**Best Reference Books on Complex Analysis**, which are used by students of top universities, and colleges. This will help you choose the right book depending on if you are a beginner or an expert. Here is the complete list of

**Complex Analysis Books**with their authors, publishers, and an unbiased review of them as well as links to the Amazon website to directly purchase them. If permissible, you can also download the free PDF books on Complex Analysis below.

## 1. Complex Analysis

1."Complex Variables and Applications" by R V Churchill and J W Brown
“Complex Variables and Applications” Book Review: The revised and updated book provides a comprehensive introduction to the theory and application of functions of a complex variable, featuring the latest developments in the subject. The book contains detailed discussions and proofs of complex results in advanced calculus, making it an ideal resource for students, researchers, and professionals in engineering and the sciences seeking to deepen their understanding of complex variables. The efficient theory presented in the book is widely applicable and reflective of current trends in the field.
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2."Complex Analysis" by J M Howie
“Complex Analysis” Book Review: The book provides an up-to-date and practical view of complex analysis, covering both theory and application. It covers basic concepts and classical theories, which are clearly explained with the help of worked examples, illustrations, and problem sets. This approach helps readers develop an intuitive understanding of the material. The book emphasizes technology and is an excellent resource for students and professionals of mathematics and engineering.
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3."Complex Variables- Introduction and Applications" by M J Ablowitz and A S Fokas
“Complex Variables- Introduction and Applications” Book Review: The book is about complex analysis and provides methods for solving related problems. It starts with an explanation of complex variables and covers topics such as analytic functions, integration, series, residue calculus, transform methods, ODEs in the complex plane, and numerical methods. The book also covers more advanced topics, including conformal mappings, asymptotic expansions, and Riemann-Hilbert problems. It includes many applications, examples, and exercises, making it useful for undergraduate and introductory graduate-level courses in complex variables.
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4."Complex Analysis" by Joseph Bak and Donald J Newman
“Complex Analysis” Book Review: This book presents the fundamental concepts and major aspects of complex analysis in a balanced manner, encompassing both theoretical and practical perspectives. It introduces authentic complex-variable ideas and techniques while incorporating advanced concepts from several-variable calculus and topology in the chapters. The text is enriched with various illustrations, examples, and exercises. It is designed for both students and professionals in engineering who seek a comprehensive understanding of complex analysis.
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5."Complex Analysis" by Ahlfors
“Complex Analysis” Book Review: The book focuses on the advanced and state-of-the-art aspects of functions of one complex variable. The chapters of this book cover all the major topics related to complex functions, change of length and area under conformal mapping, analytic functions, and complex Integration. The general form of Cauchy’s theorem is explained quite efficiently. The topics featured in this book are accurate, precise, and consist of many modern notations and terminologies. The book will be valuable for the students and professionals of mathematics.
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6."Introductory Complex Analysis" by Richard A Silverman
“Introductory Complex Analysis” Book Review: This book covers both theory and practical approaches in complex analysis, with updated information on problem sets. It starts with basics like complex numbers, their geometric representation, and algebra, and moves on to topics like complex integrals, power series, and conformal mapping. It also includes a chapter on the residue theorem. The book includes many examples and problems to help readers understand the material. It is suitable for graduate or undergraduate courses in complex analysis.
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7."Complex Analysis" by Elias M Stein and Rami Shakarchi
“Complex Analysis” Book Review: This book is about complex analysis and has been updated and revised to include the latest theory and practical exercises. It explains topics like the Cauchy theorems, residues, analytic continuation, and argument principle in detail, with a balanced approach between conceptual insights and technical details. The book includes many examples and applications to help readers understand the material. It will be useful for students in fields like mathematics, physics, engineering, and other sciences.
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8."A First Course in Complex Analysis" by Matthias Beck
“A First Course in Complex Analysis” Book Review: This book is an introduction to complex analysis and covers topics like complex numbers, differentiation, integration, Cauchy’s theorem, harmonic functions, power series, Taylor and Laurent series, isolated singularities, and the Residue theorem. It uses many examples to help explain the material, and highlights the applications of the topics and theorems. The book will be useful for students and professionals in research and engineering who want to learn about these topics.
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9."Visual Complex Analysis" by Tristan Needham
“Visual Complex Analysis” Book Review: This book is a good mix of theory and practical aspects of complex analysis. It starts with topics like geometry, complex arithmetic, complex functions as transformations, and differentiation. The final section covers topics like non-Euclidean geometry, complex integration, and harmonic functions. There are many diagrams to help readers understand the material, and each chapter has exercises at the end. The book is suitable for undergraduate students in math and sciences as well as professional mathematicians who want to learn about complex analysis.
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10."Complex Analysis: Conformal Inequalities and the Bieberbach Conjecture" by Prem K Kythe
“Complex Analysis: Conformal Inequalities and the Bieberbach Conjecture” Book Review: This book is suitable for graduate students who are pursuing analytical research on complex analysis in one or multiple complex variables, as well as for researchers working on related domains. The book covers topics like analytic functions, univalent functions, conformal mapping, area principle theorems, and Lowner theory. It also discusses Schiffer’s variation method, subclasses of univalent functions, and generalized convexity. The book explores orthogonal polynomials and the de Branges theorem. It also addresses current and emerging developments related to the de Branges theorem.
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11."Real and Complex Analysis" by W Rudin
“Real and Complex Analysis” Book Review: This book provides a comprehensive overview of both real analysis and complex analysis, covering all the major topics in these fields. The text presents the basic techniques and theorems in a clear and concise manner, highlighting the connections between different branches of analysis. The book includes chapters on differentiation and functional analysis, with precise and clear proofs of the theorems presented. The chapters progress from basic to more complex topics, with exercises provided at the end of each chapter. This book will be an invaluable resource for students of mathematics, science, computer science, and electrical engineering.
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## 2. Advance Complex Analysis

