The Schwarz Function and Its Generalization to Higher Dimensions

Based on a series of lectures delivered at the University of Arkansas on the Schwarz function - a tool in the geometry theory of complex analysis - this text explores the relationship between the Schwarz function of an analytic curve and complex analysis, operator theory and differential equations.
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.