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Composite Asymptotic Expansions and Difference Equations

Abstract : 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.
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Augustin Fruchard, Reinhard Schäfke. Composite Asymptotic Expansions and Difference Equations. Revue Africaine de la Recherche en Informatique et Mathématiques Appliquées, INRIA, 2015, 20, pp.63-93. ⟨hal-01320625⟩



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