Summary
A classical result (see R.Nevanlinna, Acta Math.,58 (1932), p. 345) states that for a second-order linear differential equation, w″ + P(z) w′ + Q(z) w=0, where P(z) and Q(z) are polynomials, there exist finitely many rays, arg z=ϕj, for j=1,..., m, such that for any solution w=f(z) ≢ 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 ε>0, 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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This research was supported in part by the National Science Foundation (DMS-84-20561 and DMS-87-21813).
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Bank, S.B. On the complex zeros of solutions of linear differential equations. Annali di Matematica 161, 83–112 (1992). https://doi.org/10.1007/BF01759633
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DOI: https://doi.org/10.1007/BF01759633