Potential Theory in the Complex Plane [Hardcover]

Potential theory is the broad area of mathematical analysis encompassing such topics as harmonic and subharmonic functions.Potential theory is the broad area of mathematical analysis encompassing harmonic and subharmonic functions, the Dirichlet problem, harmonic measure, Green's functions, potentials and capacity. This introduction concentrates on the important case of two dimensions, and emphasizes its links with complex analysis.Potential theory is the broad area of mathematical analysis encompassing harmonic and subharmonic functions, the Dirichlet problem, harmonic measure, Green's functions, potentials and capacity. This introduction concentrates on the important case of two dimensions, and emphasizes its links with complex analysis.Ransford provides an introduction to the subject, concentrating on the important case of two dimensions, and emphasizing its links with complex analysis. This is reflected in the large number of applications, which include Picard's theorem, the Phragmén-Lindelöf principle, the Radó-Stout theorem, Lindelöf's theory of asymptotic values, the Riemann mapping theorem (including continuity at the boundary), the Koebe one-quarter theorem, Hilbert's lemniscate theorem, and the sharp quantitative form of Runge's theorem. In addition, there is a chapter on connections with functional analysis and dynamical systems, which shows how the theory can be applied to other parts of mathematics and gives a flavor of some recent research in the area.Preface; A word about notation; 1. Harmonic functions; 2. Subharmonic functions; 3. Potential theory; 4. The Dirichlet problem; 5. Capacity; 6. Applications; Borel measures; Bibliography; Index; Glossary of notation. Graduate students and researchers in complex analysis will find in this book most of the potential theory that they are likely to need...this attractive book is recommended. Mathematical Reviews