
Author 
Tristan Needham 
ISBN10 
0198534469 
Year 
1998 
Pages 
592 
Language 
en 
Publisher 
Oxford University Press 
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Now available in paperback, this successful radical approach to complex analysis replaces the standard calculational arguments with new geometric ones. With several hundred diagrams, and far fewer prerequisites than usual, this is the first visual intuitive introduction to complex analysis. Although designed for use by undergraduates in mathematics and science, the novelty of the approach will also interest professional mathematicians.

Author 
Elias Wegert 
ISBN10 
9783034801805 
Year 
20120830 
Pages 
360 
Language 
en 
Publisher 
Springer Science & Business Media 
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This book provides a systematic introduction to functions of one complex variable. Its novel feature is the consistent use of special color representations – socalled phase portraits – which visualize functions as images on their domains. Reading Visual Complex Functions requires no prerequisites except some basic knowledge of real calculus and plane geometry. The text is selfcontained and covers all the main topics usually treated in a first course on complex analysis. With separate chapters on various construction principles, conformal mappings and Riemann surfaces it goes somewhat beyond a standard programme and leads the reader to more advanced themes. In a second storyline, running parallel to the course outlined above, one learns how properties of complex functions are reflected in and can be read off from phase portraits. The book contains more than 200 of these pictorial representations which endow individual faces to analytic functions. Phase portraits enhance the intuitive understanding of concepts in complex analysis and are expected to be useful tools for anybody working with special functions – even experienced researchers may be inspired by the pictures to new and challenging questions. Visual Complex Functions may also serve as a companion to other texts or as a reference work for advanced readers who wish to know more about phase portraits.

Author 
H. A. Priestley 
ISBN10 
0191583332 
Year 
20030828 
Pages 
344 
Language 
en 
Publisher 
OUP Oxford 
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Complex analysis is a classic and central area of mathematics, which is studied and exploited in a range of important fields, from number theory to engineering. Introduction to Complex Analysis was first published in 1985, and for this much awaited second edition the text has been considerably expanded, while retaining the style of the original. More detailed presentation is given of elementary topics, to reflect the knowledge base of current students. Exercise setshave been substantially revised and enlarged, with carefully graded exercises at the end of each chapter.This is the latest addition to the growing list of Oxford undergraduate textbooks in mathematics, which includes: Biggs: Discrete Mathematics 2nd Edition, Cameron: Introduction to Algebra, Needham: Visual Complex Analysis, Kaye and Wilson: Linear Algebra, Acheson: Elementary Fluid Dynamics, Jordan and Smith: Nonlinear Ordinary Differential Equations, Smith: Numerical Solution of Partial Differential Equations, Wilson: Graphs, Colourings and the FourColour Theorem, Bishop: Neural Networks forPattern Recognition, Gelman and Nolan: Teaching Statistics.

Author 
Elias Wegert 
ISBN10 
9783034801805 
Year 
20120830 
Pages 
360 
Language 
en 
Publisher 
Springer Science & Business Media 
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This book provides a systematic introduction to functions of one complex variable. Its novel feature is the consistent use of special color representations – socalled phase portraits – which visualize functions as images on their domains. Reading Visual Complex Functions requires no prerequisites except some basic knowledge of real calculus and plane geometry. The text is selfcontained and covers all the main topics usually treated in a first course on complex analysis. With separate chapters on various construction principles, conformal mappings and Riemann surfaces it goes somewhat beyond a standard programme and leads the reader to more advanced themes. In a second storyline, running parallel to the course outlined above, one learns how properties of complex functions are reflected in and can be read off from phase portraits. The book contains more than 200 of these pictorial representations which endow individual faces to analytic functions. Phase portraits enhance the intuitive understanding of concepts in complex analysis and are expected to be useful tools for anybody working with special functions – even experienced researchers may be inspired by the pictures to new and challenging questions. Visual Complex Functions may also serve as a companion to other texts or as a reference work for advanced readers who wish to know more about phase portraits.

Author 
Joseph Bak 
ISBN10 
1441972889 
Year 
20100802 
Pages 
328 
Language 
en 
Publisher 
Springer Science & Business Media 
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This unusual and lively textbook offers a clear and intuitive approach to the classical and beautiful theory of complex variables. With very little dependence on advanced concepts from severalvariable calculus and topology, the text focuses on the authentic complexvariable ideas and techniques. Accessible to students at their early stages of mathematical study, this full first year course in complex analysis offers new and interesting motivations for classical results and introduces related topics stressing motivation and technique. Numerous illustrations, examples, and now 300 exercises, enrich the text. Students who master this textbook will emerge with an excellent grounding in complex analysis, and a solid understanding of its wide applicability.

