On closed-form integration of some Fuchsian differential equations related to a conformal mapping of circular pentagons with a cut

To solve the problem on a conformal mapping of some circular pentagons with a cut, we suggest to use special methods designed for a class of polygons in polar grids (i.e., bounded by arcs of concentric circles and segments of lines passing through the origin) and based on finding particular solutions of Fuchs type equations in the form of linear combinations of known particular solutions of some simpler equations with three singular points with indeterminate coefficients. The obtained results are first used to solve problems on a conformal mapping of circular quadrangles with a cut which belong to the class of polygons in polar grids, and then, with regard of found solutions, to pentagons of a more complicated structure, which are not polar. In all cases, we present the complete solution of the problem on parameters.