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The Cauchy transform potential theory and conformal mapping/ Steven R Bell

By: Bell, Steven R [author].

Publisher: Boca Ratan: CRC, 2015Edition: 2nd.Description: xii, 209 pages, 23.5 cm.ISBN: 9781498727204.Subject(s): Mathematics | Complex Analysis | Cauchy TransformDDC classification: 515.9

Contents:

Preface -- Table of Symbols -- 1. Introduction -- 2. The Improved Cauchy Integral Formula -- 3. The Cauchy transform -- 4. The Hardy Space Szegő Projection and Kerzman-Stein Formula -- 5. The Kerzman-Stein Operator and Kernel -- 6. The Classical Definition of the hardy Space -- 7. The Szegő Kernel Function -- 8. The Riemann Mapping Function -- 9. A Density Lemma and Consequences -- 1-. Solution of the Dirichlet Problem in Simply Connected Domains -- 11. The Case of Real Analytic Boundary -- 12. The Transformation Law for the Szegő Kernel -- 13. The Ahlfors Map of a Multiply Connected Domain -- 14. The Dirichlet Problem in Multiply Connected Domains -- 15. The Bergman Space -- 16. Proper Holomorphic mappings and the Bergman Projection -- 17. The Solid Cauchy Transform -- 18. The Classical Neumann problem -- 19. Harmonic Measure and the Szegő Kernel -- 20. The Neumann problem in Multiply Connected Domains -- 21. The Dirichlet Problem Again -- 22. Area Quadrature Domains 23. Arc Length Quadrature Domains -- 24. The Hilbert Transform -- 25. The Bergman Kernel and the Szegő Kernel -- 26. Pseudo-local Property of the Cauchy transform -- 27. Zeros of the Szegő Kernel -- 28. The Kerzman -Stein Integral Equation -- 29. Local Bundary Behavior of Holomorphic Mapping -- 30. The Dual Space of Aᾳ (Ω) -- 31. The green's Function and he Bergman Kernel -- 32. Zeros of the Bergman Kernel -- 33. Complexity in Complex Analysis -- 34. Area Quadrature Domains and the Double -- A The cauchy-Kovalevski Theorem for the cauchy-riemann Operator -- Bibliographic Notes -- Bibliography -- Index

Summary: this book explores the most central result in all of classical function theory, the Cauchy integral formula, in a new and novel way based on an advance made by Kerzman and Stein in 1976. It provides a fast track to understanding the Riemann Mapping Theorem. The Dirichlet and Neumann problems for the Laplace operator are solved, the Poisson kernel is constructed, and the inhomogenous Cauchy-Reimann equations are solved concretely and efficiently using formulas stemming from the Kerzman-Stein result.These explicit formulas yield new numerical methods for computing the classical objects of potential theory and conformal mapping, and the book provides succinct, complete explanations of these methods. Four new chapters have been added to this second edition: two on quadrature domains and another two on complexity of the objects of complex analysis and improved Riemann mapping theorems.

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ISI Library, Kolkata

515.9 B433 (Browse shelf)

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Includes bibliographical references and indexes

Preface --
Table of Symbols --
1. Introduction --
2. The Improved Cauchy Integral Formula --
3. The Cauchy transform --
4. The Hardy Space Szegő Projection and Kerzman-Stein Formula --
5. The Kerzman-Stein Operator and Kernel --
6. The Classical Definition of the hardy Space --
7. The Szegő Kernel Function --
8. The Riemann Mapping Function --
9. A Density Lemma and Consequences --
1-. Solution of the Dirichlet Problem in Simply Connected Domains --
11. The Case of Real Analytic Boundary --
12. The Transformation Law for the Szegő Kernel --
13. The Ahlfors Map of a Multiply Connected Domain --
14. The Dirichlet Problem in Multiply Connected Domains --
15. The Bergman Space --
16. Proper Holomorphic mappings and the Bergman Projection --
17. The Solid Cauchy Transform --
18. The Classical Neumann problem --
19. Harmonic Measure and the Szegő Kernel --
20. The Neumann problem in Multiply Connected Domains --
21. The Dirichlet Problem Again --
22. Area Quadrature Domains
23. Arc Length Quadrature Domains --
24. The Hilbert Transform --
25. The Bergman Kernel and the Szegő Kernel --
26. Pseudo-local Property of the Cauchy transform --
27. Zeros of the Szegő Kernel --
28. The Kerzman -Stein Integral Equation --
29. Local Bundary Behavior of Holomorphic Mapping --
30. The Dual Space of Aᾳ (Ω) --
31. The green's Function and he Bergman Kernel --
32. Zeros of the Bergman Kernel --
33. Complexity in Complex Analysis --
34. Area Quadrature Domains and the Double --
A The cauchy-Kovalevski Theorem for the cauchy-riemann Operator --
Bibliographic Notes --
Bibliography --
Index

this book explores the most central result in all of classical function theory, the Cauchy integral formula, in a new and novel way based on an advance made by Kerzman and Stein in 1976. It provides a fast track to understanding the Riemann Mapping Theorem. The Dirichlet and Neumann problems for the Laplace operator are solved, the Poisson kernel is constructed, and the inhomogenous Cauchy-Reimann equations are solved concretely and efficiently using formulas stemming from the Kerzman-Stein result.These explicit formulas yield new numerical methods for computing the classical objects of potential theory and conformal mapping, and the book provides succinct, complete explanations of these methods.

Four new chapters have been added to this second edition: two on quadrature domains and another two on complexity of the objects of complex analysis and improved Riemann mapping theorems.

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Indian Statistical Institute Library, Kolkata

The Cauchy transform potential theory and conformal mapping/ Steven R Bell

By: Bell, Steven R [author].