Complex Analysis

Today, the theory of complex-valued functions finds widespread applications in various areas of mathematical research, as well as in electrical and mechanical engineering, aeronautics, and other disciplines. Complex analysis has become a basic course in mathematics, physics, and select engineering departments. This concise textbook provides a thorough introduction to the function theory of one complex variable. It presents the fundamental concepts with clarity and rigor, offering concise proofs that avoid lengthy and tedious arguments commonly found in mathematics textbooks. It goes beyond traditional texts by exploring less common topics, including the different approaches to constructing analytic functions, the conformal mapping criterion, integration of analytic functions along arbitrary curves, global analytic functions and their Riemann surfaces, the general inverse function theorem, the Lagrange–Bürmann formula, and Puiseux series. Drawing from several decades of teaching experience, this book is ideally suited for one or two semester courses in complex analysis. It also serves as a valuable companion for courses in topology, approximation theory, asymptotic analysis, and functional analysis. Abundant examples and exercises make it suitable for self-study as well.

Table of Contents:
1 Complex Numbers and Complex Plane.- 2  Analytic Functions.- 3 Elementary Analytic Functions.- 4 Integration of Functions of a Single Complex Variable.- 5 Complex Power Series.- 6 Laurent Series. Isolated Singularities of Analytic Functions.- 7 Residue Calculus.- 8 Analytic Continuations.- 9 Qualitative Properties of Analytic Functions.

About the Author :
Taras A. Mel'nyk is a professor of the department of mathematical physics at Taras Shevchenko National University of Kyiv. He is a fellow of the Alexander von Humboldt Foundation and a member of the American Mathematical Society. In his teaching, he has developed special courses on a range of topics such as asymptotic methods in mathematical physics, the theory of homogenization, the theory of Sobolev spaces, complex analysis and nonlinear analysis. He has published widely and has authored the monograph Multiple-Scale Analysis of Boundary-Value Problems in Thick Multi-Level Junctions of Type 3:2:2 (Springer, 2019) and two textbooks Complex Analysis (2015, in Ukrainian) and Sobolev Space Theory and Weak Solutions of Boundary Value Problems (2018, in Ukrainian). His research interests are related to asymptotic analysis of boundary-value problems, spectral problems, variational inequalities, optimal control problems, convection-diffusion problems in domains with complex micro-inhomogeneous structures (perforated materials, composite materials, thick multi-structures, domains with rapidly oscillating boundaries, domains with concentrated masses, thin domains, thin graph-like junctions).

Review :
"The exposition in the textbook is very clear and easy to read, all facts are presented with complete and detailed proofs. The book contains many helpful illustrations. When presenting the material, the author gives many useful exercises of theoretical character. ... The textbook will be useful not only for mathematics students, but also to physicists and engineers who want to gain a deeper understanding of the theory of functions of a complex variable." (Anna Vishnyakova, zbMATH 1546.30001, 2024)