A parametric method of finding accessory parameters for the generalized Schwarz–Christoffel integrals

An approximate method of finding accessory parameters for the generalized Schwarz–Christoffel integrals has been suggested. The integrals provide conformal mappings of a half-plane onto the polygonal Riemann surfaces with inner branch points. The method is based on including the desired map into a one-parametric family of conformal mappings of the upper half-plane onto the Riemann surfaces which are obtained from some fixed Riemann surface by cutting it along an elongated polygonal slit. A system of ordinary differential equations for parameters of the Schwarz–Christoffel integrals, i.e., for the preimages of their vertexes and branch points, has been deduced. Application of the method consists in solving a number of successive Cauchy problems describing the process of moving of the end of the slit along the chains of the polygon. The solution obtained in the previous step forms the initial data for the Cauchy problem in the next step. A numeric example illustrating the method has been considered. For univalent mappings, a similar method was first suggested by P. P. Kufarev.