Charaterization of Holomorphic function using power series

Abstract

Complex analysis is a subfield of mathematical analysis that focuses on the operations of complex numbers. The theory of functions of a complex variable is another name for it. Holomorphic functions, is the key object studied in the filed of complex analysis. These functions can be differentiated, have negative values, and are defined on the complex plane. Some of the ideas, concepts, and techniques frequently utilized in research include the residue theorem, Cauchy integral formula, Laurent series expansion, etc.In particular, algebraic geometry, fluid dynamics, quantum mechanics, and other related sciences have made substantial use of complex analysis in mathematics, physics, and engineering over the years. Girolamo Cardano and Raphael Bombelli, two Italian mathematicians, initially recognized complex numbers while attempting to solve a particular algebra in the 16th century, while Cauchy and Riemann further improved it in the 19th century.This essay explores the arithmetic features of complex numbers firstly. Secondly, a thorough interpretation of the characteristics of holomorphic functions is provided. Finally, using power series methods, the holomorphic function on the disc is characterized.