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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1970-0255909-X
PII: S 0002-9939(1970)0255909-X
Keywords: Zeros of special functions, asymptotic expansions, approximation of zeros, Sturm comparison theorem, Airy functions, Bessel functions
Article copyright: © Copyright 1970 American Mathematical Society