Complex Analysis (Universitext) Review

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(More customer reviews)This book was originally written in German, and the German version is just incredible: a real gem. Good reason to translate it! Unfortunately, this is one of the worst translations from German I have seen. Some of it is just awkward grammar, which the reader may be able to ignore. But, there are also some words and phrases which are translated incorrectly.
For example "Paragraphen" in German does not mean paragraph in English, it means section. But in this book it is translated as paragraph. Try looking for something at the end of a paragraph or in the previous paragraph, when you should actually be looking at the end of the section or in the previous section. An example of this can be found in the explanation of the addition theorem for complex exponents (p. 27). The English text claims there is a remark concerning this at the end of the "paragraph." The paragraph ends and there is no remark. Turn to the end of the section (p. 31) and you will find the remark just above the exercises.
My advice is, if you can read German, get the German version! If you can't read German, you can still get the English version, but you will have to be very patient with the mistakes incurred in the translation (not to be found in the German original). If you own this book, you should systematically go through it and replace "paragraph" everywhere with "section." Most of the other translation mistakes can be figured out by context.

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All needed notions are developed within the book: with the exception of fundamentals which are presented in introductory lectures, no other knowledge is assumedProvides a more in-depth introduction to the subject than other existing books in this areaOver 400 exercises including hints for solutions are included

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Complex Analysis Review

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(More customer reviews)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.

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The book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. The first part comprises the basic core of a course in complex analysis for junior and senior undergraduates. The second part includes various more specialized topics as the argument principle the Poisson integral, and the Riemann mapping theorem. The third part consists of a selection of topics designed to complete the coverage of all background necessary for passing Ph.D. qualifying exams in complex analysis.

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Introduction to Complex Analysis Review

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(More customer reviews)The existing reviews refer to the 1st edition of this book, which I agree was a difficult read, though still accessible to undergraduates. The new book has been revised substantially to make it more readable, with a much more leisurely introduction and better partitioning of tougher material (which can be omitted by those such as physicists and engineers who require only a working knowledge of the subject). If anything I feel the result is too dumbed down; Priestley is loathe even to make use of such basic tools from real analysis as uniform convergence. Nevertheless, the second half of the book is more adventurous, making the totality a guide for an excellent undergrad class, such as the Oxford one on which the book was based. Beware: there is a large number of typos, which one must hope will be corrected in subsequent printings. Usually it will not be too challenging to circumnavigate these.

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Complex analysis is a classic and central area of mathematics, which is studies 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 sets have been substantially revised and enlarged, with carefully graded exercises at the end of each chapter.

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Algebraic Curves and Riemann Surfaces (Graduate Studies in Mathematics, Vol 5) Review

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(More customer reviews)If you want to learn the basic properties of compact Riemann surfaces this is the book to read. If you want to know the "motivations" of modern Algebraic geometry this is again a book to read.
First of all the pace and the style are very casual. You really don't feel overwhelm by a mountain of definitions. The author always favor simplicity and concreteness instead of abstractions and generality. This is really a book that I should have read before taking a class on Schemes. For exemple in the context of Riemann surfaces an "very ample divisor" is simply a linear system without fixed base point that gives rise to an holomorphic embedding. This definition (at least for me) is much much more satisfactory and illuminating than the definition of a very ample sheaf that you can find in Hartshorne (even though his definition is much more general).
There is a very nice chapter on meromorphic differentials which explains how those object can be used to define line integral on any riemann surface. Topics like divisors, Riemann-Roch and curves are treated with a lot of depth. There are not a lot of pictures but having pictures supported by an unclear text is quite useless. Here the writing is so clear (not to say flawless) that on the first reading you really get the idea of what's going on.
There are very few mistakes in this book which is another reason why I like it. I'm really pissed off by those mathematicians
that are rushing to publish their books crowded by mistakes.
But don't get me wrong, I don't have anything against mathematicians that are writing books (this is a learning experience) but don't feel force to publish them unless they are very polished and "innovative".
Finally the last chapters treat of Abel's theorem ( which tells us exactly when a divisor is principal), Sheaves, Cech cohomologies and line bundles. Again the exposition is very well
motivated with a good supply of interesting exemples.
This is the best book that I read on subject and honestly if professor Miranda is writing another book related to my field of research you can be sure that I will have it my collection.
Hugo Chapdelaine,
McGill



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In this book, Miranda takes the approach that algebraic curves are best encountered for the first time over the complex numbers, where the reader's classical intuition about surfaces, integration, and other concepts can be brought into play. Therefore, many examples of algebraic curves are presented in the first chapters. In this way, the book begins as a primer on Riemann surfaces, with complex charts and meromorphic functions taking center stage. But the main examples come from projective curves, and slowly but surely the text moves toward the algebraic category. Proofs of the Riemann-Roch and Serre Duality Theorems are presented in an algebraic manner, via an adaptation of the adelic proof, expressed completely in terms of solving a Mittag-Leffler problem. Sheaves and cohomology are introduced as a unifying device in the latter chapters, so that their utility and naturalness are immediately obvious. Requiring a background of a one semester of complex variable! theory and a year of abstract algebra, this is an excellent graduate textbook for a second-semester course in complex variables or a year-long course in algebraic geometry.

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