Properties of the series solution for Painlevé I

Abstract

We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painlevé equation. For the general solution, explicit recursive formulae for the Taylor expansion of the tau-function around a zero are given, which are natural extensions of analogous formulae for the elliptic sigma function, as given by Weierstrass. Numerical and exact results on the symmetric solution which is singular at the origin are also presented.

Keywords:

• Painlevé equation
• tau-function
• sigma function

2000 Mathematics Subject Classification:

• 33E05
• 33E17
• 34M55