# What is the best way to supplement a complex variables class to make it more complete for a math major?

For the upcoming semester I plan on a taking a “complex variables” course that many people, including myself, would not consider a true complex analysis class. I know that the course will likely use a text similar those by Saff & Snider or Brown & Churchill because it is more of a survey class meant to give the basics for leading into to true complex analysis classes and giving the appropriate tools for physicists and engineers. As someone interested in theoretical mathematics, I naturally want to expand my knowledge beyond what is taught, see a more rigorous presentation of the material, have applications leaning more toward number theory than physics, and see topological constructions in action. I know that Ahlfors’ Complex Analysis is the a very common text instructors and students turn to for what I am looking for, but it is very expensive ($200 USD + for a ~300 page text), and I have heard people describe it as “difficult” for independent study unless you really know what you’re doing beforehand. Is there a better text for me to follow? I see that MIT has its 18.112 course (Functions of a Complex Variable), an undergraduate level course based on Ahlfors, listed on OCW, so I would have something to follow and test myself on, but I would prefer to not use Ahlfors. I have seen recommendations to other people to use Visual Complex Analysis for self-study, but this book is still more directed at undergraduate physics students and the like. What are the best alternatives to a text like Ahlfors? Which are the best suited for independent study for someone working alongside a less mathematically rigorous course? Which are the more comprehensive? Are there any that follow naturally from where books like that by Brown & Church leave off? Which are the most comprehensive, and are there any that lead into analytic number theory or give a taste of complex analysis in several variables? • If you want to see connections between complex analysis and number theory, look at Lang's Complex Analysis, which at the end has a treatment of the prime number theorem. I think his treatment of integrals using the residue theorem is very nice (varied worked examples). – KCd Dec 9 '13 at 1:44 • If you've learned real analysis, I might strongly recommend Rudin's Real and Complex Analysis. – user98602 Dec 9 '13 at 1:45 • I personally found learning from Rudin to be difficult, although it is very popular. You might try Conway's Complex Functions of One Variable. – D Wiggles Dec 9 '13 at 2:14 • I was just about to ask about Conway’s book. I saw that it covered more than Ahlfors and is only a quarter the market price. I know Ahlfors has become almost a classic text. What is the general consensus on Conway’s books? I see that it comes in two volumes and appears highly comprehensive while only assuming basic knowledge of real analysis. This seems ideal, so I think that this may be the way to go. After working through a large part of Conway’s books would Rudin then be redundant? I know that this is a lot of advanced planning, but I like to have a plan to tackle a broad subject. – Galois' Canada Goose Dec 9 '13 at 2:59 • If you want Ahlfors, you can get it for about \$20, not \$200: amazon.com/Complex-Analysis-L-Ahlfors/dp/0070850089/… – Potato Dec 9 '13 at 3:45 ## 4 Answers I recommend Stein and Shakarchi's Complex Analysis. It's clear, easy to read, and gives a proof of the Prime Number Theorem. It also has a little more material on analytic number theory in the last chapter, on representations as sums of squares (via theta functions). I would supplement this with the material on Cauchy's theorem in Ahlfors' text. Stein and Shakarchi give a handwavey proof of a simple version in their text, which I think is appropriate for a first read, and leave some remarks on more general versions to their appendix. I don't remember if they actually prove a more general version in that appendix, but Ahlfors definitely does. • I have heard of the Elias Stein book, but I am not too familiar with it, but that you suggest supplementing it with Ahlfors’ book makes me almost feel that Ahlfors is the best way to go. In a way, my first read will be whichever of the application-oriented texts my instructor ends up choosing, which should help me get through Ahlfors in the way you describe Stein’s doing so. I think that I’ll go with Ahlfors. And in the end I can always do a little independent study on analytic number theory just for fun once I am ready for that, which would open many new options for me. – Galois' Canada Goose Dec 9 '13 at 3:56 • Why is it a poor choice for a second text? I know that this is terrible evidence, but I see that some universities use it as a first text for advanced undergraduates. I definitely see where Ahlfors falls short, and that is why I am so hesitant to use a supplementary text. But that it is so widely used and praised makes me confused, and—this may be just me—but the Stein book is not something I ever really hear discussed. Could you elaborate on what Stein does that Ahlfors doesn’t? – Galois' Canada Goose Dec 9 '13 at 4:06 • @Galois'CanadaGoose Stein explains concepts. His proofs are clear. Both of these things make it greatly preferable to Ahlfors for a first book. I'm as big a fan of Ahlfors as anyone, but it's a horrible book if you don't already know a good bit of complex analysis. Everything is as condensed and opaque as possible. Nontrivial details are left to the reader. Trust me, get Stein. Alhfors is a great book, but you'll like it much better if you read Stein first. – Potato Dec 9 '13 at 4:10 • Oh, I see what you mean, but my using it as a first book doesn’t completely apply to me because I have done some complex analysis before independently and will be enrolled in a mixed advanced undergraduate and introductory graduate course that likely won’t help me get where I want to be because of its focus and whatnot. I have the basics down, so maybe working through Ahlfors would be feasible. – Galois' Canada Goose Dec 9 '13 at 10:33 I can't recommend Visual Complex Analysis enough -- I would say it is one of the ten best mathematics textbooks ever written. I'm not sure why you think this book is directed towards physics majors. I read it as a graduate student in pure math, after having taken two graduate courses in complex analysis, and I felt like it provided me with significant insight into complex analysis that I hadn't gained from either course. • Well, I am just not very familiar with the book, and—to be honest—I was more than anything else going by its preface and how the text is marketed. Still, I did see the text’s table of contents of the text, and the book is just not as advanced as what I am looking for. The geometric approach the book takes seems to get great praise with it and other books following similar methods. I suppose that it’s not the insight that I looking for, i.e., what I hope to gain the course itself, but rather a deeper, more formal way of doing complex analysis.… What other texts did you use for complex analysis? – Galois' Canada Goose Dec 9 '13 at 3:47 • @Galois'CanadaGoose Fair enough. One of the courses used Conway, and the other used Rudin's Real & Complex Analysis. I wasn't fond of either book, but I suppose I prefer Conway to Rudin -- Rudin's treatment just isn't very comprehensive, and seems secondary to his treatment of real analysis and measure theory. Frankly, Ahlfors' book might be the best choice. Amazon.com has new copies of the third edition of Ahlfors for$25. – Jim Belk Dec 9 '13 at 4:02
• Yeah, I was looking at Rudin’s Real and Complex Analysis it seems to be all about measure theory, and I imagine there are more direct ways of studying that when the time comes. – Galois' Canada Goose Dec 9 '13 at 10:39

