Transactions of the American Mathematical Society

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

Author(s): C. K. Qu; R. Wong
Journal: Trans. Amer. Math. Soc. 351 (1999), 2833-2859.
MSC (1991): Primary 41A60, 33C45
Posted: March 18, 1999
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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''.

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Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 41A60, 33C45

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

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