**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 book is an updated and revised piece of work featuring the latest aspects and recent developments in the theory and application of functions of a complex variable. The book reflects an efficient theory that is prominent in applications of the subject. The chapters of the book contain proofs and discussions of complex results in advanced calculus. The book will be suitable for the students, researchers, and professionals of engineering and sciences.
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2."Complex Analysis" by J M Howie
“Complex Analysis” Book Review: The book is written from both theoretical as well as practical point-of-view and aims at presenting the latest information on complex analysis. The book lays an emphasis on basic points and classical theories. The topics and ideologies featured in this book are clearly explained with the help of worked examples. The book highlights technology, illustrations, and problem sets, hence enabling the readers in developing an intuitive understanding of the material. This book is an excellent resource for the 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 aims at presenting the relevant methods for dealing with the problems faced during complex analysis. It begins with an introduction of complex variables, and features topics like analytic functions, integration, series, residue calculus, transform methods, ODEs in the complex plane, and numerical methods. Moving on, the advanced topics like conformal mappings, asymptotic expansions, and Riemann–Hilbert problems are discussed in detail. The book is supported by an array of applications, illustrative examples, and homework exercises. . This book will be a valuable source for the 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: The book aims at presenting the major aspects and basic concepts of complex analysis from both theoretical as well as practical perspective. The book is a balanced text featuring complex analysis from both theoretical as well as practical point-of-view. The authentic complex-variable ideas and techniques are introduced in this text. The chapters of this book consist of advanced concepts from several-variable calculus and topology. The book is elaborated and enhanced with the help of several illustrations, examples, and exercises. This book will be suitable for the students and professionals of engineering.
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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: The book is written from both theoretical as well as practical point-of-view and aims in presenting latest information on practical approach on problem sets in complex analysis. The book is an updated and revised piece of work presenting latest information on practical approaches on problem sets in complex analysis. The initial chapters of the book cover basics of complex analysis along with the definition of complex numbers, their geometric representation, their algebra, powers and roots of complex numbers, set theory, and complex functions and sequences. The remaining chapters describe topics like circle-preserving property, exponentials and logarithms, complex integrals and the Cauchy theorem, complex series, power series, Laurent series, harmonic functions, partial fraction expansions, conformal mapping, and analytic continuation. A chapter dedicated to residue theorem and its implications is added to this book. For better understanding of the readers many examples and problems are included in this text. The book will be an asset for a graduate or undergraduate course in complex analysis.
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7."Complex Analysis" by Elias M Stein and Rami Shakarchi
“Complex Analysis” Book Review: The book is an updated and revised piece of writing reflecting the latest theory and practical exercises in complex analysis. The major topics like the Cauchy theorems, residues, analytic continuation, and argument principle are thoroughly explained. The chapters of this book are balanced between conceptual insights and the technical underpinnings. The content of this book is supported by several examples and applications. The book will be a great source of information for the students of 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: The book focuses on presenting a suitable introduction of complex analysis. The major topics related to complex numbers, differentiation, functions, integration, Cauchy’s theorem, harmonic functions, power series, Taylor and Laurent series, isolated singularities, and Residue theorem are described in detail. The book is illustrated with numerous relatable examples. The applications of featured topics and theorems are highlighted in this text. The book will be beneficial for the students and professionals related to research and engineering.
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9."Visual Complex Analysis" by Tristan Needham
“Visual Complex Analysis” Book Review: The book is an excellent blend of both theoretical and practical aspects of complex analysis. The initial section of the book covers topics like geometry and complex arithmetic, complex functions as transformations, mobulus transformations and inversion, and differentiation. The final section features non-Euclidean geometry, winding number and topology, complex integration, Cauchy’s theorem, vector fields, and flow and harmonic functions. The text is supported with several diagrams in order to provide a visual insightful introduction of complex analysis. Each chapter ends with a number of exercises. The book will be suitable for the undergraduate students of mathematics and sciences. The professional mathematicians will also find this text valuable.
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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 suits the graduate students pursuing analytical research on the topics and researchers working on related domains of complex analysis in one or multiple complex variables. The book first talks about the theory of analytic functions, univalent functions, and conformal mapping before eventually covering various theorems related to the area principle and discussing Lowner theory. The author also discusses about the Schiffer’s variation method, bounds for 4rth and higher-order coefficients, numerous subclasses of univalent functions, generalized convexity and the class of α-convex functions, and numerical estimates of the coefficient problem. The book further summarises the orthogonal polynomials, explores the de Branges theorem, and addresses current and emerging developments through the de Branges theorem.
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11."Real and Complex Analysis" by W Rudin
“Real and Complex Analysis” Book Review: The book is an excellent blend of all the major topics of real analysis and complex analysis. The basic techniques and theorems of analysis are clearly presented. It also highlights the link between different branches of analysis. The chapters based on differentiation and functional analysis are included in this book. The proofs of the theorems featured in this book are precise and clear. The chapters of this book are ascended from basic topics to complex topics. Each chapter concludes with a bunch of exercises. The book will be beneficial for graduating 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 is for researchers and postgraduate students. It includes the latest research papers on various aspects of this subject. Topics included are relations to operator theory and hypercomplex analysis. Schrödinger equation, subelliptic operators, Lie algebras and superalgebras are also included.
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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 | |

