Harmonic Function TheoryHarmonic functions - the solutions of Laplace's equation - play a crucial role in many areas of mathematics, physics, and engineering. Avoiding the disorganization and inconsistent notation of other expositions, the authors approach the field from a more function-theoretic perspective, emphasizing techniques and results that will seem natural to mathematicians comfortable with complex function theory and harmonic analysis; prerequisites for the book are a solid foundation in real and complex analysis together with some basic results from functional analysis. Topics covered include: basic properties of harmonic functions defined on subsets of Rn, including Poisson integrals; properties bounded functions and positive functions, including Liouville's and Cauchy's theorems; the Kelvin transform; Spherical harmonics; hp theory on the unit ball and on half-spaces; harmonic Bergman spaces; the decomposition theorem; Laurent expansions and classification of isolated singularities; and boundary behavior. An appendix describes routines for use with MATHEMATICA to manipulate some of the expressions that arise in the study of harmonic functions. |
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assume ball Bôcher's Theorem Borel measurable boundary data bounded harmonic function chapter compact subset completing the proof computation continuous function converges uniformly Corollary decomposition defined denote desired differentiable dim Hm Dirichlet problem equals equation Exercise exists finite formula function harmonic function on Q Green's identity harmonic on Q harmonic polynomials Harnack's Inequality Hm Rn holomorphic function homogeneous of degree homogeneous polynomial hyperplane implies inner product inversion isolated singularity Kelvin transform L¹(S Laplacian Lemma linear Liouville's Theorem maximum principle mean-value property multi-index nontangential limit Note parallels orthogonal Poisson integral Poisson kernel polynomial on Rn positive and harmonic positive harmonic functions power series Proposition 5.5 Prove radial real analytic real-valued harmonic function removable singularity result sequence Show spherical harmonics subharmonic Suppose symmetric tion uniformly on compact unique zonal harmonics