Lectures on Complex Analysis
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.
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Person
Norbert Steinmetz is a professor emeritus at the Mathematical Institute of the Technical University of Dortmund. His field of work encompasses complex analysis, in particular Nevanlinna theory, the field of ordinary differential equations in the complex domain, and the theory of complex dynamic systems.
Content
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.
