Harmonic function theory
This is a book about harmonic functions in Euclidean space. Readers with a background in real and complex analysis at the beginning graduate level will feel comfortable with the material presented here. The authors have taken unusual care to motivate concepts and simplify proofs. Topics include: basic properties of harmonic functions, Poisson integrals, the Kelvin transform, spherical harmonics, harmonic Hardy spaces, harmonic Bergman spaces, the decomposition theorem, Laurent expansions, isolated singularities, and the Dirichlet problem. The new edition contains a completely rewritten chapter on spherical harmonics, a new section on extensions of Bocher?'s Theorem, new exercises and proofs, as well as revisions throughout to improve the text. A unique software package-designed by the authors and available by email-supplements the text for readers who wish to explore harmonic function theory on a computer.
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apply assume ball Bocher's Theorem Borel measurable boundary bounded harmonic function chapter compact subset complete the proof complex continuous function converges uniformly Corollary define denote differentiable Dirichlet problem equals everywhere Exercise exists Fatou Theorem finite formula function harmonic function on H harmonic at oo harmonic func harmonic on Cl harmonic polynomials Harnack's Inequality Hm(Rn Hm(S holomorphic function homogeneous expansion hp(B hyperplane implies isolated singularity Kelvin transform linear Liouville's Theorem maximum principle mean-value property multi-index nontangential limit parallels orthogonal pointwise Poisson integral Poisson kernel polynomial on Rn positive and harmonic positive harmonic function power series proof of Theorem Proposition Prove radial reader real analytic real valued real-valued harmonic function removable singularity result Schwarz Lemma series converges solve the Dirichlet spherical harmonics subharmonic subset of Cl Suppose symmetric tion uniformly on compact unique unit sphere upper half-space Wm(Rn zonal harmonic