Author 
Reinhold Remmert 
ISBN10 
9781461209393 
Year 
20121206 
Pages 
458 
Language 
en 
Publisher 
Springer Science & Business Media 
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A lively and vivid look at the material from function theory, including the residue calculus, supported by examples and practice exercises throughout. There is also ample discussion of the historical evolution of the theory, biographical sketches of important contributors, and citations  in the original language with their English translation  from their classical works. Yet the book is far from being a mere history of function theory, and even experts will find a few new or long forgotten gems here. Destined to accompany students making their way into this classical area of mathematics, the book offers quick access to the essential results for exam preparation. Teachers and interested mathematicians in finance, industry and science will profit from reading this again and again, and will refer back to it with pleasure.

Author 
Nathan Carter 
ISBN10 
088385757X 
Year 
20090409 
Pages 
297 
Language 
en 
Publisher 
MAA 
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A fullcolour visual guide for students of group theory.

Author 
Michael A. Brilleslyper 
ISBN10 
9780883857786 
Year 
20120101 
Pages 
373 
Language 
en 
Publisher 
MAA 
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This book is written for mathematics students who have encountered basic complex analysis and want to explore more advanced project and/or research topics. It could be used as (a) a supplement for a standard undergraduate complex analysis course, allowing students in groups or as individuals to explore advanced topics, (b) a project resource for a senior capstone course for mathematics majors, (c) a guide for an advanced student or a small group of students to independently choose and explore an undergraduate research topic, or (d) a portal for the mathematically curious, a handson introduction to the beauties of complex analysis. Research topics in the book include complex dynamics, minimal surfaces, fluid flows, harmonic, conformal, and polygonal mappings, and discrete complex analysis via circle packing. The nature of this book is different from many mathematics texts: the focus is on studentdriven and technologyenhanced investigation. Interlaced in the reading for each chapter are examples, exercises, explorations, and projects, nearly all linked explicitly with computer applets for visualization and handson manipulation. There are more than 15 Java applets that allow students to explore the research topics without the need for purchasing additional software.

Author 
Ian Stewart 
ISBN10 
0521287634 
Year 
19830310 
Pages 
290 
Language 
en 
Publisher 
Cambridge University Press 
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This is a very successful textbook for undergraduate students of pure mathematics. Students often find the subject of complex analysis very difficult. Here the authors, who are experienced and wellknown expositors, avoid many of such difficulties by using two principles: (1) generalising concepts familiar from real analysis; (2) adopting an approach which exhibits and makes use of the rich geometrical structure of the subject. An opening chapter provides a brief history of complex analysis which sets it in context and provides motivation.

Author 
Steven G Krantz 
ISBN10 
9781461215882 
Year 
20121206 
Pages 
290 
Language 
en 
Publisher 
Springer Science & Business Media 
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This book is written to be a convenient reference for the working scientist, student, or engineer who needs to know and use basic concepts in complex analysis. It is not a book of mathematical theory. It is instead a book of mathematical practice. All the basic ideas of complex analysis, as well as many typical applica tions, are treated. Since we are not developing theory and proofs, we have not been obliged to conform to a strict logical ordering of topics. Instead, topics have been organized for ease of reference, so that cognate topics appear in one place. Required background for reading the text is minimal: a good ground ing in (real variable) calculus will suffice. However, the reader who gets maximum utility from the book will be that reader who has had a course in complex analysis at some time in his life. This book is a handy com pendium of all basic facts about complex variable theory. But it is not a textbook, and a person would be hard put to endeavor to learn the subject by reading this book.

