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

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.