Resurgence-summability of ramified formal series depending of a parameter and solutions of linear differential equations
type: Item
instance of: scholarly article
title: Resurgence-summability of ramified formal series depending of a parameter and solutions of linear differential equations
zbMATH Open document ID: 1205.34122
author: Jean-Marc Rasoamanana
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2010-10-12T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/115879
review text: The author investigates the Borel-resummable and resurgent properties of formal power series, solutions of linear differential equations of the form \[ Q_n(x,b)d^2\Phi (x)/dx^2=P_m(x,a)\Phi (x), \] where \[ P_m=x^m+a_1x^{m-1}+\cdots +a_m,\quad Q_n=x^n+b_1x^{n-1}+\cdots +b_n. \] In particular, he studies the analytic dependence of the associated Borel sum in the parameters of the equations. He also analyses the whole resurgent structure associated with these formal power series thanks to the tool of microfunctions and the so-called alien derivatives. This analysis allows to extend some results due to \textit{Y. Sibuya} in [Global theory of a second order linear differential equation with a polynomial coefficient. Amsterdam-Oxford: North-Holland Publishing Company; New York: American Elsevier Publishing Company (1975; Zbl 0322.34006)], and to illustrate the powerful alien calculus of J. Ecalle.
Mathematics Subject Classification ID: 34M30
Mathematics Subject Classification ID: 34M03
Mathematics Subject Classification ID: 34M25
zbMATH DE Number: 5799093
zbMATH Keywords: linear differential equations
zbMATH Keywords: resurgence
zbMATH Keywords: Borel-summability
zbMATH Keywords: alien calculus
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
OpenAlex ID: W2028143818
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DOI: 10.5802/AFST.1245
description: scientific article; zbMATH DE number 5799093