Complex analysis is one of the most beautiful subjects in mathematics, and one of the most surprising: functions that are merely differentiable in the complex sense turn out to be infinitely differentiable, rigid, and full of structure that has no analog on the real line. That richness makes it a joy, but it also means the theorems come fast and interconnected, so learning from a text that suits your background matters more than usual.
The order that works starts with a solid first course, deepens into the core theory through a couple of complementary classics, and then reaches the advanced topics — Riemann surfaces, entire functions, and number-theoretic applications — where the subject shows its full power. Each step builds directly on the theorems before it.
The first course
Start with Complex Analysis by Ahlfors, the revered classic that has trained generations, demanding but deeply rewarding for its rigor and elegance. If you want a more modern and motivated pace, Complex analysis by Stein is superb, weaving the theory together with applications and a strong narrative sense. Complex Analysis by Gamelin is the most accessible of the three, a clear and well-organized text that many find the gentlest genuine entry point. Choose the one matching your background; all cover holomorphic functions, Cauchy's theorem, and residues.
Deepen the theory
Next, go further into the core. Functions of One Complex Variable I by Conway is the standard graduate text, thorough and precise, and the natural place to solidify and extend a first course. Its sequel Functions of one complex variable II pushes into more advanced material for readers ready to specialize. Working through this stage gives you command of the theorems — conformal mapping, the argument principle, analytic continuation — that the advanced topics assume.
Advanced topics
The final arc branches into the subject's deep applications. Riemann surfaces by Ahlfors opens the geometric theory that reframes complex functions in terms of the surfaces they live on. Entire functions studies the growth and distribution of the zeros of functions analytic on the whole plane, a rich classical theory. And Analytic Number Theory shows how complex analysis proves deep facts about the integers, including approaches to the distribution of primes — a striking demonstration of the subject's reach.
Read in this order and complex analysis unfolds from a set of surprising theorems into a coherent, powerful theory. Follow the full path to go from Cauchy's theorem to analytic number theory.
