Complex Analysis in One Variable

This textbook provides a concise introduction to complex analysis, analytic functions, and Riemann surfaces, along with several more recent developments motivated by the theory of several complex variables.

The first two chapters present classical material: the basic properties of analytic functions, complex integration and residue calculus, the Riemann Mapping Theorem, and the theory of harmonic functions. The theory of elliptic functions is introduced and used to prove Montel s theorem on the universal cover of the twice punctured plane. This result becomes a key tool in the classical theory of complex dynamics developed in the book. As another application of complex analysis, the text includes selected results from analytic number theory, in particular the prime number theorem and a reformulation of the Riemann Hypothesis in terms of the distribution of prime numbers.

Potential theory is introduced in the following chapter and later becomes the central tool for developing the theory of Riemann surfaces, whose main result is the uniformization theorem. The final chapter presents methods from the theory of several complex variables. A key result here is Hörmander s L2-estimate for the -operator, which is applied to solve the Beltrami equation. The chapter also includes a quantitative version of Carleson s theorem characterizing domains with nontrivial Bergman spaces, giving an optimal lower bound (conjectured by Suita) for the Bergman kernel in terms of logarithmic capacity.

Based on courses taught over many years, the material in this book has been refined for decades and the presentation is intentionally succinct, favoring the simplest available proofs. This approach makes it possible to include material that would typically require two separate volumes. Each chapter concludes with a collection of exercises ranging from standard problems to more advanced ones.