Skip to main content
SpringerLink
Log in

Value distribution of Painlevé transcendents of the first and the second kind

  • Published:
Journal d’Analyse Mathématique Aims and scope

Abstract

We examine value distribution properties of the first and the second Painlevé transcendents. For every transcendental meromorphic solution ϕ(z) (resp. ψ(z)) of the first (resp. second) Painlevé equation, the deficiency δ(g,ϕ) (resp. δ(g, ψ)) of a small functiong(z) does not exceed 1/2. Furthermore, for ϕ(z), the ramification index satisfies ϑ(gφ)≤5/12.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. Hille,Ordinary Differential Equations in the Complex Domain, Wiley, New York, London, Sydney, 1976.

    MATH  Google Scholar 

  2. A. Hinkkanen and I. Laine,Solutions of the first and second Painlevé equations are meromorphic, J. Analyse Math.79 (1999), 345–377.

    Article  MATH  MathSciNet  Google Scholar 

  3. H. Kießling,Zur Werteverteilung der Lösungen algebraischer Differentiagleichungen, Dissertation, Fachbereich 3 Mathematik, Technischen Universität Berlin, 1996.

  4. I. Laine,Nevanlinna Theory and Complex Differential Equations, de Gruyter, Berlin, New York, 1993.

  5. A. Z. Mohon'ko and V. D. Mohon'ko,Estimates for the Nevanlinna characteristics of some classes of meromorphic functions and their applications to differential equations, Sibirsk Mat. Zh.15 (1974), 1305–1322 [Russian]. Engl. Transl.: Siberian Math. J.15 (1974), 921–934.

    MathSciNet  Google Scholar 

  6. Y. Murata,Rational solutions of the second and the fourth Painlevé equations, Funkcial. Ekvac.28 (1985), 1–32.

    MATH  MathSciNet  Google Scholar 

  7. H. Schubart,Zur Wertverteilung der Painlevéschen Transzendenten, Arch. Math.7 (1956), 284–290.

    Article  MATH  MathSciNet  Google Scholar 

  8. H. Schubart and H. Wittich,Über die Lösungen der beiden ersten Painlevéschen Differentialgleichungen, Math. Z.66 (1957), 364–370.

    Article  MATH  MathSciNet  Google Scholar 

  9. Y. Sibuya,Global Theory of a Second Order Linear Ordinary Differential Equation with a Polynomial Coefficient, North-Holland, Amsterdam, Oxford, 1975.

    MATH  Google Scholar 

  10. N. Steinmetz,Zur Wertverteilung der Lösungen der vierten Painlevéschen Differentiagleichung, Math. Z.181 (1982), 553–561.

    Article  MATH  MathSciNet  Google Scholar 

  11. H. Wittich,Eindeutige Lösungen der Differentialgleichungen w″=P(z,w), Math. Ann.125 (1953), 355–365.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Keio University, 3-14-1, Hiyoshi, 223-8522, Yokohama, Japan

    Shun Shimomura

Authors
  1. Shun Shimomura
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shun Shimomura.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shimomura, S. Value distribution of Painlevé transcendents of the first and the second kind. J. Anal. Math. 82, 333–346 (2000). https://doi.org/10.1007/BF02791233

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02791233

Keywords

Search

Navigation