Customer Reviews

Complex Variables and Applications

5 star:		(22)
4 star:		(9)
3 star:		(8)
2 star:		(4)
1 star:		(4)

 

 

This book is simply clearer than any other complex analysis book I've read, although it's not particularly advanced or concise.

This book is a great text for undergraduates studying complex analysis for the first time. It does not assume a strong background in rigorous analysis, making the material accessible to a wider audience.

At times I find that this book moves a bit slow for my personal taste, but what it loses in speed it makes up for in clarity. The explanations are always clear. I find that I never get stuck in a proof in this book. If there is a certain topic that I absolutely must understand, and I want to understand in a straightforward, useful way, as quick as possible, I turn to this book.

I would recommend this book for self-study as well as a textbook at the introductory level. It is not a particularly advanced book, and is not comprehensive as a reference for more advanced students, nor would it be a great choice for a graduate or advanced course.

This course was my first exposure to the mathematical field of analysis at the undergraduate level, and our school ditched Gamelin's book used two years ago in favor of this book. Just to give you an idea of the difference a book makes (it was the same teacher for both courses, mind you): when Gamelin was used, EVERYONE dropped out of the course; when Brown/Churchill was used, only one person dropped the course and half the class received A's!

Truly, this is a remarkable shift, and this book had a lot to do with it. I thought the organization was flawless (note: you will have to go through the book in order, as many examples depend on previous material), and starting from the beginning with the definition of a complex number was definitely the way to go, as about 1/3 of my class had never seen a complex number before. I loved the fact that there were many examples worked out (never explicitly showing people how to do the end-of-section exercises, but showing them the methods for where to go) and the major theorems were alloted many pages for clear proofs with diagrams and detailed explanations (an entire section was devoted to a proof of the Cauchy-Goursat theorem!). Also, the choices of problems were superb, with some routine exercises meant to get you thinking along the right tracks followed by some very difficult ones. Basically, enough to challenge even the ablest math student, but enough for the average one to get a grasp on the concepts as well.

The book also provides an advantage for the instructor as to what applications to teach. Granted, chapters 1-6 cover almost all the theory, but 7-12 are all applications (7 is "usually" considered theoretical as well, but it is called "applications of residues!") in physics, advanced calculus and geometry, and engineering. So, a professor could choose to emphasize only the theoretical parts and save the apps. for independent study (which my prof. did) or could teach the relevant theories coupled with some of the applications (conformal mapping with fluid flow and heat flow, for example). It truly is a versatile book.

I noticed a complaint on here about not having enough examples or worked-out proofs. Well, to that individual (and any others who might be having the same problem), this book is meant for an upper-level undergraduate course, which means that there are going to be less examples worked out in great detail, the proofs may just be thumbnail sketches, and the problems will not have a quick reference page in the chapter for a formula or method like in calculus, for example; even though the book is versatile, a lot of the learning still falls on the student's shoulders.

My one and only gripe is that the book didn't take a lot of time to spell out how to perform a delta-epsilon proof for limits, which is one of the basic proofs in analysis. But, luckily, I had a very patient instructor who was willing to walk it through with me (most of the rest of the class had already had real analysis, so they didn't need to go over it). But, still, it's not enough to take it down a star, in my opinion.

They say this book is among the canon of undergraduate mathematics, and I can certainly see why. What a great introduction to complex analysis! This book will definitely be accompanying me to grad school!

I read this book in preparation for an analysis qualifying exam, and found that the examples, exercises, and explanations provided made the entire subject seem both easy and interesting. For a beginning student of complex analysis, I do not see any better option. Moreover, I believe every future mathematics-book author should study this book as an exercise on "what to do". Finally part II of Lang's "Complex Analysis" has alot of interesting advanced material related to geometric function theory, and would make a good follow-up to this book.

This is a nice book for understanding the basic concepts of early Complex Analysis. The integral formulas, residue theorems, Fourier Analysis, infinite series/sequences, etc. are all covered. There are plenty of exercises and examples. Everything is so clearly presented, that it is easy for people with very little background in analysis to read (ie, you don't really need real analysis before reading this book).

My problems with the book are thus: There are very, very few proofs to any of the theorems. I'd rather have more proofs than examples. The problems are almost all computational. Almost none of the exercises require much thought, although some of them will take a while to do. There is no discussion on the importance of certain topics to the wider context of math. No discussion of certain standard complex-valued functions like elliptic functions, zeta functions, or gamma functions.

