Painlevé Differential Equations in the Complex Plane

This book is the first comprehensive treatment of Painlevé differential equations in the complex plane. Starting with a rigorous presentation for the meromorphic nature of their solutions, the Nevanlinna theory will be applied to offer a detailed exposition of growth aspects and value distribution o...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Hauptverfasser: Gromak, Valerii I, Laine, Ilpo, Shimomura, Shun
Format: E-Book Buch
Sprache:Englisch
Veröffentlicht: Germany De Gruyter 2008
Walter de Gruyter
Walter de Gruyter GmbH
Ausgabe:1
Schriftenreihe:De Gruyter Studies in Mathematics
Schlagworte:
Functions of complex variables
Komplexe Funktion
MATHEMATICS
MATHEMATICS / General
MATHEMATICS / Mathematical Analysis
Painlevé equations
Painlevé-Gleichung
Komplexe Funktion
Painlevé-Gleichung
ISBN:3110198096, 9783110198096, 9783110173796, 3110173794
Online-Zugang:Volltext
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Inhaltsangabe:
  • 35 Rational and classical transcendental solutions of (P3) when γδ ≠ 0 -- 8 The fifth Painlevé equation (P5) -- 36 Preliminary observations -- 37 The behavior of solutions near z = 0 and z = ∞ -- 38 The special case δ = 0, γ ≠ 0 -- 39 The Bäcklund transformations of (P5) -- 40 Rational and one-parameter families of solutions -- 41 Connection between (P3) and (P5) revisited -- 9 The sixth Painlevé equation (P6) -- 42 General properties of solutions -- 43 Pairs of differential equations equivalent to (P6) -- 44 A Riccati differential equation related to (P6) -- 45 A first order algebraic differential equation related to (P6) -- 46 Singular points of solutions of (P6) -- 47 Connection formulae between solutions of (P6) -- 48 Elementary solutions of (P6) -- 10 Applications of Painlevé equations -- 49 Partial differential equations related to Painlevé equations -- 50 Discrete Painlevé equations -- Appendix A Local existence and uniqueness of solutions of complex differential equations -- Appendix B Basic notations and facts in the Nevanlinna theory -- Bibliography -- Index
  • Intro -- Introduction -- 1 Meromorphic nature of solutions -- 1 The first Painlevé equation (P1) -- 2 The second Painlevé equation (P2) -- 3 The third Painlevé equation (P3) -- 4 The fourth Painlevé equation (P4) -- 5 The fifth Painlevé equation (P5) -- 6 The sixth Painlevé equation (Pe) -- 2 Growth of Painlevé transcendents -- 7 Growth of first Painlevé transcendents -- 8 Growth of second and fourth Painlevé transcendents -- 9 Growth of third and fifth Painlevé transcendents -- 3 Value distribution of Painlevé transcendents -- 10 Deficiencies and ramification indices -- 11 The second main theorem for Painlevé transcendents -- 12 Value distribution with respect to small target functions -- 4 The first Painlevé equation (P1) -- 13 Nonexistence of the first integrals -- 14 Representation of solutions as quotient of entire functions -- 15 Special expansions of solutions -- 16 Higher order analogues of (P1) -- 5 The second Painlevé equation (P2) -- 17 Canonical representation of solutions -- 18 Poles of second Painlevé transcendents -- 19 The Bäcklund transformations of (P2) -- 20 Rational solutions of (P2) -- 21 The Airy solutions of (P2) -- 22 Higher order analogues of (P2) -- 6 The fourth Painlevé equation (P4) -- 23 Preliminary remarks -- 24 Poles of fourth Painlevé transcendents -- 25 Connection formulae between solutions of (P4) -- 26 Rational solutions of (P4) -- 27 The complementary error function hierarchy -- 28 A half-integer hierarchy of solutions -- 7 The third Painlevé equation (P3) -- 29 Preliminary remarks -- 30 Behavior of solutions around z = 0 and z = ∞ -- 31 Poles of third Painlevé transcendents -- 32 Canonical representation of solutions -- 33 The special case γ = 0, αδ ≠ 0 -- 34 Connection between solutions of (P3) and (P5)
  • Chapter 5. The second Painlevé equation (P2) --
  • Contents --
  • Chapter 2. Growth of Painlevé transcendents --
  • Chapter 3. Value distribution of Painlevé transcendents --
  • Chapter 1. Meromorphic nature of solutions --
  • Chapter 6. The fourth Painlevé equation (P4) --
  • Chapter 10. Applications of Painlevé equations --
  • Appendix B. Basic notations and facts in the Nevanlinna theory --
  • Appendix A. Local existence and uniqueness of solutions of complex differential equations --
  • Chapter 8. The fifth Painlevé equation (P5) --
  • Backmatter
  • Chapter 4. The first Painlevé equation (P1) --
  • Chapter 7. The third Painlevé equation (P3) --
  • Frontmatter --
  • Chapter 9. The sixth Painlevé equation (P6) --