Function Theory of One Complex Variable: Third Edition (Graduate Studies in Mathematics, 40)

This book has great exercises, and covers all of the important stuff. If you plan on reading it cover to cover it is a great choice since it has solid, readable prose that explains the theorems. If you want a book with just important theorems and concise results, this is not a good choice.

An amazing book on learning the basics on Complex Analysis of one variable.It is written in a modern and comprehensive way.

I recently used this book, as a student, for a graduate complex analysis course. It seems that it is becoming more popular, and ranks as one of the standard complex analysis texts, and it's not hard to see why: a casual perusal makes it seem like a sensible, gentle-yet-rigorous introductory text which, unlike some of its competitors, doesn't presume any but the most basic knowledge of topology, which some beginning graduate students are weak in. It starts out with the most elementary fundamentals of the subject, not even presuming the reader knows what a complex number is, and in the first few chapters the exposition seems very thorough. It also has some fairly modern material in the later chapters. At the start, I myself was very optimistic about it.However, after reading a bit further I was disappointed to find it severely flawed. It seems that the beginning was written very carefully, but by the middle chapters it becomes unforgivably sloppy. It is riddled with errors of a kind that don't belong in the second edition of anything, and more importantly, the presentation is disorganized and uneven. Most of the proofs are inelegant and could be shortened considerably; many of them contain completely unnecessary statements that serve no apparent purpose, formal or intuitive, and it seems as though the authors simply didn't put much thought into cleaning them up. The prose similarly contains a lot of hemming and hawing and little material of use. The presence of elementary material alongside advanced topics is incongruous. Far from easing the novice into difficult subject matter, the authors present basic, even inane material right next to serious theorems with hardly any segue, and to make matters worse, skip the essential points of many proofs on the basis of an assumption that the reader has an extremely thorough knowledge of analysis at the level of Rudin; this in a book that asks the reader to perform such puerile tasks as differentiating polynomials and deliberately avoids topology. Speaking of which, the postponement of the topological content, which is pretty elementary anyway, to a sloppy chapter almost halfway through ends up being utterly pointless. Through most of the book we are doing everything on circles instead of general curves, and it is implicit that the reader is simply supposed to understand that any homotopic curve would do just as well - or that it wouldn't, depending on the context. What's worse, the omission of rigorous proofs using general curves deprives the novice of an appreciation of the interplay between complex analysis and topology, which he apparently is supposed to possess innately. The result of all this is a jagged jumble of text that beginners will find incomprehensible, advanced students will find obnoxious, and nobody will find instructive.I would strongly advise anyone considering assigning this book for a course to think twice. There are dozens of highly polished, intelligently written books on complex analysis. This is not one of them.

The hardcover, and the book itself are good. We use it as a textbook; the content is sort of hard yet sufficient motivation is provided.

i love this boojk

It's riddled with terrible pedagogy and insufficient to provide even a third course in complex analysis.

This book is rather unorthodox in a number of respects, but it has become one of my favourite texts in complex analysis. The authors claim that their motivation for their presentation of the subject is to emphasize the interconnectedness of complex function theory with multivariable calculus, and de-emphasize the connection with topology. While I do not exactly agree with these goals, I think they do an excellent job of acheiving them. My only complaint about the book is that a few proofs in early chapters result in a sea of differential operators that is resolved by a plug-and-chug computation, something I'd always rather avoid.The level of the book is elementary, especially for a graduate text, and I appreciate the authors for making honest and reasonable claims about the accessibility of their book. This book would probably even work well for someone who has not had a prior course in complex analysis, such as senior undergraduates. Some of the more advanced topics are presented in clearer ways in this book than I have seen elsewhere.This book has a wealth of exercises, and the difficulty level is somewhat inconsistent. Some of the exercises are outright inane--possibly inappropriate for a graduate-level text, but useful for rote practice. Others are more interesting. I appreciate, however, the inclusion of more elementary exercises: many graduate texts have the problem of not including enough such exercises, which can make it hard for students to master the fundamentals. This book avoids this pitfall.The best part about this book is the prose. This book is well-written and is a pleasure to read. Theorems and results are well-motivated, and necessary nuances are effectively communicated through the text. The authors do not over-emphasize equations: they use words in a proof when they are clearer.The book is well-indexed and comprehensive. A student with prior background in Complex analysis might want to read the first chapter to get familiar with the authors' incessant use of differential operators, and then feel free to skip around. I think this book would make an excellent textbook for a course in complex analysis, although the unusual development in the first chapter might annoy some professors.

I had this textbook during my first experience with Complex Analysis (or variables for that matter!). For most of the semester I had to use other books to understand HW problems. The authors make too many assumptions of the readers, leave many proofs to exercises, and constantly say that missing steps are "obvious". Totally unreadable. I don't recommend this book to anyone.