On the complex zeros of solutions of linear differential equations
The author investigates the distribution of zeros of higher-order differential equations \(w^{(n)}+R_{n-1}(z)w^{(n-1)}+\dots+R_ 0(z)w=0\), where \(n>2\), and the coefficients \(R_ j(z)\) are polynomials. It is proved that one of the following two properties must hold: (a) for any ray, \(\arg z=\varphi\), and any \(\varepsilon>0\), there exists a solution \(w\not\equiv 0\) with infinite number of zeros in the sector \(|\arg z-\varphi|<\varepsilon\), or (b) there exist finitely many rays, \(\arg z=\varphi_ j\), for \(j=1,\dots,m,\) such that for any \(\varepsilon>0\), all but finitely many zeros of any solution \(w\not\equiv 0\) must lie in the union of the sectors \(|\arg z-\varphi_ j|<\varepsilon\) for \(j=1,\dots,m\). Notice that, according to a classical result due to R. Nevanlinna, only the case (b) is possible for \(n=2\).
distribution of zeros of higher-order differential equations
scientific article; zbMATH DE number 94992
scientific article