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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{H}^ + $\end{document} and the solution ℘ of the corresponding Riemann-Hilbert problem in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{H}^ + $\end{document}. Methods for finding this mapping are described, and a technique for constructing an analytic function ℘+ in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{H}^ + $\end{document} 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.