The Schwarz function and its application to two dimensional flows

Abstract

In this thesis, we will survey the Schwarz function and its applications. The Schwarz function of an analytic arc is first defined and numerous examples are given. We develop the relationship of the Schwarz function to Schwarzian reflection and deduce important functional identities. Secondly we relate the derivatives of the Schwarz function to an analytic arc to the slope and curvature of the arc. The operators of partial differential equations are expressed in terms of conjugate coordinates, then Green’s theorem, in its complex analytic form, is used to derive a number of integral identities, some of which are of interest in the theory of complex analysis. Finally, we apply the theory of Schwarz function to two-dimensional flows including some free boundary problems in a Hele-Shaw cell.