Abstract
We derive a Schwarz-Christoffel formula for the conformal mapping of an arbitrary n-connected domain D bounded by mutually disjoint circles ¦z − a k¦ = rk, k = 1,2,…,n, onto the exterior of mutually disjoint polygons. The derivation is based on the exact solution to a Riemann-Hilbert problem for D without any geometric restriction imposed upon the location of the non-overlapping disks ¦z − ak¦ ≤ rk.
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D. G. Crowdy The Schwarz-Christoffel mapping to bounded multiply connected polygonal domains, Proc. Roy. Soc. A 461 (2005), 2653–2678.
D. G. Crowdy, Schwarz-Christoffel mappings to unbounded multiply connected polygonal regions, Math. Proc. Camb. Phil. Soc. 142 (2007), 319–339.
D. G. Crowdy, Conformal Mappings between Canonical Multiply Connected Domains, Comput. Methods Funct. Theory 6 no.1 (2006), 59–76.
D. G. Crowdy and A. S. Fokas Conformal mappings to a doubly connected polycircular arc domain, Proc. Roy. Soc. A 463 (2007), 1885–1907.
D. G. Crowdy, A. S. Fokas and C. C. Green, Conformal mappings to multiply connected polycircular arc domains, Comput. Methods Funct. Theory 11 no.2 (2011), 685–706.
T. K. DeLillo, Schwarz-Christoffel mapping of bounded, multiply connected domains, Comput. Methods Funct. Theory 6 no.2 (2006), 275–300.
T. K. DeLillo, T. A. Driscoll, A. R. Elcrat and J. A. Pfaltzgraff, Computation of multiply connected Schwarz-Christoffel map for exterior domains, Comput. Methods Funct. Theory 6 no.2 (2006), 301–315.
T. K. DeLillo, A. R. Elcrat and J. A. Pfaltzgraff, Schwarz-Christoffel mapping of multiply connected domains, J. d’Analyse Math. 94 (2004), 17–47.
T. A. Driscoll and L. N. Trefethen, Schwarz-Christoffel Mapping, Cambridge University Press, Cambridge, 2002.
F. D. Gakhov, Boundary Value Problems, Nauka, Moscow, 1977 (3rd edition) (in Russian); Engl. transl. of 1st ed.: Pergamon Press, Oxford, 1966.
V. V. Mityushev, Solution of the Hilbert boundary value problem for a multiply connected domain, Slupskie Prace Mat.-Przyr. 9a (1994), 37–69.
V. V. Mityushev, Convergence of the Poincaré series for classical Schottky groups, Proc. Amer. Math. Soc. 126 no.8 (1998), 2399–2406.
V. V. Mityushev, Hilbert boundary value problem for multiply connected domains, Complex Variables 35 (1998), 283–295.
V. V. Mityushev, Scalar Riemann-Hilbert problem for multiply connected domains, in: Th. M. Rassias and J. Brzdek (eds.), Functional Equations in Mathematical Analysis, Springer-Verlag, New York, 2011, 599–632.
V. V. Mityushev, Riemann-Hilbert problem for multiply connected domains and circular slit map, Comput. Methods Funct. Theory 11 no.2 (2011), 575–590.
V. V. Mityushev and S.V. Rogosin, Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions. Theory and Applications, Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall / CRC, Boca Raton etc., 2000.
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Mityushev, V. Schwarz-Christoffel Formula for Multiply Connected Domains. Comput. Methods Funct. Theory 12, 449–463 (2012). https://doi.org/10.1007/BF03321837
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DOI: https://doi.org/10.1007/BF03321837
En]Keywords
- Schwarz-Christoffel formula
- Riemann-Hilbert problem
- conformal mapping functional equation
- Poincaré series