Über das Anwachsen der Lösungen linearer Differentialgleichungen mit rationalen Koeffizienten in Winkelräumen

Abstract

Let z=∞ be the only irregular singular point of the linear differential equation

$$(D)w^{(n)} + P_{n - 1} (z)w^{(n - 1)} + \cdots + P_o (z)w = 0$$

with rational coefficients p_j(z). If w is a multivalent and irregular solution of (D), we define the index I(τr,W), τ ∈ (0,1), of a branch W of w in the plane cut along a half ray. If r→∞, I(τr,W) possesses a finite number of aymptotic directions being exactly the asymptotic directions of the points z with maximal possible modulus of W(z). It follows that each branch W is of mean type σ(W)=|d|/λ in each sector containing an asymptotic direction of I. The possible values of the order of growth λ=λ(W)<∞ and the constant d are given by the PUISEUX-diagram of (D).