The Schwarz Function and Its Generalization to Higher Dimensions

The Schwarz function originates in classical complex analysis and potential theory. In this text, the author presents the advantages of the function, favouring a mode of treatment which unites the subject with modern theory of distributions and partial differential equations - thus bridging the gap between two-dimensional analysts. The author 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 partial differential equations.