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?