Author(s): Kunihiko Kodaira
This textbook is an introduction to the classical theory of functions of a complex variable. The author's aim is to explain the basic theory in an easy to understand and careful way. He emphasizes geometrical considerations, and, to avoid topological difficulties associated with complex analysis, begins by deriving Cauchy's integral formula in a topologically simple case and then deduces the basic properties of continuous and differentiable functions. The remainder of the book deals with conformal mappings, analytic continuation, Riemann's mapping theorem, Riemann surfaces and analytic functions on a Riemann surface. The book is profusely illustrated and includes many examples. Problems are collected together at the end of the book. It should be an ideal text for either a first course in complex analysis or more advanced study.
Kunihiko Kodaira (1915-1997) worked in many areas including harmonic integrals, algebraic geometry and the classification of compact complex analytic surfaces. He held faculty positions at many universities including Tokyo, Harvard, Stanford, and Johns Hopkins, and the Institute for Advanced Study in Princeton. He was awarded a Fields medal in 1954 and a Wolf Prize in 1984.
1. Holomorphic functions; 2. Cauchy's theorem; 3. Conformal mappings; 4. Analytic continuation; 5. Riemann's mapping theorem; 6. Riemann surfaces; 7. The structure of Riemann surfaces; 8. Analytic functions on a closed Riemann surface.