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Harmonic Function Theory

  • Textbook
  • © 1992

Overview

Authors:
  1. Sheldon Axler

    1. Department of Mathematics, Michigan State University, East Lansing, USA

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  2. Paul Bourdon

    1. Department of Mathematics, Washington and Lee University, Lexington, USA

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    You can also search for this author in PubMed   Google Scholar

  3. Wade Ramey

    1. Department of Mathematics, Michigan State University, East Lansing, USA

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Part of the book series: Graduate Texts in Mathematics (GTM, volume 137)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xii
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  2. Basic Properties of Harmonic Functions

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 1-29

  3. Bounded Harmonic Functions

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 31-44

  4. Positive Harmonic Functions

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 45-58

  5. The Kelvin Transform

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 59-71

  6. Spherical Harmonics

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 73-96

  7. Harmonic Hardy Spaces

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 97-124

  8. Harmonic Functions on Half-Spaces

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 125-149

  9. Harmonic Bergman Spaces

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 151-168

  10. The Decomposition Theorem

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 169-182

  11. Annular Regions

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 183-195

  12. The Dirichlet Problem and Boundary Behavior

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 197-212

  13. Back Matter

    Pages 213-233
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Keywords

About this book

Harmonic 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.

Reviews

From the reviews of the second edition:

"There are several major changes in this second edition … . Many exercises have been added and several photographs of mathematicians related to harmonic functions are included. The book is a nice introduction to the fundamental notions of potential theory." (European Mathematical Society Newsletter, June, 2002)

"We warmly recommend this textbook to graduate students interested in Harmonic Function Theory and/or related areas. We are sure that the reader will be able to appreciate the lively and illuminating discussions in this book, and therefore, will certainly gain a better understanding of the subject." (Ferenc Móricz, Acta Scientiarum Mathematicarum, Vol. 67, 2001)

"This is a new edition of a nice textbook … on harmonic functions in Euclidean spaces, suitable for a beginning graduate level course. … New exercises are added and numerous minor improvements throughout the text are made." (Alexander Yu. Rashkovsky, Zentralblatt MATH, Vol. 959, 2001)

Authors and Affiliations

  • Department of Mathematics, Michigan State University, East Lansing, USA

    Sheldon Axler, Wade Ramey

  • Department of Mathematics, Washington and Lee University, Lexington, USA

    Paul Bourdon

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