The Schwarz Function and Its Generalization to Higher Dimensions

The Schwarz function originates in classical complex analysis and potential theory. Here the author presents the advantages favoring a mode of treatment which unites the subject with modern theory of distributions and partial differential equations thus bridging the gap between two-dimensional geometric and multi-dimensional analysts. Examines the Schwarz function and its relationship to recent investigations regarding inverse problems of Newtonian gravitation, free boundaries, Hele-Shaw flows and the propagation of singularities for holomorphic p.d.e. The Schwarz Principle of Reflection.

The Logarithmic Potential, Balayage, and Quadrature Domains.

Examples of ``Quadrature Identities''.

Quadrature Domains: Basic Properties, 1.

Quadrature Domains: Basic Properties, 2.

Schwarzian Reflection, Revisited.

Projectors from L? (dOmega) to H? (dOmega).

The Friedrichs Operator.

Concluding Remarks.

Bibliography.

Index.

Harold Seymour Shapiro is a professor emeritus of mathematics at the Royal Institute of Technology in Stockholm, Sweden, best known for inventing the so-called Shapiro polynomials also known as Golay-Shapiro polynomials or Rudin-Shapiro polynomials and for pioneering work on quadrature domains.