Proof of Cauchy’s theorem on Disc

Abstract

Mathematical analysis known as "complex analysis" examines characteristics of functions, sequences, derivatives of complex numbers, and other objects. When Italian mathematicians Girolamo Cardano and Raphael Bombelli attempted to solve a unique algebra, that is when people first learned about complex numbers. After hundreds of years of growth, complex analysis is now a crucial component of mathematics, physics, and engineering, particularly in the areas of algebraic geometry, fluid dynamics, quantum mechanics, and other related subjects. Complex analysis is a fascinating and esoteric field of study. Mathematicians have created groundbreaking work in this area during the past few centuries. Complex numbers were first defined by mathematicians, who subsequently gradually looked into their algebraic and geometric properties. After hundreds of years, they discovered and studied complex computations. Convergent power series local representations exist for holomorphic functions. This amazing discovery, made by Cauchy between 1830 and 1840, helps to explain the fascinating properties of holomorphic functions. The Cauchy integral theorem is one of the most important concepts in complex analysis. In this paper, a detailed proof of Cauchy’s theorem on a local disc is given.