Complex Analysis [Hardcover]

Theodore W. Gamelin (Author)

Book Description

ISBN-10: 0387950931 | ISBN-13: 978-0387950938 | Publication Date: April 15, 2001 | Edition: 1

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.

Editorial Reviews

Review

From the reviews:

"More than 800 well-chosen exercises with 20 pages of hints and solutions, together with clear and concise expositions of many results, makes this book enjoyable even for specialists in the field. The book is recommended for libraries, students, and teachers of both undergraduate and graduate courses."
Newsletter of the EMS, Issue 42, December 2001

"This is a wonderful book about the fundamentals of complex analysis. It touches all essential parts of complex function theory, and very often goes deeper into the subject than most elementary texts. a ] I find the pace, style and didactics in the presentation perfect. a ] All in all, this is one of the best if not the best book on the elementary theory of complex functions, and it can serve as a textbook as well as a reference book." (V. Totik, Mathematical Reviews, Issue 2002 h)

"This is a well organized textbook on complex analysis a ] . Each part of the book contains some interesting exercises which give many new insights into further developments and enhance the usefulness of the book. At the end of the book there are hints and solutions for selected exercises." (F. Haslinger, Internationale Mathematische Nachrichten, Vol. 56 (191), 2002)

"As the book begins with the rudiments of the subject and goes upto an advanced level, it will be equally useful to the undergraduates and to students at the pre Ph. D. level. The numerous applications to Physics found in the book redound to its value in the hands of students of Mathematics, Physics and Engineering alike." (K. S. Padmanabhan, Journal of the Indian Academy of Mathematics, Vol. 24 (1), 2002)

"This is a beautiful book which provides a very goodintroduction to complex analysis for students with some familiarity with complex numbers. a ] The book is clearly written, with rigorous proofs, in a pleasant and accessible style. It is warmly recommended to students and all researchers in complex analysis." (Gabriela Kohr, The Mathematical Gazette, Vol. 86 (506), 2002)

"The author of this fine textbook is a prominent function theorist. He leads the reader in a careful but far-sighted way from the elements of complex analysis to advanced topics in function theory and at some points to topics of modern research. a ] We found, however, that a special feature of the book is the wealth of exercises at the end of each section a ] . Altogether, the author has given us a wonderful textbook for use in classroom and in seminars." (Dieter Gaier, Zentralblatt MATH, Vol. 978, 2002)

"The author wishes to provide beginners with standard material in a rather flexible course. a ] The preface contains useful hints for both instructors and students. More than 800 well-chosen exercises with 20 pages of hints and solutions, together with clear and concise expositions of many results, make this book enjoyable even for specialists in the field. The book is recommended for libraries, students, and teachers of both undergraduate and graduate courses." (Acta Scientiarum Mathematicarum, Vol. 68, 2002)

"This book has the somewhat unusual aim of providing a primer in complex analysis at three different levels - a basic undergraduate introduction, a course for those who have decided to specialise as part of their first degree and a more demanding treatment of postgraduate topics. a ] I can certainly recommend this book to all those who wish to experience(in the authora (TM)s own words) the a ~fascinating and wonderful worlda (TM) of complex analysis, a ~filled with broad avenues and narrow backstreets leading to intellectual excitement.a (TM)" (Gerry Leversha, European Mathematical Society Newsletter, Issue 42, December 2001) --This text refers to the Paperback edition.

Product Details

• Hardcover: 496 pages
• Publisher: Springer; 1 edition (April 15, 2001)
• Language: English
• ISBN-10: 0387950931
• ISBN-13: 978-0387950938
• Product Dimensions: 9.6 x 6.4 x 1.1 inches
• Shipping Weight: 1.9 pounds
• Average Customer Review: 4.1 out of 5 stars  See all reviews (14 customer reviews)
• Amazon Best Sellers Rank: #1,205,946 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

28 of 30 people found the following review helpful:

4.0 out of 5 stars Interesting Book, December 12, 2002

By A Customer

This review is from: Complex Analysis (Paperback)

Gamelin's book covers an interesting and wide range of topics in a somewhat unorthodox manner. Examples: Riemann surfaces are introduced in the first chapter, whereas winding numbers don't make an appearance until halfway into the book. Cauchy's theorem and its kin are instead developed in the context of piecewise-smooth boundaries of domains (in particular, simple closed curves) and only later generalized to arbitrary closed paths, almost as an afterthought.

