UID:
almafu_9959234632702883
Umfang:
1 online resource (x, 232 pages) :
,
digital, PDF file(s).
ISBN:
1-316-08716-6
,
1-107-36696-8
,
1-107-36205-9
,
0-511-94468-3
,
0-511-62377-1
,
1-299-40907-5
,
1-107-36450-7
Serie:
London Mathematical Society student texts ; 28
Inhalt:
Potential theory is the broad area of mathematical analysis encompassing such topics as harmonic and subharmonic functions, the Dirichlet problem, harmonic measure, Green's functions, potentials and capacity. This is an introduction to the subject suitable for beginning graduate students, concentrating on the important case of two dimensions. This permits a simpler treatment than other books, yet is still sufficient for a wide range of applications to complex analysis; these include Picard's theorem, the Phragmén-Lindelöf principle, the Koebe one-quarter mapping theorem and a 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 flavour of some recent research. Exercises are provided throughout, enabling the book to be used with advanced courses on complex analysis or potential theory.
Anmerkung:
Title from publisher's bibliographic system (viewed on 05 Oct 2015).
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Cover; Title; Copyright; Dedication; Contents; Preface; A Word about Notation; 1 Harmonic Functions; 1.1 Harmonic and Holomorphic Functions; 1.2 The Dirichlet Problem on the Disc; 1.3 Positive Harmonic Functions; Notes on Chapter 1; 2 Subharmonic Functions; 2.1 Upper Semicontinuous Functions; 2.2 Subharmonic Functions; 2.3 The Maximum Principle; 2.4 Criteria for Subharmonicity; 2.5 Integrability; 2.6 Convexity; 2.7 Smoothing; Notes on Chapter 2; 3 Potential Theory; 3.1 Potentials; 3.2 Polar Sets; 3.3 Equilibrium Measures; 3.4 Upper Semicontinuous Regularization; 3.5 Minus-Infinity Sets
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3.6 Removable Singularities3.7 The Generalized Laplacian; 3.8 Thinness; Notes on Chapter 3; 4 The Dirichlet Problem; 4.1 Solution of the Dirichlet Problem; 4.2 Criteria for Regularity; 4.3 Harmonic Measure; 4.4 Green's Functions; 4.5 The Poisson-Jensen Formula; Notes on Chapter 4; 5 Capacity; 5.1 Capacity as a Set Function; 5.2 Computation of Capacity; 5.3 Estimation of Capacity; 5.4 Criteria for Thinness; 5.5 Transflnite Diameter; Notes on Chapter 5; 6 Applications; 6.1 Interpolation of Lp-spaces; 6.2 Homogeneous Polynomials; 6.3 Uniform Approximation; 6.4 Banach Algebras
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6.5 Complex DynamicsNotes on Chapter 6; A Borel Measures; A.I Supports; A.2 Regularity; A.3 Radon Measures; A.4 Weak*-Convergence; Bibliography; Index; Glossary of Notation
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English
Weitere Ausg.:
ISBN 0-521-46120-0
Weitere Ausg.:
ISBN 0-521-46654-7
Sprache:
Englisch
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