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: The book introduces new methods in complex analysis. It is based on investigations in the theory of integral representations of analytic and entire functions. Theory of harmonic analysis in the complex domain is also the basis. This study was based on asymptotics of the entire function (p, J1 > 0). The results of the studies have been expanded and evaluated.
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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 | |

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 for undergraduate and postgraduate students, researchers and practitioners. This book includes case studies on numerical computing. This is for industrial processes. Minimal previous knowledge is required. Topics like fundamentals of industrial computing and finite difference methods are included. The Wavelet-Collocation Method and the Wavelet-Galerkin Method are also included. It also includes High Resolution Methods and comparative studies of various methods. The book included various examples. Step wise procedures are described. Various applications of these procedures are also studied.
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6."Advances in Complex Function Theory: Proceedings of Seminars held at Maryland, University" by W E Kirwan and L Zalcman | |

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: This book has the latest research results of the subject. Author’s solution of discrete optimal control problems in networks is included. solving game variants of Markov decision problems for computational networks is also given. The limited state space of Markov processes and reviews are studied in the beginning. New approaches based on dynamic programming and combinatorial methods are discussed later. Infinite horizon stochastic discrete optimal control models are given in the second chapter. It also included Markov decision problems. Chapter 3 has a theoretical approach to Markov decision processes. It includes stochastic discrete optimal control problems. Finite horizon stochastic control problems are discussed in chapter 4.
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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 encompasses material that combines both disciplines. It studies the interactions apparent in this field. Contributions are given in survey style. This makes it easy to understand by a large audience. Better understanding can be expected with a background in analysis or geometry.
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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 has survey papers of lectures. They were held at the 3rd International Winter School “Modern Problems of Mathematics and Mechanics”. Problems of modern analysis were discussed in this book. Modern problems of applied analysis are also discussed. Results and their applications are discussed. Applications of composite materials, anomalous diffusion, and fluid dynamics are discussed.
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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: Mathematics and engineering topics are discussed in this book. Simple explanation of mathematical concepts and their applications are included. Topics like real and complex analysis and special functions are included. Analytic number theory, q-series, Ramanujan’s mathematics are studied. It also includes graph theory, complex analysis and complex dynamical systems. Complex function spaces and operator theory are also included. Geometric analysis of complex manifolds, geometric function theory, Riemannian surfaces have been included. Teichmüller spaces and Kleinian groups are studied as well. It also has engineering applications of complex analytic methods. It includes nonlinear analysis, inequality theory and potential theory. Partial differential equations, numerical analysis and fixed-point theory is also included. Variational inequality, equilibrium problems and optimization problems have been discussed. Also,stability of functional equations, and mathematical physics is given.
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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 exercises book at the beginning graduate level, whose aim is to illustrate some of the connections between functional analysis and the theory of functions of one variable. A key role is played by the notions of positive definite kernel and of reproducing kernel Hilbert space. A number of facts from functional analysis and topological vector spaces are surveyed. Then, various Hilbert spaces of analytic functions are studied.
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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 aim of the book was to bring together scientists from pure and applied mathematics as well as computer scientists. The main topics were problems dealing with approximation and interpolation by polynomial and rational functions (in particular Pade approximation), numerical methods for the solution of algebraic equations and differential equations, the large field of conformal mapping, aspects of computer imple mentation of complex arithmetic and calculations based on complex variable techniques. The sessions on short communications not only provided a platform for the presentation of contributions by the participants of the ASI but also the opportunity to discuss the material more thoroughly, to bring up open problems and to point out the inter relationship of the above mentioned 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 separation between kinds of courses has unhealthy effects. Mathematics students reverse the historical development of analysis, learning the unifying abstractions first and the examples later (if ever). Science students learn the examples as taught generations ago, missing modern insights. A choice between encountering Fourier series as a minor instance of the representation theory of Banach algebras, and encountering Fourier series in isolation and developed in an ad hoc manner, is no choice at all. It is easy to recognize these problems, but less easy to counter the legitimate pressures which have led to a separation.
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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 Presents in this volume are the theory of conformal metrics (including the Poincare metric, the ahlfors-robinson proof of picard’s theorem, and Bells proof of the painlevé smoothness theorem), topics in analytic number theory (including jacobi’s two- and Foursquare theorems, the Dirichlet prime progression theorem, the prime number theorem, and the hardy-littlewood Asymptotic for the number of partitions), the theory of fuchsian differential equations, Asymptotic methods (including euler’s method, stationary Phase, the saddle-point method, and the web method), univalent functions (including an introduction to sle), and nevanlinna theory.
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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 function. Main topics included are absolute convergence test, boundary set, cauchy Integral formula, closed curve theorem. Other topics included are Fixed point fundamental theorem, identity theorem for power series, Jordan’s inequality, Laplace equation. All the topics and methods are described in detail with proper examples. Multiple illustrations are discussed that shows the students can get the most out of the chapters. This book is beneficial for undergraduate mathematics and engineering students.
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2."The Theory of Functions" by E C Titchmarsh | |

