Harmonic function theory

Axler S., Bourdon P., Ramey W.

The authors have taken unusual care to motivate concepts and simplify proofs in this 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. Topics include basic properties of harmonic functions, Poisson integrals, the Kelvin transform, harmonic polynomials, spherical harmonics, harmonic Hardy spaces, harmonic Bergman spaces, the decomposition theorem, Laurent expansions, isolated singularities, and the Dirichlet problem. This new edition contains a completely rewritten chapter on harmonic polynomials and spherical harmonics, as well as new material on Bocher’s Theorem, norms for harmonic Hardy spaces, the Dirichlet problem for the half space, and the relationship between the Laplacian and the Kelvin transform. in addition, the authors have included new exercises and have made numerous minor improvements throughout the text. The authors have developed a software package, available electronically without charge, that uses results from this book to calculate many of the expressions that arise in harmonic function theory. For example, the Poisson integral of any polynomial can be computed exactly.