Wachstumsordnung und Index der Lösungen linearer Differentialgleichungen mit rationalen Koeffizienten

Abstract

Let z=∞ be an irregular singular point of the differential equation wⁿ+p_n−1(z)w⁽ⁿ⁻¹⁾+...+p₀(z)w=0 with rational coefficients. The functions of the canonical set of solutions relative to z=∞ are of the form

$$w(z) = z^\rho \cdot \sum { d_m (z) (\log z)^m , } \rho \varepsilon \mathbb{C}$$

with univalent functions d_m(z) in a neighbourhood of z=∞. Let λ(w)=max {λ(d_m)} denote the maximal order of growth of an irregular solution relative to z=∞, then it is shown that there exists a branch of w in the plane cut along a half ray, which attains the maximal order λ(w). An important tool for the proof is the index of the branches of w.