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El. knyga: Harmonic Function Theory

3.67/5 (6 ratings by Goodreads)
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  • Formatas: PDF+DRM
  • Serija: Graduate Texts in Mathematics 137
  • Išleidimo metai: 11-Nov-2013
  • Leidėjas: Springer-Verlag New York Inc.
  • Kalba: eng
  • ISBN-13: 9781475781373
Kitos knygos pagal šią temą:
Harmonic Function TheoryDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Graduate Texts in Mathematics 137
  • Išleidimo metai: 11-Nov-2013
  • Leidėjas: Springer-Verlag New York Inc.
  • Kalba: eng
  • ISBN-13: 9781475781373
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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.

Recenzijos

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)

Daugiau informacijos

2nd edition
Preface ix
Acknowledgments xi
Basic Properties of Harmonic Functions
1(30)
Definitions and Examples
1(1)
Invariance Properties
2(2)
The Mean-Value Property
4(3)
The Maximum Principle
7(2)
The Poisson Kernel for the Ball
9(3)
The Dirichlet Problem for the Ball
12(5)
Converse of the Mean-Value Property
17(2)
Real Analyticity and Homogeneous Expansions
19(6)
Origin of the Term ``Harmonic''
25(1)
Exercises
26(5)
Bounded Harmonic Functions
31(14)
Liouville's Theorem
31(1)
Isolated Singularities
32(1)
Cauchy's Estimates
33(2)
Normal Families
35(1)
Maximum Principles
36(2)
Limits Along Rays
38(2)
Bounded Harmonic Functions on the Ball
40(2)
Exercises
42(3)
Positive Harmonic Functions
45(14)
Liouville's Theorem
45(2)
Harnack's Inequality and Harnack's Principle
47(3)
Isolated Singularities
50(5)
Positive Harmonic Functions on the Ball
55(1)
Exercises
56(3)
The Kelvin Transform
59(14)
Inversion in the Unit Sphere
59(2)
Motivation and Definition
61(1)
The Kelvin Transform Preserves Harmonic Functions
62(1)
Harmonicity at Infinity
63(3)
The Exterior Dirichlet Problem
66(1)
Symmetry and the Schwarz Reflection Principle
67(4)
Exercises
71(2)
Harmonic Polynomials
73(38)
Polynomial Decompositions
74(4)
Spherical Harmonic Decomposition of L2(S)
78(4)
Inner Product of Spherical Harmonics
82(3)
Spherical Harmonics Via Differentiation
85(7)
Explicit Bases of Hm (Rn) and Hm(S)
92(2)
Zonal Harmonics
94(3)
The Poisson Kernel Revisited
97(3)
A Geometric Characterization of Zonal Harmonics
100(4)
An Explicit Formula for Zonal Harmonics
104(2)
Exercises
106(5)
Harmonic Hardy Spaces
111(32)
Poisson Integrals of Measures
111(4)
Weak* Convergence
115(2)
The Spaces hp(B)
117(4)
The Hilbert Space h2(B)
121(2)
The Schwarz Lemma
123(5)
The Fatou Theorem
128(10)
Exercises
138(5)
Harmonic Functions on Half-Spaces
143(28)
The Poisson Kernel for the Upper Half-Space
144(2)
The Dirichlet Problem for the Upper Half-Space
146(5)
The Harmonic Hardy Spaces hp(H)
151(2)
From the Ball to the Upper Half-Space, and Back
153(3)
Positive Harmonic Functions on the Upper Half-Space
156(4)
Nontangential Limits
160(1)
The Local Fatou Theorem
161(6)
Exercises
167(4)
Harmonic Bergman Spaces
171(20)
Reproducing Kernels
172(4)
The Reproducing Kernel of the Ball
176(5)
Examples in bp(B)
181(4)
The Reproducing Kernel of the Upper Half-Space
185(3)
Exercises
188(3)
The Decomposition Theorem
191(18)
The Fundamental Solution of the Laplacian
191(2)
Decomposition of Harmonic Functions
193(4)
Bocher's Theorem Revisited
197(3)
Removable Sets for Bounded Harmonic Functions
200(3)
The Logarithmic Conjugation Theorem
203(3)
Exercises
206(3)
Annular Regions
209(14)
Laurent Series
209(1)
Isolated Singularities
210(3)
The Residue Theorem
213(2)
The Poisson Kernel for Annular Regions
215(4)
Exercises
219(4)
The Dirichlet Problem and Boundary Behavior
223(16)
The Dirichlet Problem
223(1)
Subharmonic Functions
224(2)
The Perron Construction
226(1)
Barrier Functions and Geometric Criteria for Solvability
227(6)
Nonextendability Results
233(3)
Exercises
236(3)
Appendix A Volume, Surface Area, and Integration on Spheres 239(8)
Volume of the Ball and Surface Area of the Sphere
239(2)
Slice Integration on Spheres
241(3)
Exercises
244(3)
Appendix B Harmonic Function Theory and Mathematica 247(2)
References 249(2)
Symbol Index 251(4)
Index 255
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