Modified method of successive conformal mappings of polygonal domains

The method for solving the inverse Schwarz–Christoffel problem — conformal mapping of a given polygonal area on the canonical domain, the unit circle — is developed in the paper. The method is based on the use of a sequence of conformal mappings related to the mapping of the polygon onto the upper half with discarded segments by a Cayley linear fractional transformation followed by sequential addition of the discarded segments to the upper half-plane. Model problems consider conformal mappings of circular, elliptical, and hyperbolic lunes to the upper half-plane. The solution of the presented model problems is based on the generalized Zhukovsky function; the obtained results expand the applicability of the Zhukovsky function as compared to existing methods of its application in the implementation of conformal mappings. The working ability of the solution was tested by specific examples in solving the problem of conformal mapping of quadrilateral and heptagonal domains to the unit circle. Recommendations for the algorithmic implementation of the method are presented.