Conformally mapping a curvilinear quadrangle to a half plane is a classical problem in analysis and occurs during the analytical solution of free boundary problems involving groundwater flows. Apart from degenerate cases it is, in general, 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 2 pi. By implication this suggests that there are degenerate classes of Fuchsian equations involving accessory parameters and free points; these are discussed.
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