Conformal mappings involving curvilinear quadrangles

The conformal mapping of a curvilinear quadrangle to a half-plane is a classical problem in analysis; it occurs during the analytical solution of free-boundary problems involving groundwater flows. Apart from degenerate cases, in general, it is not known how to perform such mappings; the difficulty arises because the mapping functions are given by the solutions of a Fuchsian differential equation. For a quadrangle this Fuchsian equation involves both accessory parameters and free points that are unknown a priori; the analysis of such equations is therefore difficult, and there are usually no obvious solutions. In this paper conformal mappings involving a special class of curvilinear quadrangles are constructed, and a general approach is devised in the special cases when one (or more) vertex angle is equal to 2 pi. By implication this suggests that there are degenerate classes of Fuchsian equations involving accessory parameters and free points; these classes are discussed.