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Proceedings of the American Mathematical Society

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Bounds for zeros of some special functions


Author: Herbert W. Hethcote
Journal: Proc. Amer. Math. Soc. 25 (1970), 72-74
MSC: Primary 34.42; Secondary 33.00
DOI: https://doi.org/10.1090/S0002-9939-1970-0255909-X
MathSciNet review: 0255909
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Abstract: For $n \geqq 1$ let ${b_n}$ and ${c_n}$ be zeros (ordered by increasing values) of $u(x)$ and $v(x)$, respectively, which are non-trivial solutions of $u'' + p(x)u = 0$ and $v'' + q(x)v = 0$ with continuous $p(x)$ and $q(x)$. It is shown that if ${b_n} - {c_n} \to 0$ as $n \to \infty ,\;p(x) \geqq q(x)$, and either $p(x)$ or $q(x)$ is nonincreasing, then ${b_n} \geqq {c_n}$ for $n \geqq 1$. Inequalities related to asymptotic expansions are obtained for the negative zeros ${a_n}$ of the Airy function $Ai(z)$ and the zeros ${j_{v,n}}$ of the Bessel function ${J_v}(x)$.


References [Enhancements On Off] (What's this?)

  • E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. MR 0010757
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Keywords: Zeros of special functions, asymptotic expansions, approximation of zeros, Sturm comparison theorem, Airy functions, Bessel functions
Article copyright: © Copyright 1970 American Mathematical Society