If anything, I see this as mostly a how-to book for engineers and physicists who come across complex variables in their work. For math students, I'd recommend the books by Shakarchi/Stein, Lang, Conway, Ahlfors.

I am a Ph.D. student in physics and used this book for an undergraduate course in mathematical methods for physics majors. This book is an excellent introduction to complex variables for physics and mathematics students. It is clear, concise and well-written. The proofs are easy to follow (but that also reflects the subject-matter). The problems are very good too and the answers are provided right in the text, which is very helpful for independent study.

Complete with the physicists' rough-and-ready kinda proofs. Discussions are easy to follow. (I have tried a few other books earlier.) Examples and exercises are balanced & comprehensive. Could be complemented with Spiegel's Complex Variables.

I chose this for self-study after some frustrating experience with a course at the math department. It worked wonderfully.

Brown and Churchill's book is neither rigorous nor intuitive; it is a true pedagogical nightmare. The authors are extremely sloppy with their exposition, structure and rigor.
Some trivial results are "proved" with pedantic detail, but even there the proof is not exhaustive. As an example, to prove sufficient conditions for differentiability (p. 63-5), two pages are devoted to setting up some elaborate structure, but the real meat (basically Taylor's theorem) is not even mentioned, rather the authors cite another text. Similarly, equality of mixed partial derivatives is waved off as "a theorem in advanced calculus." What is complex analysis but advanced calculus, and why do the authors here devote space to prove thoroughly trivial results (e.g. limit of sums converges to sum of limits), while leaving out other important foundations?
A similar sloppiness is shown even in those results that are more fully proved in the book. For example, the theorem presented in Section 26 depends on a theorem in Section 68! Pedagogically this is inexcusable, as the authors introduce these results willy-nilly, not as a coherent whole; the book must be read at least twice to check its consistency!
The layout of the book is awfully confusing. There is practically no white space, and a single font and font size are used throughout the book, for explanations, theorems, examples and exercises. Examples sometimes are placed within the section they illustrate, and sometimes bizarrely they are given their own section. This means that the table of contents cannot indicate the relative importance of book content. Likewise proofs are sometimes given their own sections, sometimes buried in an overly large section.
The exercises are mostly computational, and usually they are spoiled by "hints" that are so exhaustive that the only thing left for the student to do is to move some symbols around as directed by the book.
In general, the book causes both my mathematically rigorous colleagues and my application oriented colleagues to cringe in pain. Compared to some other works on analysis, this volume is a true abomination. Walter Rudin's Principles of Mathematical Analysis is an exquisite, mathematically thorough and rigorous treatise on the subject, in which practically every exercise is meaningful. That the publisher of the present book dares to charge as much money for this seventy-year old volume as Rudin's book costs is farcical.

This text is for use in an undergrad course or an early grad course. It is written very clearly and has LOTS of problems. These authors' problems fit along perfectly with the text; the examples lead to the early problems and the earlier problems lead smoothly to the more advanced problems. I am very pleased with this text and its clear setup, and I have been using it for self-study. I hardly miss the teacher!

I used this text for an undergrad course, and I thoroughly enjoyed it. The book, as the title suggests, is ideal for physicists, engineers, and applied mathematicians, in that it hits the important and powerful features of complex analysis which are tremendously useful for applied work. What's more, it succeeds in doing so without sacrificing mathematical rigor, and without getting bogged down on stodgy formalism.

As for the scope of the book, I believe it can be fairly stated that just about everything in the book should be studied and mastered by readers doing applied work. From my own experience, everything covered in the book has turned out to be relevant at one point or another. It can be said without exaggeration that this book is a gold mine.

Buy it. It's worth every penny. As George Foreman would say: I guarantee it.

This book manages to provide an in depth treatise of some of the main application of complex analysis. The pace is fast, but the exercises and their solutions are well done, so self-study is quite a legitimate undertaking. I would not recommend this for someone who is new to the study of complex numbers, due to the fast paced nature. The book is more aimed at those who have already had some exposure to complex numbers, but haven't dealt with the analytical side. An undergraduate at University, for instance. (I have used this book as a textbook, doing undergraduate complex analysis.) The only gripe I had with the text is that the last part of the book delves into applications of complex analysis, when I was wanting more theory. So if you are a theory junkie, there maybe be better books out there, however this book does provide good tutelage in basic complex analysis.