Singularities of the First Kind

Abstract

In this chapter, we consider a system of differential equations $$x\frac{{d\vec{y}}}{{dx}} = \vec{f}\left( {x,\vec{y}} \right)$$ (E) assuming that the entries of the \({{\mathbb{C}}^{n}}\)-valued function \(\vec{f}\) are convergent power series in complex variables \(\left( {x,\vec{y}} \right) \in {{\mathbb{C}}^{{n + 1}}}\) with coefficients in ℂ, where x is a complex independent variable and \(\vec{y} \in {{\mathbb{C}}^{n}}\) is an unknown quantity. The main tool is calculation with power series in x. In §I-4, using successive approximations, we constructed power series solutions. However, generally speaking, in order to construct a power series solution \(\vec{y}\left( x \right) = \sum\limits_{{m = 1}}^{\infty } {{{x}^{m}}{{{\vec{a}}}_{m}}}\) this expression is inserted into the given differential equation to find relationships among the coefficients \({{\vec{a}}_{m}}\), and the coefficients \({{\vec{a}}_{m}}\) are calculated by using these relationships. In this stage of the calculation, we do not pay any attention to the convergence of the series. This process leads us to the concept of formal power series solutions (cf. §V-1). Having found a formal power series solution, we estimate \(|{{\vec{a}}_{m}}|\) to test its convergence. As the function \({{x}^{{ - 1}}}\vec{f}\left( {x,\vec{y}} \right)\) is not analytic at x = 0, Theorem I-4-1 does not apply to system (E). Furthermore, the existence of formal power series solutions of (E) is not always guaranteed. Nevertheless, it is known that if a formal power series solution of (E) exists, then the series is always convergent. This basic result is explained in §V-2 (cf. [CL, Theorem 3.1, pp. 117-119] and [Wasl, Theorem 5.3, pp. 22-25] for the case of linear differential equations]). In §V-3, we define the S-N decomposition for a lower block-triangular matrix of infinite order. Using such a matrix, we can represent a linear differential operator $$ \mathcal{L}\left[ {\vec y} \right]{\text{ = }}x\frac{{d\vec y}}{{dx}}{\text{ + }}\Omega {\text{(}}x{\text{)}} \vec y, $$ (LDO) Where \(\Omega \left( x \right)\) is an n x n matrix whose entries are formal power series in x with coefficients in ℂ n . In this way, we derive the S-N decomposition of L in §V-4 and a normal form of L in §V-5 (cf. [HKS]). The S-N decomposition of L was originally defined in [GerL]. In §V-6, we calculate the normal form of a given operator L by using a method due to M. Hukuhara (cf. [Si17, §3.9, pp. 85-89]). We explain the classification of singularities of homogeneous linear differential equations in §V-7. Some basic results concerning linear differential equations given in this chapter are also found in [CL, Chapter 4].