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Nevanlinna functions with real zeros. (English) Zbl 1098.34072
Let \(\mathbb{C}\) be a field of complex numbers and \(P\in \mathbb{C}[x]\). For the equation \(w'' + Pw = 0\) the following problem is investigated. When has this equation a solution with all roots real? The question of describing equations with this property was proposed by D. A. Brannan and W. K. Hayman [Bull. London Math. Soc. 21, 1-35 (1989; Zbl 0695.30001)]see Probl. 2.71). It is easy to construct family of such equations solvable in quadratures. For irreducible equations (when \(deg(P)\) is odd, for example), similar results are not available. It is proved here, that in this case a countable set of examples of such equations exists.

34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
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