3."Functions of One Complex Variable" by J B Conway
“Functions of One Complex Variable” Book Review: This book covers the basic concepts of theory of functions of one complex variable. Main topics mentioned are the complex number system, metric spaces in the topology, elementary properties and examples of analytical function. Other chapters mentioned are complex integration, singularities, the maximum models’ theorem, compactness and convergence in the space of analytical functions. All the theorems and functions are mentioned in a detailed manner with proper figures. Bibliography and a list of symbols is added at the end of the book. This book is suitable for undergraduate mathematics students.
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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 of a complex variable. Main chapters included are complex numbers, the convergence of infinite series, functions of complex variable, uniform convergence. Other topics included are integral functions, the calculus of residues, Conformal representation, the gamma function, Jacobi’s elliptic function. Total 15 chapters are discussed in detail in this book. proper equations and figures are provided to explain the text easily. Students studying graduate level mathematics and engineering can use this book.
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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 covers the introductory course on complex analysis with applications. Main topics discussed are complex numbers in the complex plane, complex functions and mappings, analytical functions. Other chapters mentioned are integration in the complex plane, series, conformal mapping and residues. Each chapter contains extensive exercises for the practice of students. Proper examples in illustrations are provided to explain the main topics and its applications. at the end of each chapter review quiz is also provided for students to test their knowledge. This book is beneficial for graduate level mathematics and engineering students.
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7."The Elements of Set Theory" by Kumar | |

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 describes the concepts regarding set theory. Main topics included are elementary set theory, growth of set theory, various systems of axiomatic set theory. Chapters such as Classes and sets, functions, relations, partially ordered classes and the axiom of choice are also included. Proper illustrations in examples are provided on several topics. students studying advanced level mathematics can refer to this book.
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9."Foundations Of Complex Analysis" by S Ponnusamy
“Foundations of Complex Analysis” Book Review: This book explains the core principles of complex analysis. Chapters included are basic analysis, concepts in metric spaces, banach spaces, normed spaces, Approximation in function spaces. Other topics included are orthogonal family of actors, cardinality theorem for orthonormal basis, Hilbert spaces. End of chapter exercises are added for students to practice. Well-structured examples and their solutions are provided for students to understand the topic easily. This book is beneficial for undergraduate mathematics and engineering students.
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10."Complex Analysis" by Vasishtha
“Complex Analysis” Book Review: This book discusses the main and fundamental concepts of complex analysis. Main topics mentioned are analytical function, complex functions and mappings, integration in the complex plane. Other chapters included are conformal mapping, Taylor series, independence of path, trigonometric and hyperbolic functions. applications of each function and theorems are also provided in each chapter. proper illustrations and equations are provided to make the text easily understandable. Students studying undergraduate mathematics and engineering can refer to this book.
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