A constructive solution to the Riemann–Hilbert boundary value problem as well as to the 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.
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Vladimir V. Mityushev, Sergei V. Rogosin, On the Riemann–Hilbert Problem with a Piecewise Constant Matrix. Z. Anal. Anwend. 27 (2008), no. 1, pp. 53–66DOI 10.4171/ZAA/1343