“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

Full-text PDF Free Access

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, $\ldots$, \[ \nu - \frac {a_k}{2^{1/3}} \nu ^{1/3} < j_{\nu ,k} < \nu - \frac {a_k}{2^{1/3}} \nu ^{1/3} + \frac {3}{20} a_k^2 \frac {2^{1/3}}{\nu ^{1/3}} . \] 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

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