Harmonic functions--the solutions of Laplace's equation--play a crucial role in many areas of mathematics, physics, and engineering. But learning about them is not always easy. At times the authors have agreed with Lord Kelvin and Peter Tait, who wrote (, Preface) There can be but one opinion as to the beauty and utility of this analysis of Laplace; but the manner in which it has been hitherto presented has seemed repulsive to the ablest mathematicians, and difficult to ordinary mathematical students.
Preface Acknowledgments CHAPTER 1 Basic Properties of Harmonic Functions Definitions and Examples Invariance Properties The Mean-Value Property The Maximum Principle The Poisson Kernel for the Ball The Dirichlet Problem for the Ball Converse of the Mean-Value Property Real Analyticity and Homogeneous Expansions Origin of the Term "Harmonic" Exercises CHAPTER 2 Bounded Harmonic Functions Liouvfile‘s Theorem Isolated Singularities Cauchy‘s Estimates Normal Families Maximum Principles Limits Along Rays Bounded Harmonic Functions on the Ball ExercisesCHPATER 3 Positive Harmonic FunctionsCHPATER 4 The Kelvin TransformCHPATER 5 Harmonic PolynomialsCHPATER 6 Harmonic Hardy SpacesCHPATER 7 Harmonic Funtions on Half-SpacesCHAPTER 8 Harmonic Bergaman SpacesCHAPTER 9 The Decomposition TheoremCHAPTER 10 Annular RegionsCHAPTER 11 The Dirichlet Problem and Boundary BehaviorAPPENDIX A Volume,Surface Area,and Interation on SpheresAPPENDIX B Harmonic Function Theory and MathematicaReferencesSymbol IndexIndex
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