Weighted products and parametric resurgence. (English) Zbl 0834.34067

Ecalle, Jean

Boutet de Monvel, Louis (ed.), Analyse algébrique des perturbations singulières. I. Méthodes résurgentes. Conférences du symposium franco-japonais sur l’analyse algébrique des perturbations singulières, CIRM, Marseille-Luminy, France, October 20-26, 1991. Paris: Hermann. Trav. Cours. 47, 7-49 (1994).

The article is an introduction to the subject of resurgent monomials (expressed as weighted products or alternatively is weighted convolutions). A more exhaustive treatment will appear. Two types of divergence matched by two types of resurgence, i.e. equational and coequational ones, are discussed. Eight fundamental products including weighted multiplications sem, lem, som, lom, and the corresponding convolutions sec, lec, soc, loc, are defined. The monomials \(\text{sem}^A\) and \(\text{lem}^A\) were already dealt with by the author [Ann. Inst. Fourier 42, No. 1-2, 73–164 (1992; Zbl 0940.32013)]. The construction of tessellation coefficients \(\text{tes}^w\) and their vanishing was dealt with. Fourier analyses relate to alien differentiation of weighted products, canonical resurgence monomials, interpretation of the tessellation coefficients as sums of hyperlogarithms, moulds, resurgent functions, and alien calculus.
For the entire collection see [Zbl 0824.00035].

Reviewer: V.Burjan (Praha)

MSC:

34E05	Asymptotic expansions of solutions to ordinary differential equations
34M30	Asymptotics and summation methods for ordinary differential equations in the complex domain
39A10	Additive difference equations

Keywords:

resurgent monomials; resurgence; tessellation coefficients

Citations:

Zbl 0940.32013