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Asymptotic solutions of linear ordinary differential equations at an irregular singularity of rank 1. (English) Zbl 0896.34049
New existence theorems are constructed for asymptotic solutions of linear ordinary differential equations of arbitrary order in the neighborhood of an irregular singularity of rank 1 with distinct characteristic values. Let (1) $L(y)= 0$ be an equation of the above type and $\Lambda$ be the set of characteristic values of (1). The author builds on the base of $\Lambda$ a set of canonical sectors $S= \{S_\lambda: \lambda\in\Lambda\}$ and proves the following main theorem. $\forall\lambda\in \Lambda$ there exists a unique solution $w_\lambda$ of (1) such that $$w_\lambda\sim e^{\lambda z}z^{\mu_\lambda} \sum^\infty_{s= 0} {a_{s\lambda}\over z^s}$$ as $z\to\infty$, uniformly in any closed sector properly interior to $S_\lambda$. Furthermore, this asymptotic expansion can be differentiated $n-1$ times under the same circumstances, and the $n$ solutions $w_\lambda$ are linearly independent.

34E05Asymptotic expansions (ODE)
34A30Linear ODE and systems, general