ABSTRACT

The Cauchy Transform, Potential Theory and Conformal Mapping 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.The book provides a fast track to understanding the Riemann Mapping Theorem. The Dirichlet and Neumann problems f

part 1|2 pages

- Introduction

chapter 2|6 pages

- The improved Cauchy integral formula

chapter 3|4 pages

- The Cauchy transform

chapter 5|4 pages

- The Kerzman-Stein operator and kernel

chapter 7|6 pages

The Szego˝ kernel function

part 8|2 pages

- The Riemann mapping function

chapter 9|8 pages

- A density lemma and consequences

chapter 11|4 pages

- The case of real analytic boundary

chapter 15|8 pages

- The Bergman space

chapter 17|6 pages

- The Solid Cauchy transform

chapter 18|4 pages

- The classical Neumann problem

chapter 19|10 pages

Harmonic measure and the Szego˝ kernel

part 21|2 pages

- The Dirichlet problem again

chapter 22|10 pages

- Area quadrature domains

chapter 23|8 pages

- Arc length quadrature domains

chapter 24|4 pages

- The Hilbert transform

chapter 25|6 pages

The Bergman kernel and the Szego˝ kernel

chapter 27|4 pages

Zeroes of the Szego˝ kernel

chapter 28|6 pages

- The Kerzman-Stein integral equation

chapter 30|12 pages

- The dual space of A∞(Ω)

chapter 32|4 pages

- Zeroes of the Bergman kernel

chapter 33|4 pages

- Complexity in complex analysis

chapter 34|6 pages

- Area quadrature domains and the double