Simplification of a system of linear ordinary differential equations about a singular point.

Abstract

Preliminary remarks: Let us start with a general survey of the theory of asymptotic solutions of linear ordinary differential equations about an irregular singular point.Systems of equations which will be considered in this section are of the following type:(1. 1)where $r$ is a nonnegative integer, $y$ is an $mathrm{n}$ -dimensional vector, and $A(x)$ is an $n$ by $n$ matrix whose components are one-valued, bounded and holomorphic with respect to a complex variable $x$ for $|x|geqq N_{0}$ , $N_{0}$ being a positive constant.The point at $ x=infty$ is in general an irregular singular point of solutions of (1. 1).As it is known already [2, 5, 11], asymptotic solutions of the system (1. 1) are constructed in the following manner: In the first step, by a formal transformation, say $T$ , the system (1. 1) is reduced to another system similar to (1. 1), but with a negative integer $r$ .Since $T$ is formal, the system which is derived from (1. 1) by $T$ is considered only in a formal sense.It should be remarked that, if $r$ is negative, then the point at $ x=infty$ is in general a regular singular point.Therefore, the system of formal equations is integrated formally by the use of the same technique as in the theory of solutions about regular singular points.By combining these formal solutions with the formal transformation $T$ , we can construct formal solutions of the system (1.1).In the second step, these formal solutions of (1. 1) are proved to be asymptotic forms of certain actual solutions of (1. 1).The construction of the formal transformation $T$ is also not given in a single step.Actually $T$ is a product of a finite number of transformations, say $T_{1}$ , $T_{2}$ , $cdots$ , $T_{N}$ .Namely, $T=T_{1}T_{2}cdots T_{N}$ .Among $T_{j}$ there are five types of transformations: