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.
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Book Description
Editorial Reviews
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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)
--This text refers to an alternate
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1 of 1 people found the following review helpful:
5.0 out of 5 stars
Excellent text,
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This review is from: Harmonic Function Theory (Hardcover)
We used this text in class for a topics course on Harmonic Analysis. The book is very well written and has a plethora of problems and examples to make sure the reader knows that he is on the right track. An excellent text.
1.0 out of 5 stars
harmonic functions done wrong,
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This review is from: Harmonic Function Theory (Graduate Texts in Mathematics) (Paperback)
Too many lines of the "it's easily checked that ..." kind
detract from the value this might have had and waste the reader's time. It would be smarter to purchase either the Schaums outlines (pde) or one of the many Dover pde reprints.
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First Sentence:
Harmonic functions, for us, live on open subsets of real Euclidean spaces. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
nonconstant bounded harmonic function, conjugation theorem, positive harmonic functions, nontangential limit, multiple power series, bounded harmonic functions, symmetry lemma, harmonic function theory, fundamental pole, deleted neighborhood, zonal harmonics, homogeneous expansion, inversion map, bounded holomorphic function, subharmonic functions, ball condition, harmonic polynomials, removable singularity, singular with respect, parallels orthogonal, radial limit, harmonic extension, bounded components, linear isometry, harmonic conjugate Key Phrases - Capitalized Phrases (CAPs): (learn more)
Liouville's Theorem, Harnack's Inequality, Harnack's Principle, Cauchy's Estimates, Generalized Liouville Theorem, Local Fatou Theorem New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Harmonic functions, for us, live on open subsets of real Euclidean spaces. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
nonconstant bounded harmonic function, conjugation theorem, positive harmonic functions, nontangential limit, multiple power series, bounded harmonic functions, symmetry lemma, harmonic function theory, fundamental pole, deleted neighborhood, zonal harmonics, homogeneous expansion, inversion map, bounded holomorphic function, subharmonic functions, ball condition, harmonic polynomials, removable singularity, singular with respect, parallels orthogonal, radial limit, harmonic extension, bounded components, linear isometry, harmonic conjugate Key Phrases - Capitalized Phrases (CAPs): (learn more)
Liouville's Theorem, Harnack's Inequality, Harnack's Principle, Cauchy's Estimates, Generalized Liouville Theorem, Local Fatou Theorem New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 100
books:
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100
books
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- Functions of One Complex Variable (Graduate Texts in Mathematics - Vol 11) (v. 1) by John B Conway in Back Matter (1), and Back Matter (2)
- Elliptic Curves: Diophantine Analysis (Grundlehren der mathematischen Wissenschaften) by S. Lang in Back Matter (1), and Back Matter (2)
- Galois Theory (Graduate Texts in Mathematics) by Harold M. Edwards in Front Matter, and Back Matter
- An Introduction to Algebraic Topology (Graduate Texts in Mathematics) by Joseph J. Rotman in Back Matter (1), and Back Matter (2)
- The Arithmetic of Elliptic Curves (Graduate Texts in Mathematics) (v. 106) by Joseph H. Silverman in Back Matter (1), and Back Matter (2)
See all 100 books this book cites
- Geometric Applications of Fourier Series and Spherical Harmonics (Encyclopedia of Mathematics and its Applications) by Helmut Groemer in Back Matter
- Foundations of Hyperbolic Manifolds (Graduate Texts in Mathematics) by John G. Ratcliffe in Back Matter
- An Introduction to Number Theory (Graduate Texts in Mathematics) by G. Everest in Back Matter
- Galois Theory (Graduate Texts in Mathematics) by Harold M. Edwards in Back Matter
- Probability Theory II (Graduate Texts in Mathematics) by M. Loeve in Back Matter
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