The First Painlevé Equation

  • Eric Delabaere
Part of the Lecture Notes in Mathematics book series (LNM, volume 2155)


This chapter aims at introducing the reader to properties of the first Painlevé equation and its general solution. The definition of the first Painlevé equation is recalled (Sect. 2.1). We precise how the Painlevé property translates for the first Painlevé equation (Sect. 2.2), a proof of which being postponed to an appendix. We explain how the first Painlevé equation also arises as a condition of isomonodromic deformations for a linear ODE (Sect. 2.3 and Sect. 2.4). Some symmetry properties are mentioned (Sect. 2.5). We spend some times describing the asymptotic behaviour at infinity of the solutions of the first Painlevé equation and, in particular, we introduce the truncated solutions (Sect. 2.6). We eventually briefly comment the importance of the first Painlevé transcendents for models in physics (Sect. 2.7).


Singular Point Meromorphic Function Linear Ordinary Differential Equation Divergent Series Transcendental Meromorphic Function 
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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Eric Delabaere
    • 1
  1. 1.Département de MathématiquesUniversité d’AngersAngersFrance

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