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Introduction à la multisommabilité. (Introduction to multisummability). (French) Zbl 0722.34005
The paper is an informal introduction into the “multisummability theory” [J. Martinet and J.-P. Ramis, Computer algebra and differential equations, Colloq., Comput. Math. Appl., 117-214 (1988; Zbl 0722.12007)] and its applications in the theory of linear differential equations in the complex domain. From the fundamental formal matrix solution \(\hat Y(t)=\hat H(t)t^ L \exp Q(1/t)\) of a linear differential system \(Y'=A(x)Y\) where \(t=x^{1/p}\) is a suitable ramification, one may obtain a “true solution” by summation of the formal matrix \(\hat H.\) It is suggested that there exists a factorization of the matrix \(\hat H=\hat H_ 1\hat H_ 2...\hat H_ r\) such that \(\hat H_ i\) is \(k_ i\)- summable for every \(i=1,2,...,r\). The theory is illustrated by several simple examples that are analyzed in some detail.

MSC:
34M99 Ordinary differential equations in the complex domain
40G10 Abel, Borel and power series methods
12H05 Differential algebra
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