On integration of Fuchs-class equations with four singular points in connection with a conformal mapping problem

Integration of an ordinary differential equation of Fuchs class with a large number of singular points (more than 3) is known to be a difficult problem [1]–[3]. It has a great theoretical and applied significance and is related, in particular, to the problem of conformal mapping of circular polygons [4]–[6]. One of the basic difficulties that arise here is the relation between the so-called auxiliary parameters of the equation and the geometric parameters of the circular polygon [2]–[10].