Complex Variables and Applications

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!


Very clear, great for learning and understanding quickly, a bit slow at times   June 15, 2006
Alexander C. Zorach (Lancaster, PA)
16 out of 16 found this review helpful

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.


One of the best math textbooks I've ever read   February 16, 2002
Todd Ebert (Long Beach California)
21 out of 24 found this review helpful

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.


Great text, great problems, great all around book   January 18, 2004
David Diez (Los Angeles, CA)
13 out of 15 found this review helpful

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!


a wowser for the "application" people   October 2, 2001
Jihwan Myung (Seattle, WA USA)
7 out of 7 found this review helpful

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.