Lectures on Complex Analysis

This elegant textbook offers a comprehensive course on one-dimensional complex analysis. It includes many topics that, in this scope, are not covered in most other textbooks, such as a detailed investigation of the Schwarzian derivative and its associated differential equation, with applications to conformal mappings of circular polygons; various proofs of the uniformisation theorem for planar domains; an introduction to the theory of ordinary differential equations in the complex domain, culminating in a proof of the Cauchy-Kovalevskaya theorem; an introduction to the theory of normal families, including Zalcman's lemma; a proof of the Paley-Wiener theorem; a complete discussion of the Laguerre-Pólya class; solution of the Dirichlet problem, with special emphasis on harmonic measure and Green's function, and applications to conformal mappings of multiply connected domains; a detailed description of the dynamics of polynomials; and the consistent use of the theory of proper mappings whenever possible.

Chapter 1. Complex Numbers and Functions. - Chapter 2. Two Theorems of Cauchy. - Chapter 3. The Local Theory. - Chapter 4. The Residue Theorem. - Chapter 5. Entire Functions. - Chapter 6. Special Functions. - Chapter 7. Periodic and Elliptic Functions. - Chapter 8. Conformal Mappings of Simply Connected Domains. - Chapter 9. Harmonic Functions. - Chapter 10. Conformal Mappings of Multiply Connected Domains. - Chapter 11. Analytic Continuation. - Chapter 12. Ordinary Differential Equations in the Complex Domain. - Chapter 13. Iterations of Polynomials.