functions and explains the Cauchy-Riemann equations as a necessary and sufficient condition for complex differentiability. The paper also presents the representation of these functions using power series and Taylor series, in addition to studying some basic complex functions and the properties of entire functions and Liouville's theorem and its related results. Furthermore, it presents the concept of complex integration, the Cauchy integral formula, and some important theorems. This study highlights the theoretical importance of holomorphic functions and their role in understanding many fundamental results in complex analysis