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Full Description
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.
Contents
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.
