On the complex zeros of solutions of linear differ

Authors: Steven B Bank

Publish Date: 1992/12/01

Volume: 161, Issue: 1, Pages: 83-112

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Abstract

A classical result see RNevanlinna Acta Math58 1932 p 345 states that for a secondorder linear differential equation w″ + Pz w′ + Qz w=0 where Pz and Qz are polynomials there exist finitely many rays arg z=ϕj for j=1 m such that for any solution w=fz ≢ 0 and any ε 0 all but finitely many zeros off lie in the union of the sectors ¦ arg z ϕj¦ ε for j=1 m In this paper we give a complete answer to the question of determining when the same result holds for equations of arbitrary order having polynomial coefficients We prove that for any such equation one of the following two properties must hold a for any ray arg z=ϕ and any ε 0 there is a solution f ≢ 0 of the equation having infinitely many zeros in the sector ¦arg z ϕ¦ ε or b there exist finitely many rays arg z=ϕj for j= 1 m such that for any ε0 all but finitely many zeros of any solution f ≢ 0 must lie in the union of the sectors ¦ arg z ϕj¦ ε for j=1 m In addition our method of proof provides an effective procedure for determining which of the two possibilities holds for a given equation and in the case when b holds our method will produce the rays arg z=ϕj We emphasize that our result applies to all equations having polynomial coefficients without exception In addition we mention that if the coefficients are only assumed to be rational functions our results will still give precise information on the possible location of the bulk of the zeros of any solution

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