For an introductory course or independent study I recommend Ash & Novinger for math majors. Unlike Brown & Churchill, for instance, it is not afraid of using aspects of real analysis (beyond open/closed sets, limit points, etc) such as compactness in its proofs. Apparently, you will enjoy the absence of applications to problems in heat conduction. The final chapter proves the Prime Number Theorem, so there's your number theory. I doubt it will meet your topology demands, but for that you probably should be in a second course in complex analysis in the first place.

Ash & Novinger is also clearly written. Perhaps a superficial problem with Ahlfors that people don't like to discuss is the old school typeset and endless strings of paragraphs with no intervening figures. For this modern reader I find it fatiguing to read. Like others, I also find Ahlfors opaque, and simple concepts are drenched in unnecessary and often unclear verbiage. Contrast this with Rudin's Real Analysis. For all my problems with that book's terseness, it is beautifully written and pleasant to look at. Is it so wrong to want a math book to be aesthetically pleasing to the eye? I like chummy exposition and proofs clearly delineated.

Lastly @jmracek: Gamelin refers to Goursat's Theorem as aesthetically pleasing but useless, not the extremely useful Cauchy-Gorsat Theorem.

I have always really liked Gamelin as a reference on complex analysis. All of the basics will be found in that book. I think it is an entertaining read, as Gamelin's sense of humour often shows. For example, he refers to the Cauchy-Goursat theorem as "aesthetically pleasing as it is useless". In all seriousness, there are a lot of problems, together with hints and partial solutions in case you get stuck.

One complaint I have heard about the book is that proofs are incorporated into the text, and not part of a separate proof latex environment. Some people seem to think this makes it unclear when a proof is beginning or ending, but I personally have had no problem with this.

• Although I would certainly prefer Gamelin over the sort of book the course will use, I don’t think that it’s what I am looking for. – Galois' Canada Goose Dec 9 '13 at 10:34