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A note on the zeros of solutions of \(w''+P(z)w=0\), where P is a polynomial. (English) Zbl 0589.34008
For the class of equations described in the title, there is a classical result which states that there are (deg P)\(+2\) critical rays, arg z\(=\theta_ j\), such that for any \(\epsilon >0\), all but finitely many zeros of any solution f(z)\(\not\equiv 0\) must lie in the union of the sectors \(| \arg z-\theta_ j| <\epsilon\). We prove that any infinite set of zeros in such a sector (for sufficiently small \(\epsilon)\) must actually approach a definite ray, which is a translate of the critical ray (and which can be explicitly calculated). In addition, we estimate the rate at which the zeros approach the ray.

MSC:
34M99 Ordinary differential equations in the complex domain
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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