Singular Riemann-Hilbert problem in complex-shaped domains

In simply connected complex-shaped domains a"not sign 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 a"not sign onto the half-plane and the solution a"similar to of the corresponding Riemann-Hilbert problem in . Methods for finding this mapping are described, and a technique for constructing an analytic function a"similar to(+) in in the terms of a modified Cauchy-type integral. In the case of piecewise constant data of the problem, a fundamentally new representation of a"similar to(+) 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.