Augustin Fruchard ; Reinhard Schäfke - Composite Asymptotic Expansions and Difference Equations

arima:1986 - Revue Africaine de Recherche en Informatique et Mathématiques Appliquées, August 1, 2015, Volume 20 - 2015 - Special issue - Colloquium in Honor of Éric Benoît - https://doi.org/10.46298/arima.1986
Composite Asymptotic Expansions and Difference Equations

Authors: Augustin Fruchard 1,2; Reinhard Schäfke ORCID-iD3

  • 1 Laboratoire de Mathématiques Informatique et Applications
  • 2 Laboratoire de Mathématiques Informatique et Applications [UHA]
  • 3 Institut de Recherche Mathématique Avancée

Difference equations in the complex domain of the form y(x+ϵ)−y(x)=ϵf(y(x))/y(x) are considered. The step size ϵ>0 is a small parameter, and the equation has a singularity at y=0. Solutions near the singularity are described using composite asymptotic expansions. More precisely, it is shown that the derivative v′ of the inverse function v of a solution (the so-called Fatou coordinate) admits a Gevrey asymptotic expansion in powers of the square root of ϵ, denoted by η, involving functions of y and of Y=y/η. This also yields Gevrey asymptotic expansions of the so-called Écalle-Voronin invariants of the equation which are functions of epsilon. An application coming from the theory of complex iteration is presented.

https://doi.org/10.46298/arima.1986
Source: HAL:hal-01320625v1
Volume: Volume 20 - 2015 - Special issue - Colloquium in Honor of Éric Benoît
Published on: August 1, 2015
Submitted on: January 20, 2015
Keywords: Difference equation with small step size, composite asymptotic expansion, Gevrey asymptotic, Fatou coordinate, Écalle-Voronin invariant.,Équation aux différences à petit pas, développement asymptotique combiné, asymptotique Gevrey, coordonnée de Fatou, invariant d'Écalle-Voronin.,[INFO] Computer Science [cs],[MATH] Mathematics [math]

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