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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.

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