Resurgent methods and the first Painlevé equation

Eric Delabaere (1)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

The paper is and extended form of a course given at a CIMPA master class held in LIMA, Perù, in the summer of 2008. It aims at showing the resurgent methods acting on an example and, along that line, to extend the presentation of the resurgence theory of Jean Ecalle provided that the need. The example that we follow along this paper is the first Painlevé differential equation. Among various results, we detail the formal transform that brings Painlevé I to its normal form and we analyze its 1-summability. The complete resurgent structure is eventually shown for Painlevé I, as an upshot from the resurgence theory. The paper is self-contained and essentially aims at graduate students.

Mots clés: Resurgence; Asymptotics; Painlevé equation

Domaines: Analyse classique [math.CA]

Eric Delabaere. Resurgent methods and the first Painlevé equation. 2014. ⟨hal-01067086v1⟩