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Harmonic Function Theory

  • Textbook
  • © 2001

Overview

Authors:
  1. Sheldon Axler

    1. Mathematics Department, San Francisco State University, San Francisco, USA

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    You can also search for this author in PubMed   Google Scholar

  2. Paul Bourdon

    1. Mathematics Department, 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. Berkeley, 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-xi
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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 45-57

  5. The Kelvin Transform

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 59-72

  6. Harmonic Polynomials

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 73-109

  7. Harmonic Hardy Spaces

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 111-142

  8. Harmonic Functions on Half-Spaces

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 143-169

  9. Harmonic Bergman Spaces

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 171-190

  10. The Decomposition Theorem

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 191-207

  11. Annular Regions

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 209-221

  12. The Dirichlet Problem and Boundary Behavior

    • Sheldon Axler, Paul Bourdon, Wade Ramey
    Pages 223-238

  13. Back Matter

    Pages 239-263
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Keywords

About this book

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 e-mail - supplements the text for readers who wish to explore harmonic function theory on a computer.

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

  • Mathematics Department, San Francisco State University, San Francisco, USA

    Sheldon Axler

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

    Paul Bourdon

  • Berkeley, USA

    Wade Ramey

Bibliographic Information

  • Book Title: Harmonic Function Theory

  • Authors: Sheldon Axler, Paul Bourdon, Wade Ramey

  • Series Title: Graduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4757-8137-3

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 2001

  • Hardcover ISBN: 978-0-387-95218-5Published: 25 January 2001

  • eBook ISBN: 978-1-4757-8137-3Published: 11 November 2013

  • Series ISSN: 0072-5285

  • Series E-ISSN: 2197-5612

  • Edition Number: 2

  • Number of Pages: XII, 264

  • Topics: Potential Theory

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