Advertisement

Theory of Computing Systems

, Volume 50, Issue 4, pp 579–588 | Cite as

An Effective Carathéodory Theorem

  • Timothy H. McNicholl
Article

Abstract

By means of the property of effective local connectivity, the computability of finding the Carathéodory extension of a conformal map of a Jordan domain onto the unit disk is demonstrated.

Keywords

Computable analysis Constructive analysis Complex analysis Conformal mapping Effective local connectivity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bishop, E., Bridges, D.: Constructive Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 279. Springer, Berlin (1985) zbMATHCrossRefGoogle Scholar
  2. 2.
    Brattka, V.: Plottable real number functions and the computable graph theorem. SIAM J. Comput. 38(1), 303–328 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Conway, J.: Functions of One Complex Variable I, 2nd edn. Graduate Texts in Mathematics, vol. 11. Springer, Berlin (1978) CrossRefGoogle Scholar
  4. 4.
    Conway, J.: Functions of One Complex Variable II. Graduate Texts in Mathematics, vol. 159. Springer, Berlin (1995) zbMATHCrossRefGoogle Scholar
  5. 5.
    Daniel, D., McNicholl, T.: Effective local connectivity properties. Submitted. Preprint available at http://www.cs.lamar.edu/faculty/mcnicholl
  6. 6.
    Garnett, J., Marshall, D.E.: Harmonic Measure. New Mathematical Monographs, vol. 2. Cambridge University Press, Cambridge (2005) zbMATHCrossRefGoogle Scholar
  7. 7.
    Gordon, B.O., Julian, W., Mines, R., Richman, F.: The constructive Jordan curve theorem. Rocky Mt. J. Math. 5, 225–236 (1975) CrossRefGoogle Scholar
  8. 8.
    Greene, R., Krantz, S.: Function Theory of One Complex Variable. Graduate Studies in Mathematics. AMS, Providence (2002) zbMATHGoogle Scholar
  9. 9.
    Hertling, P.: An effective Riemann Mapping Theorem. Theor. Comput. Sci. 219, 225–265 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Miller, J.: Effectiveness for embedded spheres and balls. In: Brattka, V., Schröder, M., Weihrauch, K. (eds.) CCA 2002, Computability and Complexity in Analysis. Electronic Notes in Computer Science, vol. 66, pp. 127–138. Elsevier, Amsterdam (2002) Google Scholar
  11. 11.
    Miller, J.: Degrees of unsolvability of continuous functions. J. Symb. Log. 69, 555–584 (2004) zbMATHCrossRefGoogle Scholar
  12. 12.
    Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987) zbMATHGoogle Scholar
  13. 13.
    Weihrauch, K.: Computable Analysis. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2000) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsLamar UniversityBeaumontUSA

Personalised recommendations