Asymptotic developments of analytic functions in cones

Article type: Research Article

Authors: Meana, Jorge Jiménez

Affiliations: Department of Mathematics, University of Oviedo, Edificio departamental de Energía Campus de Viesques, 33204‐Gijón, Asturias, Spain Tel.: ++34985182474; Fax: ++34985182014; E‐mail: meana@etsiig.uniovi.es

Abstract: In this paper we give a definition of asymptotic development of an analytic function of several complex variables. So far, there existed two different definitions which are joined by the definition given in this paper. This definition generalizes, in a natural sense, the one given for one variable. The generalization is made by changing the object polysector by a conoidal domain which is more general because all the polysectors are conoidal domains but not all the conoidal domains are polysectors. It is given a characterization of the analytic functions that admit asymptotic developments in a cone as those which restricted to a closed subcone admit an extension of class 𝒞∞. It is proven that, through this definition, the asymptotic developments of analytic functions of several variables verify all the good properties of the asymptotic developments of analytic functions of one variable.

Journal: Asymptotic Analysis, vol. 26, no. 3-4, pp. 239-256, 2001