COMPLEX ANALYSIS

Abstract

This course provides an introduction to the theory of functions of a complex variable and presents the fundamental concepts and techniques of Complex Analysis. It aims to develop a solid understanding of differentiable complex functions, their principal properties, and some important applications in mathematics. The course begins with a review of the topology of the complex plane, including basic notions and geometric properties. It then introduces complexvalued functions, their limits, continuity, and differentiability. Special attention is given to elementary complex functions such as exponential, logarithmic, trigonometric, and hyperbolic functions. The theory of complex integration is subsequently developed, covering contour integrals and the main results associated with analytic functions. The course also studies power series representations, including Taylor and Laurent expansions, which play a central role in the local analysis of complex functions. Finally, the residue theorem and its applications are presented. These applications include the evaluation of certain definite and improper integrals, as well as the summation of series. Throughout the course, theoretical results are illustrated with examples and exercises to facilitate understanding and practical mastery of the subject. Keyword : Topology in the comp