Potential Theory in the Complex Plane

Höfundur: Thomas Ransford (Útgáfa: 1)
Potential Theory in the Complex Plane

Kaup valmöguleikar

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.

Nánar um bókina

Útgefandi
Cambridge University Press
ISBN
9780511885259
Print ISBN
9780521461207
Format
Page Fidelity (PDF)
Útgáfa
1
Höfundar
Thomas Ransford
Tungumál
English
Útgefið
1995-03-16
Prent takmörkun á líftíma
10
Prent takmörkun
5
Afritunar takmörkun
5

Kaflar

  • 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
  • 3.6 Removable Singularities
  • 3.7 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
  • 6.5 Complex Dynamics
  • Notes 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