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Computational Mathematics and Mathematical Physics

, Volume 54, Issue 12, pp 1826–1875 | Cite as

Singular Riemann-Hilbert problem in complex-shaped domains

  • S. I. Bezrodnykh
  • V. I. Vlasov
Article

Abstract

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 \(\mathbb{H}^ + \) and the solution ℘ of the corresponding Riemann-Hilbert problem in \(\mathbb{H}^ + \). Methods for finding this mapping are described, and a technique for constructing an analytic function ℘+ in \(\mathbb{H}^ + \) 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.

Keywords

Riemann-Hilbert problem Cauchy-type integral conformal mappings Schwarz-Christoffel integral hypergeometric functions 

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© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Dorodnicyn Computing CenterRussian Academy of SciencesMoscowRussia
  2. 2.Sternberg Astronomical InstituteMoscow State UniversityMoscowRussia

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