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On solutions to differential equations with “common zero” at infinity. (English) Zbl 0914.34006

The authors investigate the differential equation \[ z''+q(t,\nu)z=0,\quad t\in (0,\infty), \] where \(\lim _{t \to \infty }q(t,\nu)=1\), \(\int ^{\infty }| q(t, \nu)-1| dt<\infty \), \(q\) has some monotonicity properties for large \(t\) and \(\nu \) is a parameter. They define the \(\kappa\)th zero of a solution \(z(t,\nu)\) for real \(\kappa \) as a function \(c_{\kappa }(\nu)\) of \(\nu \) and derive some of its properties. In particular, solutions “close” to \(\sin t\), \(\cos t\) for large \(t\) are studied.
The authors investigate applications to the Bessel differential equation, e.g. the concavity of \(c_{\kappa }(\nu)\), and introduce a new pair of linearly independent solutions instead of the usual one \(J_{\nu }(t)\), \(Y_{\nu }(t)\).
Reviewer: F.Neuman (Brno)

MSC:

34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
34A30 Linear ordinary differential equations and systems
34M99 Ordinary differential equations in the complex domain
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