Singular Riemann-Hilbert problem in complex-shaped domains

In simply connected complex-shaped domains ℬ a Riemann-Hilbert problem with discontinuous data and growth condidions of a solution at some points of the boundary is considered. The desired analytic function ℱ(z) is represented as the composition of a conformal mapping of ℬ onto the half-plane H^ + and the solution ℘ of the corresponding Riemann-Hilbert problem in H^ + . Methods for finding this mapping are described, and a technique for constructing an analytic function ℘+ in H^ + in the terms of a modified Cauchy-type integral. In the case of piecewise constant data of the problem, a fundamentally new representation of ℘+ in the form of a Christoffel-Schwarz-type integral is obtained, which solves the Riemann problem of a geometric interpretation of the solution and is more convenient for numerical implementation than the conventional representation in terms of Cauchytype integrals.