On the Order of Solutions of Analytic Linear Differential Equations

(1) d/dxX(z) = A{z)X{z), where X(z) is an n x 1 column vector and A(z) is an n x n matrix of singlevalued meromorphic functions in a neighbourhood of the (isolated) singular point 0. Sometimes we write A instead of A(z) but we always consider a matrix of functions with such a notation unless explicitly stated otherwise. Each fundamental solution matrix for (1) can be represented near zero as (2) 0(s) = 8(z)z*, where S(z) is an n x n matrix of single-valued analytic functions in some deleted neighbourhood of the origin and R is an n x n constant matrix (called the monodromy matrix) which displays the multivaluedness of the fundamental set of solutions as they are analytically continued about the origin. (See [1] 108-9.) In order to study the growth of solutions near zero, we define the norm of a matrix as the sum of the moduli of its components. Since z consists of finite sums of products of powers of z and logz, and if we make z remain in some sector of fixed angle as \z\ -> 0, then \\z\\ = Odzl"") for some real number r ([1] 113). The more interesting factor in terms of growth is \\S(z) ||. There are two cases which can arise. The first is the classically discussed case of the regular singular point in which (3) \\S(z)\\ = 0(\z\-z) as |*|->0