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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
DOI: https://doi.org/10.1090/S0002-9947-99-02165-0
Published electronically: March 18, 1999
MathSciNet review: 1466955
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Abstract | References | Similar Articles | Additional Information

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''.


References [Enhancements On Off] (What's this?)

  • 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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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: https://doi.org/10.1090/S0002-9947-99-02165-0
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

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