On the Riemann–Hilbert Problem with a Piecewise Constant Matrix

Abstract

A constructive solution to the Riemann–Hilbert boundary value problem as well as to the $\mathbb{C}$-linear conjugation problem (or Riemann problem) with a special piecewise constant matrix for a multiply connected domain is obtained. The result is based on a vector generalization of the scalar method of functional equations in complex domain and on application of the Christoffel–Schwarz transformation of the half-plane onto a circular polygon.