Advertisement

Geometric Characterization of Open image in new window-Conformal Mappings

  • K. GürlebeckEmail author
  • J. Morais
Chapter

Abstract

In this paper we study the description of monogenic functions in ℝ3 by their geometric mapping properties. The monogenic functions are considered at first as general quasi-conformal mappings. We consider the local mapping properties of a monogenic function and show that this class of functions can be defined as a special subclass of quasiconformal mappings. It is proved that monogenic functions with nonvanishing Jacobian determinant map infinitesimal balls onto special ellipsoids. The condition on the semiaxes of these ellipsoids is explicitly given.

Keywords

Conformal Mapping Quasiconformal Mapping Mapping Property Monogenic Function Geometric Characterization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahlfors, L.V.: Lectures on Quasiconformal Mappings. Van Nostrand, Princeton (1966). Reprinted by Wadsworth Inc., Belmont (1987) zbMATHGoogle Scholar
  2. 2.
    De Almeida, R., Malonek, H.R.: On a higher dimensional analogue of the Joukowski transformation. Num. Anal. Appl. Math., AIP Conf. Proc. 1048, 630–633 (2008) Google Scholar
  3. 3.
    Bock, S., Falcão, M.I., Gürlebeck, K., Malonek, H.R.: A 3-dimensional Bergman kernel method with applications to rectangular domains. J. Comput. Appl. Math. 189, 6–79 (2006) CrossRefGoogle Scholar
  4. 4.
    Boone, B.: Bergmankern en conforme albeelding: een excursie in drie dimensies. Proefschrift, Universiteit Gent Faculteit van de Wetenshappen (1991–1992) Google Scholar
  5. 5.
    Cerejeiras, P., Gürlebeck, K., Kähler, U., Malonek, H.R.: A quaternionic Beltrami-type equation and the existence of local homeomorphic solutions. ZAA 20, 17–34 (2001) zbMATHGoogle Scholar
  6. 6.
    Delanghe, R., Kraußhar, R.S., Malonek, H.R.: Differentiability of functions with values in some real associative algebras: approaches to an old problem. Bull. Soc. R. Sci. Liège 70(4–6), 231–249 (2001) zbMATHGoogle Scholar
  7. 7.
    Gürlebeck, K., Malonek, H.R.: A hypercomplex derivative of monogenic functions in ℝn+1. Complex Var. 39, 199–228 (1999) zbMATHGoogle Scholar
  8. 8.
    Gürlebeck, K., Morais, J.: On mapping properties of monogenic functions. CUBO Math. J. 11(01), 73–100 (2009) zbMATHGoogle Scholar
  9. 9.
    Gürlebeck, K., Morais, J.: Bohr type theorems for monogenic power series. Comput. Methods Funct. Theory 9(2), 633–651 (2009) zbMATHMathSciNetGoogle Scholar
  10. 10.
    Gürlebeck, K., Sprössig, W.: Quaternionic Analysis and Elliptic Boundary Value Problems. Math. Research, vol. 56. Akademieverlag, Berlin (1989) zbMATHGoogle Scholar
  11. 11.
    Gürlebeck, K., Sprössig, W.: Quaternionic Calculus for Engineers and Physicists. Wiley, New York (1997) zbMATHGoogle Scholar
  12. 12.
    Kraußhar, R.S., Malonek, H.R.: A characterization of conformal mappings in ℝ4 by a formal differentiability condition. Bull. Soc. R. Sci. Liège 70, 35–49 (2001) zbMATHGoogle Scholar
  13. 13.
    Kravchenko, V.V.: Applied Quaternionic Analysis. Research and Exposition in Mathematics, p. 28. Heldermann, Berlin (2003) zbMATHGoogle Scholar
  14. 14.
    Kravchenko, V.V., Shapiro, M.V.: Integral Representations for Spatial Models of Mathematical Physics. Research Notes in Mathematics. Pitman, London (1996) zbMATHGoogle Scholar
  15. 15.
    Liouville, L.: Extension au cas de trois dimensions de la question du tracé géographique. Application de l’analyse à la géometrie, G. Monge, Paris, pp. 609–616 (1850). Google Scholar
  16. 16.
    Malonek, H.R.: Power series representation for monogenic functions in ℝm+1 based on a permutational product. Complex Var. Theory Appl. 15, 181–191 (1990) zbMATHMathSciNetGoogle Scholar
  17. 17.
    Malonek, H.R.: Monogenic functions and M-conformal mappings. In: Brackx, F., et al. (eds.) Clifford Analysis and its Applications. Kluwer, Dordrecht (2001) Google Scholar
  18. 18.
    Mitelman, I.M., Shapiro, M.V.: Differentiation of the Martinelli–Bochner integrals and the notion of hyperderivability. Math. Nachr. 172, 211–238 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Morais, J.: Approximation by homogeneous polynomial solutions of the Riesz system in ℝ3. Ph.D. thesis, Bauhaus-Universität Weimar (2009) Google Scholar
  20. 20.
    Sudbery, A.: Quaternionic analysis. Math. Proc. Cambr. Philos. Soc. 85, 199–225 (1979) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Van der Waerden, B.L.: Algebra. Springer, Berlin (1966) zbMATHGoogle Scholar

Copyright information

© Springer-Verlag London 2010

Authors and Affiliations

  1. 1.Institut für Mathematik/PhysikBauhaus-Universität WeimarWeimarGermany

Personalised recommendations