Complex Analysis

1. Front Matter

   Pages i-xiii

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2. Introduction

   • Andrei Bourchtein, Ludmila Bourchtein

   Pages 1-59

3. Analytic Functions and Their Properties

   • Andrei Bourchtein, Ludmila Bourchtein

   Pages 61-127

4. Singular Points, Laurent Series, and Residues

   • Andrei Bourchtein, Ludmila Bourchtein

   Pages 129-198

5. Conformal Mappings. Elementary Functions

   • Andrei Bourchtein, Ludmila Bourchtein

   Pages 199-268

6. Fundamental Principles of Conformal Mappings. Transformations of Polygons

   • Andrei Bourchtein, Ludmila Bourchtein

   Pages 269-338

7. Back Matter

   Pages 339-346

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This book discusses all the major topics of complex analysis, beginning with the properties of complex numbers and ending with the proofs of the fundamental principles of conformal mappings. Topics covered 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 evaluation of integrals, systematic study of elementary functions, analysis of conformal mappings and their applications—making this book self-sufficient and the reader independent of any other texts on complex variables. 


The book is aimed at the advanced undergraduate students of mathematics and engineering, as well as those interested in studying complex analysis with a good working knowledge of advanced calculus. The mathematical level of the exposition corresponds to advanced undergraduate courses of mathematical analysis and first graduate introduction to the discipline. The book contains a large number of problems and exercises, making it suitable for both classroom use and self-study. Many standard exercises are included in each section to develop basic skills and test the understanding of concepts. Other problems are more theoretically oriented and illustrate intricate points of the theory. Many additional problems are proposed as homework tasks whose level ranges from straightforward, but not overly simple, exercises to problems of considerable difficulty but of comparable interest.

• Complex Analysis
• Analysis
• Analytic Functions
• Singular Points
• Laurent Series
• Cauchy’s Theorem
• Goursat’s Theorem
• Conformal Mappings
• Polygons
• Joukowski Function

• Institute of Physics and Mathematics, Universidade Federal de Pelotas, Pelotas, Brazil

  Andrei Bourchtein, Ludmila Bourchtein

ANDREI BOURCHTEIN is Professor at the Institute of Physics and Mathematics, Federal University of Pelotas, Brazil. He received his Ph.D. in Mathematics and Physics from the Hydrometeorological Center of Russia. He began his academic and research career as an Associate Professor at Mathematics Institute, Far East State University, Russia, and a Research Scientist at the Hydrometeorological Institute, Russia. In 1995, he joined the Federal University of Pelotas, Brazil, as Associate Professor at the Institute of Physics and Mathematics before being promoted to Full Professor. An author of more than 100 refereed articles and 7 books, Prof. Bourchtein’s research interests include real and complex analysis, numerical analysis, computational fluid dynamics and numerical weather prediction. During his research career, he was awarded a number of grants from Brazilian science foundations and scientific societies, including the International Mathematical Union (IMU) and the International Council for Industrial and Applied Mathematics (ICIAM).

LUDMILA BOURCHTEIN is Professor Emeritus and Senior Research Scientist at the Institute of Physics and Mathematics, Federal University of Pelotas, Brazil. She received her Ph.D. in Mathematics from Saint Petersburg State University, Russia. During the span of 34 years at the Mathematics Institute, Far East State University, Russia, she held different positions, from Assistant Professor, Associate Professor, to the rank of Full Professor. In the last 15 years, she served as Associate and Full Professor at the Institute of Physics and Mathematics, Federal University of Pelotas, where she was conferred with the title of Emeritus Professor in 2019. An author of more than 80 referred articles and 5 books, her research interests include real and complex analysis, conformal mappings and numerical analysis. During her research career, she was awarded a number of grants of Russian and Brazilian science foundations and scientific societies, including the International Mathematical Union (IMU) and the International Council for Industrial and Applied Mathematics (ICIAM).

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