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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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“Best possible” upper and lower bounds for the zeros of the Bessel function $J_\nu (x)$
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by C. K. Qu and R. Wong PDF
Trans. Amer. Math. Soc. 351 (1999), 2833-2859 Request permission

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, $\ldots$, \[ \nu - \frac {a_k}{2^{1/3}} \nu ^{1/3} < j_{\nu ,k} < \nu - \frac {a_k}{2^{1/3}} \nu ^{1/3} + \frac {3}{20} a_k^2 \frac {2^{1/3}}{\nu ^{1/3}} . \] These bounds coincide with the first few terms of the well-known asymptotic expansion \[ j_{\nu ,k} \sim \nu - \frac {a_k}{2^{1/3}} \nu ^{1/3} + \frac {3}{20} a_k^2 \frac {2^{1/3}}{\nu ^{1/3}} + \cdots \] 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
  • MR Author ID: 192744
  • Email: mawong@cityu.edu.hk
  • Received by editor(s): July 22, 1996
  • Received by editor(s) in revised form: March 18, 1997
  • Published electronically: March 18, 1999
  • © Copyright 1999 American Mathematical Society
  • 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
  • MathSciNet review: 1466955