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Complex Analysis (Undergraduate Texts in Mathematics) 3rd ed. 2010 Edition

by Bak (Author)
4.4 4.4 out of 5 stars 40 ratings
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Beginning with the ?rst edition of Complex Analysis, we have attempted to present the classical and beautiful theory of complex variables in the clearest and most intuitive form possible. The changes inthisedition, which include additions to ten of the nineteen chapters, are intended to provide the additional insights that can be obtainedby seeing a little more of the “bigpicture”.This includesadditional related results and occasional generalizations that place the results inaslightly broader context. The Fundamental Theorem of Algebra is enhanced by three related results. Section 1.3 offers a detailed look at the solution of the cubic equation and its role in the acceptance of complex numbers. While there is no formula for determining the rootsof a generalpolynomial,we added a section on Newton’sMethod,a numerical technique for approximating the zeroes of any polynomial. And the Gauss-Lucas Theorem provides an insight into the location of the zeroes of a polynomial and those of its derivative. Aseries of new results relate to the mapping properties of analytic functions. Arevised proof of Theorem 6.15 leads naturally to a discussion of the connection between critical points and saddle points in the complex plane. The proof of the SchwarzRe?ectionPrinciplehasbeenexpandedtoincludere?ectionacrossanalytic arcs, which plays a key role in a new section (14.3) on the mapping properties of analytic functions on closed domains. And our treatment of special mappings has been enhanced by the inclusion of Schwarz-Christoffel transformations.
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  1. ISBN-10
    1441972870
  2. ISBN-13
    978-1441972873
  3. Edition
    3rd ed. 2010
  4. Publisher
    Springer
  5. Publication date
    August 6, 2010
  6. Part of series
    Undergraduate Texts in Mathematics
  7. Language
    English
  8. Dimensions
    6.14 x 0.99 x 9.21 inches
  9. Print length
    340 pages
  10. See all details
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Editorial Reviews

Review

From the reviews of the third edition:

“The book of the known mathematicians J. Bak and D. Newman is an excellent introduction into the theory of analytic functions of one complex variable. The book is written on an elementary level and so it supports students in the early stages of their mathematical studies. … The book also contains many illustrations, examples and exercises, which give additional information and explanations.” (Konstantin Malyutin, Zentralblatt MATH, Vol. 1205, 2011)

From the Back Cover

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 several-variable calculus and topology, the text focuses on the authentic complex-variable ideas and techniques. Notable additions to "Complex Analysis, Third Edition," include: • The solution of the cubic equation and Newton’s method for approximating the zeroes of any polynomial; • Expanded treatments of the Schwarz reflection principle and of the mapping properties of analytic functions on closed domains; • An introduction to Schwarz-Christoffel transformations and to Dirichlet series; • A streamlined proof of the prime number theorem, and more. 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.

Product details

  • Publisher ‏ : ‎ Springer; 3rd ed. 2010 edition (August 6, 2010)
  • Language ‏ : ‎ English
  • Hardcover ‏ : ‎ 340 pages
  • ISBN-10 ‏ : ‎ 1441972870
  • ISBN-13 ‏ : ‎ 978-1441972873
  • Item Weight ‏ : ‎ 1.45 pounds
  • Dimensions ‏ : ‎ 6.14 x 0.99 x 9.21 inches
  • Customer Reviews:
    4.4 4.4 out of 5 stars 40 ratings

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Joseph Bak

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Customer reviews

4.4 out of 5 stars
4.4 out of 5
40 global ratings

Top reviews from the United States

Mulling
5.0 out of 5 stars For Math students - not engineers
Reviewed in the United States on September 21, 2013
Verified Purchase
This is the most crystal clear book on the topic that is in print.
For what it's worth, I feel like I'm more of an algebraist.
After spending many years feeling inadequate about my complex analysis skills, I bought this book.
I did not feel like I had hope on my side.
I used Chapters 1 - 15, plus additional topics from later chapters.
For chapters that I worked through, I did every darn problem (checking solutions against those that exist in the back of the book).

If you can work through one section (not chapter!) every few days, then you should have a pretty good general feel for the subject.

It's a beautiful book, and a great preparation for Lars Ahlfors' "Complex Analysis" book.
11 people found this helpful
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Amazon Customer
5.0 out of 5 stars Course I took a long time ago. I guess 5 stars.
Reviewed in the United States on December 23, 2020
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Course I took a long time ago. I guess 5 stars.
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James M. Cargal
5.0 out of 5 stars Excellent!
Reviewed in the United States on June 19, 2000
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This is a brief text on complex analysis aimed at the traditional junior-senior course. As a text it may be a little too succinct for the average undergraduate. For example, I have no intention of teaching out of it. However, its clarity and presentation is absolutely refreshing. I think it is one of the best books written on complex analysis in the last twenty years. I recommend this book to any student of complex analysis.
32 people found this helpful
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Simon YAU
5.0 out of 5 stars Great
Reviewed in the United States on October 24, 2019
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It is good to read to know what complex analysis is.
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Alexander C. Zorach
3.0 out of 5 stars Hard to follow, not comprehensive enough
Reviewed in the United States on June 6, 2006
This book is disappointing, especially after encountering Newman's "Analytic Number Theory", which is a wonderful book. This book takes the readers on a concise, linear journey through Complex analysis to a few key theorems at the end, but does not do justice to the richness or diversity of the subject. This book will be especially lacking to students studying complex analysis for purposes related to applied mathematics.