1."Complex Analysis and Related Topics: Volume 114 (Operator Theory: Advances and Applications)" by E Ramirez de Arellano and M V Shapiro
“Complex Analysis and Related Topics: Volume 114 (Operator Theory: Advances and Applications)” Book Review: This book contains research papers on various aspects of a subject related to operator theory and hypercomplex analysis. It includes topics such as the Schrödinger equation, subelliptic operators, and Lie algebras and superalgebras. The book is aimed at researchers and postgraduate students who are interested in the latest developments in this field.
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2."Operator Theory and Complex Analysis: Workshop on Operator Theory and Complex Analysis Sapporo (Japan) June 1991 (Operator Theory: Advances and Applications)" by T Ando and I Gohberg | |

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3."Harmonic Analysis and Boundary Value Problems in the Complex Domain (Operator Theory: Advances and Applications)" by M M Djrbashian
“Harmonic Analysis and Boundary Value Problems in the Complex Domain (Operator Theory: Advances and Applications)” Book Review: This book presents new methods in complex analysis based on investigations in the theory of integral representations of analytic and entire functions, as well as theory of harmonic analysis in the complex domain. The study focuses on the asymptotics of entire functions with parameters p and J1 greater than 0. The results of the studies have been further developed and evaluated. The intended audience for this book is not clear from the paragraph.
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4."Advances in Harmonic Analysis and Operator Theory: The Stefan Samko Anniversary Volume (Operator Theory: Advances and Applications)" by Frank-Olme Speck and Luís Castro | |

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5."Computation of Mathematical Models for Complex Industrial Processes (Advances in Process Systems Engineering)" by Tian
“Computation of Mathematical Models for Complex Industrial Processes (Advances in Process Systems Engineering)” Book Review: This book is designed for undergraduate and postgraduate students, researchers, and practitioners who are interested in case studies on numerical computing for industrial processes. The topics covered in the book include the fundamentals of industrial computing and finite difference methods, as well as the Wavelet-Collocation Method and Wavelet-Galerkin Method. High Resolution Methods and comparative studies of various methods are also included. The book provides examples and step-by-step procedures for using these methods, as well as various applications of these procedures. No previous knowledge is required to read and understand the book.
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6."Advances in Complex Function Theory: Proceedings of Seminars held at Maryland, University" by W E Kirwan and L Zalcman | |

