Theory of Computing Systems

, Volume 50, Issue 2, pp 354–369 | Cite as

Computing Conformal Maps of Finitely Connected Domains onto Canonical Slit Domains

Article

Abstract

We continue the research initiated in Andreev et al. (Computing conformal maps onto circular domains, 2010, submitted) on the computability of conformal mappings of multiply connected domains by showing that the conformal maps of a finitely connected domain onto the canonical slit domains can be computed uniformly from the domain and its boundary. Along the way, we demonstrate the computability of finding analytic extensions of harmonic functions and solutions to Neuman problems. These results on conformal mapping then follow easily from M. Schiffer’s constructions (Dirichlet Principle, Conformal Mapping and Minimal Surfaces, Interscience, New York, 1950).

Keywords

Computable analysis Conformal mapping 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akhiezer, N.I.: Aerodynamical investigations. Ukrain. Akad. Nauk Trudi Fiz.-Mat. Viddilu 7 (1928) Google Scholar
  2. 2.
    Andreev, V., Daniel, D., McNicholl, T.H.: Computing conformal maps onto circular domains (2010, submitted) Google Scholar
  3. 3.
    Binder, I., Braverman, M., Yampolsky, M.: On the computational complexity of the Riemann mapping. Arch. Math. 45, 221–239 (2007) MATHMathSciNetGoogle Scholar
  4. 4.
    Bishop, E., Bridges, D.: Constructive Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 279. Springer, Berlin (1985) CrossRefMATHGoogle Scholar
  5. 5.
    Conway, J.: Functions of One Complex Variable I, 2nd edn. Graduate Texts in Mathematics, vol. 11. Springer, Berlin (1978) CrossRefGoogle Scholar
  6. 6.
    Crowdy, D.: Schwarz-Christoffel mappings to unbounded multiply connected polygonal regions. Math. Proc. Camb. Philos. Soc. 142(2), 319–339 (2007) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    DeLillo, T.K.: Schwarz-Christoffel mapping of bounded, multiply connected domains. Comput. Methods Funct. Theory 6(2), 275–300 (2006) MATHMathSciNetGoogle Scholar
  8. 8.
    Gaier, D.: Untersuchung zur Durchführung der konformen Abbildung mehrfach zusammenhängender Gebiete. Arch. Ration. Mech. Anal. 3, 149–178 (1959) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Garnett, J.B., Marshall, D.E.: Harmonic Measure. New Mathematical Monographs, vol. 2. Cambridge University Press, Cambridge (2005) CrossRefMATHGoogle Scholar
  10. 10.
    Halsey, N.D.: Potential flow analysis of multielement airfoils using conformal mapping. AIAA J. 17(12), 1281–1288 (1979) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Henrici, P.: Applied and Computational Complex Analysis. Pure and Applied Mathematics, vol. 3. Wiley, New York (1986). Discrete Fourier analysis—Cauchy integrals—construction of conformal maps—univalent functions, A Wiley-Interscience Publication MATHGoogle Scholar
  12. 12.
    Hertling, P.: An effective Riemann mapping theorem. Theor. Comput. Sci. 219, 225–265 (1999) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Koebe, P.: Über die konforme Abbildung mehrfach-zusammenhängender Bereiche. Jahresber. Dtsch. Math.-Ver. 19, 339–348 (1910) MATHGoogle Scholar
  14. 14.
    Koebe, P.: Abhandlungen zur Theorie der Konformen Abbildung. Acta Math. 41, 305–344 (1918) CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    McNicholl, T.: An effective Carathéodory theorem (2010, submitted) Google Scholar
  16. 16.
    Mityushev, V.V., Rogosin, S.V.: Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions. CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 108. Chapman and Hall, New York (2000) MATHGoogle Scholar
  17. 17.
    Nehari, Z.: Conformal Mapping. McGraw-Hill Book, New York (1952) MATHGoogle Scholar
  18. 18.
    Pour-El, M.B., Richards, J.I.: Computability in Analysis and Physics. Perspectives in Mathematical Logic. Springer, Berlin (1989) Google Scholar
  19. 19.
    Schiffer, M.: Some recent developments in the theory of conformal mapping. In: Courant, R. (ed.) Dirichlet Principle, Conformal Mapping and Minimal Surfaces Interscience, New York (1950) Google Scholar
  20. 20.
    Weihrauch, K.: Computable Analysis. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2000) Google Scholar
  21. 21.
    Weihrauch, K., Grubba, T.: Elementary computable topology. J. Univers. Comput. Sci. 15(6), 1381–1422 (2009) MATHMathSciNetGoogle Scholar
  22. 22.
    Ziegler, M., Brattka, V.: Computability in linear algebra. Theor. Comput. Sci. 326, 187–211 (2004) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsLamar UniversityBeaumontUSA

Personalised recommendations