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Computational Methods and Function Theory

, Volume 19, Issue 1, pp 77–96 | Cite as

Conformal Mapping onto a Doubly Connected Circular Arc Polygonal Domain

  • Ulrich Bauer
  • Wolfgang LaufEmail author
Article
  • 47 Downloads

Abstract

This paper presents a construction principle for the Schwarzian derivative of conformal mappings from an annulus onto doubly connected domains bounded by polygons of circular arcs.

Keywords

Conformal mapping Circular arc polygon Schwarzian derivative 

Mathematics Subject Classification

30C20 30C30 

Notes

References

  1. 1.
    Bjørstad, P., Grosse, E.: Conformal mapping of circular arc polygons. SIAM J. Sci. Stat. Comput. 8(1), 19–32 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Case, J.: Breakthrough in conformal mapping. SIAM News 41(1), 1–3 (2008)Google Scholar
  3. 3.
    Conway, J.B.: Functions of One Complex Variable II, Gradute Texts in Mathematics, vol. 159. Springer, New York (1995)CrossRefGoogle Scholar
  4. 4.
    Crowdy, D.G.: The Schwarz–Christoffel mapping to bounded multiply connected polygonal domains. Proc. R. Soc. 461, 2653–2678 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Crowdy, D.G., Fokas, A.S.: Conformal mappings to a doubly connected polycircular arc domain. Proc. R. Soc. A 463, 1885–1907 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Crowdy, D.G., Fokas, A.S., Green, C.C.: Conformal mappings to multiply connected polycircular arc domains. Comput. Methods Funct. Theory 11(2), 685–706 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Crowdy, D.G., Marshall, J.S.: Conformal mappings between canonical multiply connected domains. Comput. Methods Funct. Theory 6(1), 59–76 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    DeLillo, T.K.: Schwarz–Christoffel mapping of bounded, multiply connected domains. Comput. Methods Funct. Theory 6(2), 275–300 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    DeLillo, T.K., Driscoll, T.A., Elcrat, A.R., Pfaltzgraff, J.A.: Computation of Multiply connected Schwarz–Christoffel maps for exterior domains. Comput. Methods Funct. Theory 6(2), 301–315 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    DeLillo, T.K., Elcrat, A.R., Kropf, E.H., Pfaltzgraff, J.A.: Efficient calculation of Schwarz–Christoffel transformations for multiply connected domains using Laurent series. Comput. Methods Funct. Theory 13(2), 307–336 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    DeLillo, T.K., Elcrat, A.R., Pfaltzgraff, J.A.: Schwarz–Christoffel mapping of the annulus. Soc. Indus. Appl. Math. 43(3), 469–477 (2001)MathSciNetzbMATHGoogle Scholar
  12. 12.
    DeLillo, T.K., Elcrat, A.R., Pfaltzgraff, J.A.: Schwarz–Christoffel mapping of multiply connected domains. J. Anal. Math. 94(1), 14–47 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    DeLillo, T.K., Kropf, E.H.: Numerical computation of the Schwarz–Christoffel transformation for multiply connected domains. SIAM J. Sci. Comput. 33(3), 1369–1394 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Driscoll, T.A., Trefethen, L.N.: Schwarz–Christoffel Mapping. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  15. 15.
    Ford, L.R.: Automorphic Functions, 2nd edn. Chelsea Publishing Company, New York (1951)zbMATHGoogle Scholar
  16. 16.
    Henrici, P.: Applied and Computational Complex Analysis, vol. 1. Wiley Classics Library, New York (1974)zbMATHGoogle Scholar
  17. 17.
    Hille, E.: Ordinary Differential Equations in the Complex Domain. Wiley, New York (1976)zbMATHGoogle Scholar
  18. 18.
    Howell, L.H.: Numerical conformal mapping of circular arc polygons. J. Comput. Appl. Math. 46, 7–28 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mityushev, V.: Schwarz–Christoffel formula for multiply connected domains. Comput. Methods Funct. Theory 12(2), 449–463 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nehari, Z.: Conformal Mapping. McGraw-Hill, New York (1952)zbMATHGoogle Scholar
  21. 21.
    Pommerenke, C.: Boundary Behaviour of Conformal Maps. Springer, New York (1992)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer Science and MathematicsUniversity of Applied Sciences OTH RegensburgRegensburgGermany

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