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7."Optimization of Stochastic Discrete Systems and Control on Complex Networks" by Stefan Pickl and Dmitrii Lozovanu
“Optimization of Stochastic Discrete Systems and Control on Complex Networks” Book Review: The latest research results on the subject are presented in this book, including the author’s solution to discrete optimal control problems in networks. The book also covers the solution of game variants of Markov decision problems for computational networks. The limited state space of Markov processes and reviews are studied in the beginning of the book, followed by discussions on new approaches based on dynamic programming and combinatorial methods. The second chapter of the book covers infinite horizon stochastic discrete optimal control models and Markov decision problems. Chapter 3 takes a theoretical approach to Markov decision processes and includes stochastic discrete optimal control problems. Chapter 4 is dedicated to finite horizon stochastic control problems. The book is a valuable resource for researchers and professionals in the field of control theory and network optimization.
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8."Aspects of Boundary Problems in Analysis and Geometry (Operator Theory: Advances and Applications)" by Juan Gil and Thomas Krainer
“Aspects of Boundary Problems in Analysis and Geometry (Operator Theory: Advances and Applications)” Book Review: This book is a comprehensive guide to boundary problems, operator theory, and geometric analysis. Written by Juan Gil and Thomas Krainer, this book covers topics such as elliptic and parabolic equations, potential theory, Hardy spaces, and harmonic analysis. It also explores boundary value problems in Riemannian geometry and mathematical physics. With clear explanations and numerous examples, this book is ideal for graduate students and researchers in mathematics and physics.
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9."Advances in Applied Analysis (Trends in Mathematics)" by Sergei V Rogosin and Anna A Koroleva
“Advances in Applied Analysis (Trends in Mathematics)” Book Review: This book contains survey papers that were based on lectures presented at the 3rd International Winter School “Modern Problems of Mathematics and Mechanics”. The papers discuss problems related to modern analysis, as well as modern problems of applied analysis. The results and applications of the studies are also discussed, including applications of composite materials, anomalous diffusion, and fluid dynamics. The intended audience for this book is not clear from the paragraph.
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10."Advances in Real and Complex Analysis with Applications (Trends in Mathematics)" by Michael Ruzhansky and Yeol Je Cho
“Advances in Real and Complex Analysis with Applications (Trends in Mathematics)” Book Review: This book covers topics related to both mathematics and engineering and provides simple explanations of mathematical concepts and their applications. It includes topics such as real and complex analysis, special functions, analytic number theory, q-series, and Ramanujan’s mathematics. Additionally, the book covers graph theory, complex analysis, complex dynamical systems, complex function spaces, and operator theory. Furthermore, it includes discussions on geometric analysis of complex manifolds, geometric function theory, Riemannian surfaces, Teichmüller spaces, Kleinian groups, and engineering applications of complex analytic methods. The book also covers topics such as nonlinear analysis, inequality theory, potential theory, partial differential equations, numerical analysis, fixed-point theory, variational inequality, equilibrium problems, optimization problems, stability of functional equations, and mathematical physics. The book is intended for readers who are interested in exploring the applications of mathematics to engineering, but some of the content may be challenging without a solid background in mathematics.
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11."An Advanced Complex Analysis Problem Book" by Daniel Alpay
“An Advanced Complex Analysis Problem Book” Book Review: This is an exercise book for graduate-level students that demonstrates the connections between functional analysis and the theory of functions of one variable. The book focuses on key concepts such as positive definite kernel and reproducing kernel Hilbert space. It provides an overview of various facts from functional analysis and topological vector spaces. The book then goes on to study different Hilbert spaces of analytic functions. It is intended for readers who are interested in exploring the relationships between functional analysis and the theory of functions of one variable. However, it may be challenging to understand without a strong background in mathematics.
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12."Computational Aspects of Complex Analysis: Proceedings of the NATO Advanced Study Institute held at Braunlage" by K E Werner and L Wuytack
“Computational Aspects of Complex Analysis: Proceedings of the NATO Advanced Study Institute held at Braunlage” Book Review: The book aimed to unite mathematicians from various fields and computer scientists, discussing approximation and interpolation by polynomial and rational functions (especially Pade approximation), numerical methods for solving algebraic and differential equations, conformal mapping, computer implementation of complex arithmetic, and calculations based on complex variable techniques. The sessions on short communications allowed participants to present their contributions, discuss the material in greater depth, address open problems, and identify the interrelationships among the topics.
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13."Advanced Mathematical Analysis: Periodic Functions and Distributions, Complex Analysis, Laplace Transform and Applications (Graduate Texts in Mathematics)" by R Beals
“Advanced Mathematical Analysis: Periodic Functions and Distributions, Complex Analysis, Laplace Transform and Applications (Graduate Texts in Mathematics)” Book Review: The book discusses the negative effects of categorizing courses into different subjects. Mathematics students tend to learn abstract concepts first and examples later, while science students often miss out on modern insights by learning outdated examples. Offering a choice between learning Fourier series as a minor instance of representation theory versus in isolation in an ad hoc manner is not a real choice. Although the problem is easy to recognize, it is challenging to address the legitimate pressures that have led to the separation of subjects.
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14."Advanced Complex Analysis: Part 2B: A Comprehensive Course in Analysis" by Barry Simon
“Advanced Complex Analysis: Part 2B: A Comprehensive Course in Analysis” Book Review: The book discusses several mathematical concepts in this volume. These include the theory of conformal metrics, which covers the Poincare metric, the Ahlfors-Robinson proof of Picard’s theorem, and Bell’s proof of the Painleve smoothness theorem. The book also covers topics in analytic number theory, such as Jacobi’s two- and four-square theorems, the Dirichlet prime progression theorem, the prime number theorem, and the Hardy-Littlewood asymptotic for the number of partitions. It also covers the theory of Fuchsian differential equations, asymptotic methods like Euler’s method, stationary phase, saddle-point method, and the web method. Additionally, the book covers univalent functions, including an introduction to SLE, and Nevanlinna theory, which is used to study the location of the zeros of a function. The book is intended for readers who are interested in exploring these topics and significant knowledge in mathematics is required to understand the content.
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## 3. Set Theory and Complex Analysis

