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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

``Best possible'' upper and lower bounds for
the zeros of the Bessel function $J_\nu(x)$


Authors: C. K. Qu and R. Wong
Journal: Trans. Amer. Math. Soc. 351 (1999), 2833-2859
MSC (1991): Primary 41A60, 33C45
Published electronically: March 18, 1999
MathSciNet review: 1466955
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Abstract: Let $j_{\nu,k}$ denote the $k$-th positive zero of the Bessel function $J_\nu(x)$. In this paper, we prove that for $\nu>0$ and $k=1$, 2, 3, $\ldots$,

\begin{displaymath}\nu - \frac{a_k}{2^{1/3}} \nu^{1/3} < j_{\nu,k} < \nu - \frac{a_k}{2^{1/3}} \nu^{1/3} + \frac{3}{20} a_k^2 \frac{2^{1/3}}{\nu^{1/3}} \,. \end{displaymath}

These bounds coincide with the first few terms of the well-known asymptotic expansion

\begin{displaymath}j_{\nu,k} \sim \nu - \frac{a_k}{2^{1/3}} \nu^{1/3} + \frac{3}{20} a_k^2 \frac{2^{1/3}}{\nu^{1/3}} + \cdots \end{displaymath}

as $\nu\to\infty$, $k$ being fixed, where $a_k$ is the $k$-th negative zero of the Airy function $\operatorname{Ai}(x)$, and so are ``best possible''.


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Additional Information

C. K. Qu
Affiliation: Department of Applied Mathematics, Tsinghua University, Beijing, China

R. Wong
Affiliation: Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong
Email: mawong@cityu.edu.hk

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02165-0
PII: S 0002-9947(99)02165-0
Keywords: Bessel functions, zeros, inequalities, asymptotic expansions
Received by editor(s): July 22, 1996
Received by editor(s) in revised form: March 18, 1997
Published electronically: March 18, 1999
Article copyright: © Copyright 1999 American Mathematical Society