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''.

1.: M. S. Ashbaugh and R. D. Benguria, Sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions, Annals of Math., 135 (1992), 601-628. MR 93d:35105
2.: S. Breen, Uniform upper and lower bounds on the zeros of Bessel functions of the first kind, J. Math. Anal. Appl., 196 (1995), 1-17. MR 96k:33004
3.: Á. Elbert, An approximation for the zeros of Bessel function, Numer. Math., 59 (1991), 647-657. MR 92h:33008
4.: H. W. Hethcote, Asymptotic approximations with error bounds for zeros of Airy and cylindrical functions, Doctoral thesis, University of Michigan, Ann Arbor, 1968; Error bounds for asymptotic approximations of zeros of transcendental functions, SIAM J. Math. Anal., 1 (1970), 147-152. MR 42:744
5.: -, Bounds for zeros of some special functions, Proc. Amer. Math. Soc., 25 (1970), 72-74. MR 41:569
6.: A. Laforgia and M. E. Muldoon, Monotonicity and concavity properties of zeros of Bessel functions, J. Math. Anal. Appl., 98 (1984), 470-477. MR 85i:33008
7.: T. Lang and R. Wong, ``Best possible'' upper bounds for the first two positive zeros of the Bessel function $J_\nu(x)$ : the infinite case, J. Comp. Appl. Math., 71 (1996), 311-329. MR 97h:33016
8.: L. Lorch, Some inequalities for the first positive zeros of Bessel functions, SIAM J. Math. Anal., 24 (1993), 814-823. MR 95a:33010
9.: L. Lorch and R. Uberti, ``Best possible'' upper bounds for the first two positive zeros of the Bessel functions - the finite part, J. Comp. Appl. Math., 75 (1996), 249-258 MR 98b:33010
10.: F. W. J. Olver, Tables of Bessel functions of moderate or large orders, Math. Tables Nat. Phys. Lab., Vol.6, Her Majesty's Stationary Office, London, 1962. MR 26:5190
11.: -, Error bounds for asymptotic expansions in turning point problems, J. Soc. Indust. Appl. Math., 12 (1964), 200-214. MR 29:306
12.: -, Asymptotics and Special Functions, Academic Press, New York, 1974. MR 55:8655
13.: M. H. Protter, Can one hear the shape of a drum ? revisited, SIAM Rev., 29 (1987), 185-197. MR 88g:58185
14.: R. Wong and T. Lang, On the points of inflection of Bessel functions of positive order, II, Canad. J. Math., 43 (1991), 628-651. MR 92f:33008

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

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