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About The Book

This text is aimed at university students specializing in mathematics and natural sciences, and at all the readers interested in studying complex analysis. The mathematical level of the exposition corresponds to advanced undergraduate courses of mathematical analysis and first graduate introduction to the discipline.

The book covers all the traditional topics of complex analysis, starting from the very beginning — the definition and properties of complex numbers, and ending with the theorems of great importance both for analysis and applications — the fundamental properties of conformal transformations. The topics considered in the book include the study of holomorphic and analytic functions, classification of singular points and the Laurent series expansion, theory of residues and their application to the evaluation of integrals, systematic study of elementary functions, analysis of conformal mappings and their applications.

The book is designed for the reader who has a good working knowledge of advanced calculus. All the preliminary topics related to complex variables and required for the development of the complex function theory are covered in the initial part of the text and, when necessary, at the starting point of each topic. As a consequence, this work can be used both as a textbook for advanced undergraduate/graduate courses and as a source for self-study of selected (or all) topics presented in the text.

The book contains a large number of problems and exercises, which should make it suitable for both classroom use and self-study. Many standard exercises are included in each section to develop basic techniques and test understanding of concepts. Other problems are more theoretically oriented and illustrate more intricate points of the theory. Many additional problems are proposed as homework tasks at the end of each chapter. Their level ranges from straightforward, but not overly simple, exercises to problems of considerable difficulty, but of comparable interest.

Key features of the book:

  1. Self-sufficiency: all the essential topics related to complex variables are covered, and, consequently, this work can be used as both a textbook and a source for self-study.

  2. Exercises: numerous exercises include both standard examples for developing basic techniques and advanced problems for stimulating the analytical ability of the readers.

  3. Visualization: graphical representations used systematically to make the exposition clearer and develop geometric intuition.

  4. Generality: the text is designed to present all the results in a more general form while avoiding major complications of their proofs.

  5. Accessibility: all the topics are treated in a rigorous mathematical manner while keeping the exposition at a level acceptable for advanced undergraduate courses

About the Authors

Andrei Bourchtein is a Professor at the Institute of Physics and Mathematics at Pelotas State University, Brazil. He received his Ph.D. in Mathematics and Physics from Hydrometeorological Center of Russia and started his academic and research career at the Mathematics Institute at Far East State University, Russia. An author of more than 100 referred articles and 7 books, his research interests include real and complex analysis, numerical analysis, computational fluid dynamics, and numerical weather prediction. During his extensive career, he has been awarded several grants from Brazilian science foundations and scientific societies, including IMU and ICIAM.

Ludmila Bourchtein is Professor Emeritus and Senior Research Scientist at the Institute of Physics and Mathematics at Pelotas State University, Brazil. She received her Ph.D. in Mathematics from Saint Petersburg State University, Russia. Authoring more than 80 referred articles and 5 books, her research interests include real and complex analysis, conformal mappings, and numerical analysis. She is a recipient of a number of grants from Russian and Brazilian science foundations and scientific societies, including IMU and ICIAM.

Table of Contents

Chapter 1: Introduction

Chapter 2: Analytic functions and their properties

Chapter 3: Singular points. Laurent series. Residues

Chapter 4: Conformal mappings. Elementary functions

Chapter 5: Fundamental principles of conformal mappings. Transformations of polygons

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