The Cauchy Transform, Potential Theory and Conformal Mapping

Using formulas stemming from the Kerzman-Stein result, this title traces the Riemann Mapping Theorem, solves the Dirichlet and Neumann problems for the Laplace operator, constructs the Poisson kernal, and solves the inhomogenous Cauchy-Reimann equations. It is suitable for pure and applied math students.

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The Cauchy integral formula is the most central result in all of classical function theory. A recent discovery of Kerzman and Stein allows more theorems than ever to be deduced from simple facts about the Cauchy integral. In this book, the Riemann Mapping Theorem is deduced, the Dirichlet and Neumann problems for the Laplace operator are solved, the Poisson kernal is constructed, and the inhomogenous Cauchy-Reimann equations are solved concretely 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. The Cauchy Transform, Potential Theory, and Conformal Mapping is suitable for pure and applied math students taking a beginning graduate-level topics course on aspects of complex analysis. It will also be useful to physicists and engineers interested in a clear exposition on a fundamental topic of complex analysis, methods, and their application.