The Cauchy Transform, Potential Theory and Conformal Mapping

The book 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. Also, four new chapters have been added: two on quadrature domains and another two on complexity of the objects of complex analysis and improved Riemann mapping theorems.