Book ChapterDOI
Fourier Transform and Differential Equations
B. Malgrange
- pp 33-48
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TLDR
In this article, the Fourier transform is applied to the complex domain of linear differential equations with polynomial coefficients, and the Laplace method is used to make these choices in a more systematic way.Abstract:
We study, in the complex domain, the action of the Fourier transform on the solutions of ordinary linear differential equations with polynomial coefficients. In the classical “Laplace method”, there are some restrictions; also, some choice of integration contours seem rather unsystematic. We show how to remove these restrictions and how to make these choices in a more systematic way.read more
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Journal ArticleDOI
Resurgence, Stokes phenomenon and alien derivatives for level-one linear differential systems
TL;DR: In this article, a precise description of the singularities of the Borel transform of solutions of a level-one linear differential system is deduced from a proof of the summable-resurgence of the solutions by the perturbative method of J. Ecalle.
Journal ArticleDOI
Matrices de Stokes-Ramis et constantes de connexion pour les systèmes différentiels linéaires de niveau unique
TL;DR: In this paper, Loday-Richaud and Remy showed that the solutions of the new system are summable-resurgent and gave a description of singularities in the Borel plane.
Journal ArticleDOI
On the Relation between Linear Difference and Differential Equations with Polynomial Coefficients
TL;DR: In this paper, the authors presented the third part of a contribution to the "dictionary" of homogeneous linear differential equations with polynomial coefficients on one hand and corresponding difference equations on the other.
Divergent series and differential equations
TL;DR: In this paper, various known approaches to the summability of a class of series that contains all divergent series solutions of ODEs in the complex field are developed. But the authors focus on the case when the divergence depends only on one parameter (the level k or critical time) called k-summability.
Journal ArticleDOI
Resurgence and highest level’s connection-to-Stokes formulæ for some linear meromorphic differential systems
TL;DR: In this article, the authors consider a linear meromorphic differential system with several levels and prove that the Borel transforms of its highest level's reduced formal solutions are summable-resurgent and give the general form of all their singularities.