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``Best possible' upper and lower bounds for the zeros of the Bessel function

Authors:	C K Qu R Wong

Institution:	Department of Applied Mathematics, Tsinghua University, Beijing, China R. Wong ; Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong

Abstract:	Let $j_{\nu,k}$ denote the -th positive zero of the Bessel function $J_\nu(x)$ . In this paper, we prove that for $\nu>0$ and , 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 $\nu\to\infty$ , being fixed, where is the -th negative zero of the Airy function $\operatorname{Ai}(x)$ , and so are ``best possible'.

Keywords:	Bessel functions zeros inequalities asymptotic expansions

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