The prose in the book is clear, but at times, as early as chapters 2 and 3, the equations are dense for an undergraduate text, with some steps less than obvious. There is a lack of motivation for the direction of development chosen in chapters 4-6, possibly a little of 2, 7, and 8 as well. Results are proven three or more times in cases of increasing generality. While this makes the theorems easy to follow, the redundancy may be confusing for a student studying the material for the first time. The authors do not provide much of a preview of what is to come, as I think authors of an undergraduate text should (and many, such as Gamelin, do).

This book is so small and compact that I question the authors' judgment in leaving out these various explanations--little would be lost and much gained by additional explanations. This makes me wonder what the intended audience is. I think anyone who is able to follow this book without trouble would also have no trouble following a more advanced and comprehensive book. There are a number of more advanced books that are actually much easier to follow. This brings me to my next comment:

This book leaves out a lot of important topics; it is far from comprehensive. There are not very many exercises either, and the exercises are mostly related to the material in simple ways.

For those studying complex analysis for the first time, I would recommend the Gamelin book over this one; its proofs are much easier to follow, it contains much more explanatory prose. It moves slower but it is much more comprehensive and covers more advanced material, and it is better suited to students with diverse interests and different backgrounds. I also recommend the Churchhill text as a straightforward book covering the basics. Advanced students might want to use the classic Ahlfors text.
17 people found this helpful
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Angelina Schenck
5.0 out of 5 stars Great
Reviewed in the United States on January 6, 2019
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This is a great book for Complex Analysis.
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Guilherme Arcoverde
5.0 out of 5 stars Five Stars
Reviewed in the United States on March 29, 2015
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Excellent book! Direct to the point and very complete for undergraduate course. The exercises are also good.
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Alicia
2.0 out of 5 stars Difficult to understand problems because too many steps are skipped
Reviewed in the United States on December 17, 2017
Verified Purchase
I had to use this book for my course in Complex Variables at the University of Iowa. I got an A in the course, but I did not find this book to be helpful. The book does not appear to be used often enough for there to be a solution guide online, but I was able to find these homework problems discussed in online forums and sometimes on the course websites of other universities.

There are solutions to about half of the problems in the back, but only the answer is given [for example: (+/-) (1+3i)], which tells you whether or not you were correct, but it does not tell you where you went wrong in your several pages of work. If you don't know how to solve the problem, the solution alone is not helpful.

The examples given in the book are also not helpful. They always skip over many steps, and it took me at least 30 minutes to reconstruct all the missing steps in order to figure out what the book examples were trying to say. In addition, many of the examples give no explanation at all; they merely list an example problem and its solution. Without showing how to solve the problem, the book does not help you learn how to solve problems for your exams and homework.

Because of these problems, I rarely used this book in my course. I completed homework problems from the book each week, but I used online sources to figure out how to solve the problems. I used this book mainly to check that my answers were correct.
7 people found this helpful
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Top reviews from other countries

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Jason Haradyn
5.0 out of 5 stars Five Stars
Reviewed in Canada on September 8, 2017
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Great book!
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5.0 out of 5 stars Ottimo
Reviewed in Italy on March 31, 2017
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Sembra del tutto nuovo!!! Eccellente acquisto! 😊Nessuna sottolineatura, pagine ancora profumate di stampa fresca, copertina perfettamente integra.
Che altro dire?
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Martin Juul
5.0 out of 5 stars Readable but thorough complex analysis,
Reviewed in the United Kingdom on July 2, 2008
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All but the mathematical purist is going to like this book, since it is focusing on illustrating the simplicity of complex analysis, rather than giving the shortest possible account. This means that the closed curve theorem and Cauchy's integral formula are proved several times over the first 100 pages, starting with the simplest possible case and ending up with the general case. The later profs goes more or less like this: "This is the same proof as for theorem XX", and only the proof of the simple case is complete(grosely oversimplified!!). The result is a simple, easy to read, and really well structured account of the basics of complex analysis, suited for senior undergraduates(Infinite series and many other basic analytical tools are required). The book is also quite suitable for self study!
7 people found this helpful
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Easygoing
5.0 out of 5 stars Fast delivery
Reviewed in the United Kingdom on August 22, 2013
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It is a good book for Complex Analysis. I strongly recommend to people who are doing Complex Analysis; it give you a basic introduction and background reading about it.
One person found this helpful
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Davide Bondoni
3.0 out of 5 stars Delusion
Reviewed in Germany on April 8, 2011
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I read a good review of this book on the Newsletter of the European Mathematical Society, so I bought it. Frankly, I don't like very much the method used by the authors. More than a textbook is an exercises-book. The results are not very explained and the matter is compressed in little space. Albeit the number of pages, there is not much text. This book could be useful for a teacher in preparing a lecture, because is fragmentary but well structured. For a person able to read German, I suggest "Einführung in die Funktionentheorie" by Hermann Weyl. Another good book (as introduction) is Rainer Wüst's "Mathematik für Physiker und Mathematiker", volume 2. A classic is Shilov's "Elementary Real and Complex Analysis". A gap of the book in issue is the lack of an index of the symbols and the lack of explanations of the symbols. The authors take for granted too much things from Analysis and Algebra. A trivial example: what does it mean |z|? The authors don't give a definition of the module in the field of complex numbers. They can take for granted the REAL analysis, but not the complex analysis. The symbol for the neighbourhood is very strange and not intuitive at all. Summing up: the text is very condensed and not adapt for a newbie. Nevertheless, the great amount of exercises is very interesting.
4 people found this helpful
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