Complex Variables and Applications

Rating:
Summary: excellent! (as a 1st look)
Review: This book would be really good for a physics or engineering student, or for a math student as an intro since it isn't as theoretical as other books. Everything is explained very clearly, with lots of examples but still with some hard/interesting problems also. The first 7 chapters cover all the theory, which is the usual stuff like complex numbers, differentiability & Cauchy-Riemann equations, integration & Cauchy integral forumla/theorem, series, Taylor series, Laurent series, residues, poles, solving real integrals with residues, etc etc. Chapters 8-12 cover applications which I admit I don't know a lot about, but if the 1st half of the book is any indication, they are good also. The text covers everything in a CONCRETE way, as opposed to an abstract way using concepts from topology like connectedness, compactness, open/closed sets, star-shaped sets, etc. For that stuff I used GJO Jameson's Complex Functions text & I also like the one by Ahlfors & the one by Jerrold Marsden. I think every scientist or engineer would find this Brown/Churchill book very helpful. :-)

Rating:
Summary: Excellent intro. to complex analysis!
Review: 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!