A Study on Analytic Function of Complex Analysis

Abstract

To study and analyse the analytic functions in complex analysis system. A complex function is said to be analytic on a region R if it is complex differentiable at every point in R. The terms holomorphic function, differentiable function, and complex differentiable function are sometimes used interchangeably with "analytic function". Many mathematicians prefer the term "holomorphic function" (or "holomorphic map") to "analytic function", while "analytic" appears to be in widespread use among physicists, engineers, and in some older texts. If a complex function is analytic on a region R, it is infinitely differentiable in R. A complex function may fail to be analytic at one or more points through the presence of singularities, or along lines or line segments through the presence of branch cuts. A complex function that is analytic at all finite points of the complex plane is said to be entire. A single-valued function that is analytic in all but possibly a discrete subset of its domain, and at those singularities goes to infinity like a polynomial (i.e., these exceptional points must be poles and not essential singularities), is called a homomorphic function. The study also analysis different approaches to the concept of analyticity. One definition, which was originally proposed by Cauchy, and was considerably advanced by Riemann, is based on a structural property of the function the existence of a derivative with respect to the complex variable, i.e. its complex differentiability. To study the fact of the theory of analytic functions is the identity of the corresponding classes of functions in an arbitrary domain of the complex plane.