Resurgent Functions and Alien Calculus

Abstract

The present text is an introduction toÉcalle's theory of resurgent functions and alien calculus, in connection with problems of exponentially small separatrix splitting. An outline of the resurgent treatment of Abel's equation for resonant dynamics in one complex variable is included. Some proofs and details are omitted. The emphasis is on examples of nonlinear difference equations, as a simple and natural way of introducing the concepts.

Resurgence and highest level’s connection-to-Stokes formulæ for some linear meromorphic differential systems

“…-The case s " 1 is proved in [19,21,33] and the case s ě 2 stems obvious from the properties of variation (see [19,Lem. 3.6] for instance) when λ " p " 0.…”

“…Recall that C is also denoted by SING 0 by J. Écalle et al (cf. [33] for instance). Recall also that the elements of C are called micro-functions by B. Malgrange [21,22] by analogy with hyper-and micro-functions defined by Sato, Kawai and Kashiwara in higher dimensions.…”

Divergent Series, Summability and Resurgence III

“…For most of the materials presented here, we mainly refer to [Eca81-1, Eca92, Eca93-1, CNP93-1, MS016], see also [DP99, Sau006,OSS003]. Another approach to resurgence theory is provided in [SS96].…”

Alien derivatives of the WKB solutions of the Gauss hypergeometric differential equation with a large parameter

“…In this paper we apply these results to the Gauss hypergeometric differential equation with large parameters. We also refer to the works of Sauzin [15,16], where the relation between the Stokes automorphisms and the alien derivatives is clarified.…”

“…In [17] and [14], Stokes curves of Weber type are considered, and in [1] one of loop type is considered. Note that, in [8,9,15] and [16], the authors treat only the case of Weber type.…”