1."Theory of Functions Vol. I & II" by K Knopp
“Theory of Functions Vol. I & II” Book Review: This book explains the theory of functions, covering topics such as absolute convergence tests, boundary sets, the Cauchy Integral formula, and the closed curve theorem. Other topics include the fixed point fundamental theorem, identity theorem for power series, Jordan’s inequality, and Laplace equation. These concepts are illustrated with many examples and illustrations, making it accessible to students. The book can be useful for undergraduate mathematics and engineering students who want to learn about these topics in detail. However, it is important to note that this book assumes a significant amount of mathematical knowledge and may not be suitable for beginners.
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2."The Theory of Functions" by E C Titchmarsh | |

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3."Functions of One Complex Variable" by J B Conway
“Functions of One Complex Variable” Book Review: This book is about the fundamental concepts of function theory for one complex variable. It covers topics such as the complex number system, metric spaces in topology, and basic properties and examples of analytic functions. Other chapters discuss complex integration, singularities, the maximum modules theorem, compactness, and convergence in the space of analytic functions. The book explains these theorems and concepts in detail with helpful figures. It also includes a bibliography and a list of symbols at the end of the book. This book is intended for undergraduate mathematics students with some mathematical knowledge. The book covers the essential concepts and most important theories of function theory in a detailed way.
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4."Introduction to the Theory of Functions of a Complex Variable" by E T Copson
“Introduction to the Theory of Functions of a Complex Variable” Book Review: This book is an introduction to the theory of functions for one complex variable. It covers topics such as complex numbers, infinite series convergence, analytic functions, and uniform convergence. Other topics include integral functions, calculus of residues, conformal representation, the gamma function, and Jacobi’s elliptic function, with a total of 15 chapters discussed in detail. The book provides proper equations and figures to explain the text easily. This book is intended for graduate-level mathematics and engineering students. The book introduces and covers the essential concepts and theories of complex function theory.
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5."Complex Analysis" by L V Ahlfors
“Complex Analysis” Book Review: This book provides information about functions of a complex variable. Main topics included are the algebra of complex numbers, the geometric representation of complex numbers, elementary theory of power series. Other topics included are Exponential and trigonometric function, Elementary point set topology, elementary conformal mapping. A short section on Riemann Zeta function is also added in this book. Numerous examples and illustrations are provided for better understanding. Students studying graduate level mathematics can refer to this book.
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6."Complex Analysis: A First Course with Applications" by Dennis G Zill and Patrick D Shanahan
“Complex Analysis: A First Course with Applications” Book Review: This book is an introductory guide to complex analysis and its applications, covering complex numbers in the complex plane, analytical functions, mappings, integration, series, conformal mapping, and residues. With a wide range of exercises, each chapter includes proper illustrations and examples to explain the core concepts and their applications. Review quizzes are also provided at the end of each chapter to help students test their knowledge. This book is a valuable resource for graduate-level students in mathematics and engineering.
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7."The Elements of Set Theory" by Kumar | |

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8."A Book of Set Theory (Dover Books on Mathematics)" by Charles C Pinter
“A Book of Set Theory (Dover Books on Mathematics)” Book Review: This book comprehensively covers the fundamental concepts of set theory. It includes topics such as elementary set theory, the development of set theory, and various systems of axiomatic set theory. Chapters on classes and sets, functions, relations, partially ordered classes, and the axiom of choice are also featured. The book contains several well-explained examples and illustrations. This book is highly recommended for students studying advanced level mathematics.
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9."Foundations Of Complex Analysis" by S Ponnusamy
“Foundations of Complex Analysis” Book Review: The fundamentals of complex analysis are thoroughly explained in this book, covering topics such as basic analysis, metric spaces, Banach spaces, normed spaces, and function space approximations. The book also covers orthogonal families of actors, the cardinality theorem for orthonormal bases, and Hilbert spaces. Each chapter includes exercises to help students practice and solidify their understanding. The book is designed for undergraduate students in mathematics and engineering and includes well-structured examples with solutions to aid comprehension.
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10."Complex Analysis" by Vasishtha
“Complex Analysis” Book Review: The book presents the fundamental concepts of complex analysis. It covers topics such as complex functions and mappings, analytical functions, integration in the complex plane, conformal mapping, and Taylor series. The book also includes chapters on the independence of path, trigonometric and hyperbolic functions, and their applications. Each chapter provides proper illustrations and equations for better understanding. The book is suitable for undergraduate students of mathematics and engineering.
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