# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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) $L\left(y\right)=0$ 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 $S=\left\{{S}_{\lambda }:\lambda \in {\Lambda }\right\}$ and proves the following main theorem. $\forall \lambda \in {\Lambda }$ there exists a unique solution ${w}_{\lambda }$ of (1) such that

${w}_{\lambda }\sim {e}^{\lambda z}{z}^{{\mu }_{\lambda }}\sum _{s=0}^{\infty }\frac{{a}_{s\lambda }}{{z}^{s}}$

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.

##### MSC:
 34E05 Asymptotic expansions (ODE) 34A30 Linear ODE and systems, general