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On oscillation, continuation, and asymptotic expansions of solutions of linear differential equations. (English) Zbl 0743.34012

The author considers the problem of the extension to a higher-order equation of the following classical result due to Hille: For any second order differential equation \(w"+P(z)w'+Q(z)w=0\) where \(P(z)\) and \(Q(z)\) are polynomials, there exist finitely many rays, \(\arg z=\varphi_ j\) for \(j=1,2,\ldots,m\), with the property that for any \(\varepsilon>0\), all but finitely many zeros of any solution \(f\not\equiv 0\) must lie in the union of the sectors \(|\arg z-\varphi_ j|<\varepsilon\) for any \(j=1,1,\ldots,m\). A similar question has been studied by the author in some previous papers, in the case of the higher-order differential equation \(w^{(n)}+R_{n-1}(z)w^{(n-1)}+\cdots+R_ 0(z)w=0\). In the present paper the author shows some results when \(R_ 0\), \(R_ 1,\ldots,R_{n-1}\) belong to a logarithmic differential field of rank zero over \(F(a,b)\) where \(F(a,b)\) is a neighborhood system suitably defined.

MSC:

34M99 Ordinary differential equations in the complex domain
34A30 Linear ordinary differential equations and systems
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
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References:

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