Abstract
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 2π. By implication this suggests that there are degenerate classes of Fuchsian equations involving accessory parameters and free points; these classes are discussed.
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