Singularities of the Second Kind

Abstract

In this chapter, we explain the structure of asymptotic solutions of a system of differential equations at a singular point of the second kind. In §§XIII-1,XIII-2, and XIII-3, a basic existence theorem of asymptotic solutions in the sense of Poincaré is proved in detail. In §XII-4,this result is used to prove a block-diagonalization theorem of a linear system. The materials in §§XIII-1—XIII-4 are also found in [Si7]. The main topic of §XIII-5 is the equivalence between a system of linear differential equations and an n-th-order linear differential equation. The equivalence is based on the existence of a cyclic vector for a linear differential operator. The existence of cyclic vectors was originally proved in [Del]. In §XIII-6, we explain a basic theorem concerning the structure of solutions of a linear system at a singular point of the second kind. This theorem was proved independently in [Huk4] and [Tul]. In §XIII-7, the Newton polygon of a linear differential operator is defined. This polygon is useful when we calculate formal solutions of an n-th-order linear differential equation (cf. [St]). In §XIII-8, we explain asymptotic solutions in the Gevrey asymptotics. To understand materials in §XIII-8, the expository paper [Ram3] is very helpful. In §§XIII-1—XIII-4, the singularity is at x = ∞, but from §XIII-5 through §XIII-8, the singularity is at x = 0. Any singularity at x=∞ is changed to a singularity of the same kind at x=0 by the transformation $x = \frac{1}{\xi }.$