In general, the author successfully conveys the spirit of the subject, and manages to do so quite efficiently. It's not the most painstakingly rigorous text out there, and the reader is expected to fill in some of the details himself, but the payoff is that a lot of ground is covered without getting bogged down in technicalities. In many books on this subject it can be tough to see the forest for the trees. This one is a pleasant exception.

There are a lot of good complex analysis books out there: Conway, Ahlfors, Remmert, Palka, Narasimhan, the second half of big Rudin, and of course Needham's "Visual Complex Analysis." (And many others that are well-regarded but that I have not looked at, such as Lang and Jones/Singerman, as well as the old classics by Hille, Knopp, Cartan, Saks and Zygmund.) Every one of these has its own perspective, and complex analysis is a big, multifaceted subject that is perhaps best studied from multiple points of view. Anyone wanting to learn this subject well will benefit from having several books at hand.

Gamelin's contribution to the pantheon is not revolutionary, but it does collect between its pages a wide assortment of topics not generally found in a single text. The reader is whisked from the basics to the Riemann mapping theorem in 300 pages with surprising ease. The ensuing "topics" chapters include a dynamical systems-flavored section on Julia sets and fractals; special functions (gamma, zeta, etc.); the prime number theorem; and an introduction to abstract Riemann surfaces.

Overall a fun text. Certainly not the only complex analysis book one should read, but then again the the same can be said of any complex analysis book. My only real complaint is that the selection of exercises is somewhat small in some chapters.

19 of 20 people found the following review helpful:

4.0 out of 5 stars One for your reference shelf., March 3, 2006

By 

J. K. Fisher - See all my reviews
(REAL NAME)   

This review is from: Complex Analysis (Paperback)

Gamelin's 'Complex Analysis' is purported to be a text that, while it falls in the UTM series, can really be used for anything up through the Ph.D. qualifying exam level. This is true, but there are some problems with this text that would keep it from true brilliance.

The text covers a superb variety of topics, from the basic arithmetic up through graduate level complex analysis. The exercises to be found at the end of each section are likewise excellently chosen, and give students some great 'hands-on' practice using complex analysis. The exercises are often a little too easy, at least early on in the text, and can lull a reader into a great false sense of security with the field.

The approach that Gamelin takes makes for a very readable book, one that can easily give you an idea of what is happening. The problem however is that there is a level of generality--as well as rigor--that are sorely lacking from this text. The results that Gamelin presents can (here and there) be generalized without too much work, which really should be done for a graduate course. Similarly, his writing rather often seems to lack any semblence of the rigor that students of analysis would normally expect. Justifications would be a better word than proofs for many of the ways he convinces a reader a theorem must be true.

This does not detract from the value of the book however, but merely shift it to a different role in one's study of complex analysis. This is a great companion book--one that should find a well worn home on your reference shelf. It is an excellent book to go to when you want to get an idea of what a concept means, and then get a variety of doable problems that relate to that idea.

17 of 18 people found the following review helpful:

2.0 out of 5 stars Not my taste, July 18, 2007

By 

J. Katz - See all my reviews
(REAL NAME)   

This review is from: Complex Analysis (Paperback)

Although I can see what others might like in this book, I did not care for it. (To be fair, I am not sure how much of this is the book's fault and how much is the fault of the subject.) I was looking for something a bit more mathematical, and more along the lines of (say) Rudin's real analysis, and instead this book was less formal than I would have liked, seemed geared toward applications of the material in physics and engineering, and was more calculus-oriented. (Maybe the latter is inherent in the subject, I don't know.)

I prefer mathematics textbooks written in the definition-theorem-proof (followed by examples) style, and this book is not that. Terms are sometimes defined only intuitively (I don't mind intuition in addition to a formal definition, but I do mind when it is in place of a formal definition), and there are no "marked" proofs (instead, proofs are supposed to follow from the surrounding discussion, which is sometimes formal and sometimes less so).

The index was awful. I was looking for a proof of the fundamental theorem of algebra and found only one reference: to page 4, where the author promises that we will see many proofs of this theorem. (Where? Who knows!)