Author 
Jerry R. Muir, Jr. 
ISBN10 
9781118956397 
Year 
20150506 
Pages 
280 
Language 
en 
Publisher 
John Wiley & Sons 
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A thorough introduction to the theory of complex functions emphasizing the beauty, power, and counterintuitive nature of the subject Written with a readerfriendly approach, Complex Analysis: A Modern First Course in Function Theory features a selfcontained, concise development of the fundamental principles of complex analysis. After laying groundwork on complex numbers and the calculus and geometric mapping properties of functions of a complex variable, the author uses power series as a unifying theme to define and study the many rich and occasionally surprising properties of analytic functions, including the Cauchy theory and residue theorem. The book concludes with a treatment of harmonic functions and an epilogue on the Riemann mapping theorem. Thoroughly classroom tested at multiple universities, Complex Analysis: A Modern First Course in Function Theory features: Plentiful exercises, both computational and theoretical, of varying levels of difficulty, including several that could be used for student projects Numerous figures to illustrate geometric concepts and constructions used in proofs Remarks at the conclusion of each section that place the main concepts in context, compare and contrast results with the calculus of real functions, and provide historical notes Appendices on the basics of sets and functions and a handful of useful results from advanced calculus Appropriate for students majoring in pure or applied mathematics as well as physics or engineering, Complex Analysis: A Modern First Course in Function Theory is an ideal textbook for a onesemester course in complex analysis for those with a strong foundation in multivariable calculus. The logically complete book also serves as a key reference for mathematicians, physicists, and engineers and is an excellent source for anyone interested in independently learning or reviewing the beautiful subject of complex analysis.

Author 
Theodore W. Gamelin 
ISBN10 
9780387216072 
Year 
20131101 
Pages 
478 
Language 
en 
Publisher 
Springer Science & Business Media 
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An introduction to complex analysis for students with some knowledge of complex numbers from high school. It contains sixteen chapters, the first eleven of which are aimed at an upper division undergraduate audience. The remaining five chapters are designed to complete the coverage of all background necessary for passing PhD qualifying exams in complex analysis. Topics studied include Julia sets and the Mandelbrot set, Dirichlet series and the prime number theorem, and the uniformization theorem for Riemann surfaces, with emphasis placed on the three geometries: spherical, euclidean, and hyperbolic. Throughout, exercises range from the very simple to the challenging. The book is based on lectures given by the author at several universities, including UCLA, Brown University, La Plata, Buenos Aires, and the Universidad Autonomo de Valencia, Spain.

Author 
Elias M. Stein 
ISBN10 
9781400831159 
Year 
20100422 
Pages 
400 
Language 
en 
Publisher 
Princeton University Press 
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With this second volume, we enter the intriguing world of complex analysis. From the first theorems on, the elegance and sweep of the results is evident. The starting point is the simple idea of extending a function initially given for real values of the argument to one that is defined when the argument is complex. From there, one proceeds to the main properties of holomorphic functions, whose proofs are generally short and quite illuminating: the Cauchy theorems, residues, analytic continuation, the argument principle. With this background, the reader is ready to learn a wealth of additional material connecting the subject with other areas of mathematics: the Fourier transform treated by contour integration, the zeta function and the prime number theorem, and an introduction to elliptic functions culminating in their application to combinatorics and number theory. Thoroughly developing a subject with many ramifications, while striking a careful balance between conceptual insights and the technical underpinnings of rigorous analysis, Complex Analysis will be welcomed by students of mathematics, physics, engineering and other sciences. The Princeton Lectures in Analysis represents a sustained effort to introduce the core areas of mathematical analysis while also illustrating the organic unity between them. Numerous examples and applications throughout its four planned volumes, of which Complex Analysis is the second, highlight the farreaching consequences of certain ideas in analysis to other fields of mathematics and a variety of sciences. Stein and Shakarchi move from an introduction addressing Fourier series and integrals to indepth considerations of complex analysis; measure and integration theory, and Hilbert spaces; and, finally, further topics such as functional analysis, distributions and elements of probability theory.

Author 
Daniel Alpay 
ISBN10 
9783319421810 
Year 
20161127 
Pages 
596 
Language 
en 
Publisher 
Birkhäuser 
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This second edition presents a collection of exercises on the theory of analytic functions, including completed and detailed solutions. It introduces students to various applications and aspects of the theory of analytic functions not always touched on in a first course, while also addressing topics of interest to electrical engineering students (e.g., the realization of rational functions and its connections to the theory of linear systems and state space representations of such systems). It provides examples of important Hilbert spaces of analytic functions (in particular the Hardy space and the Fock space), and also includes a section reviewing essential aspects of topology, functional analysis and Lebesgue integration. Benefits of the 2nd edition Rational functions are now covered in a separate chapter. Further, the section on conformal mappings has been expanded.