د.عمر اسماعيل الحصادي2024-11-272024-11-27https://dspace.academy.edu.ly/handle/123456789/399Introduction: The theory of iteration of rational function over the Riemann sphere emerged in the papers produced, in dependently by Pierre Fatou and Gaston Julia almost 100 years ago. The study of iteration of rational mapping of a complex variable has a long history, and it has recently become a very active field of research.ABSTRACT The purpose of this study was to shed some light on rational maps. In particular, It aims to study some iterations of rational function and what results from that iteration where’s iteration of rational function the level is divided in to areas of attraction and areas of repulsion and is called Fatou set, the barrier between them moves in a confused manner and is called Julia sets. also to study the rational function, and analytic function with some discussion to understand their properties and their conditions. This research attempted to deal with the Iterations of the polynomial which included some examples of different polynomials. Moreover, the research goes deeper into the study of the Kӧenig iteration function.Kӧenig’s Method for Finding The Roots of Rational Function