Harmonic Function Theory

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

    PDF

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.

• Harmonic Function Theory
• Harmonic Functions
• Complex analysis
• integral
• integration
• Poisson integral

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)

• 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

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