"Best possible" upper and lower bounds for the zeros of the bessel function Jv(x)

C. K. Qu; R. Wong

doi:10.1090/s0002-9947-99-02165-0

"Best possible" upper and lower bounds for the zeros of the bessel function J_v(x)

C. K. Qu
, R. Wong

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

45 Citations (Scopus)

Abstract

Let j_vk denote the k-th positive zero of the Bessel function J_v(x). In this paper, we prove that for v > 0 and k = 1, 2, 3, ..., v-a_k/2^1/3v^1/3 v,k <v - a_k/2^1/3v^1/3 + 3/20a_k ²2^1/3/v^1/3. These bounds coincide with the first few terms of the well-known asymptotic expansion j_v,k ∼ v - a_k/2^1/3v^1/3 + 3/20a_k ²2^1/3/v^1/3 as v → ∞, k being fixed, where a_k is the k-th negative zero of the Airy function Ai(i), and so are "best possible". ©1999 American Mathematical Society.

Original language	English
Pages (from-to)	2833-2859
Journal	Transactions of the American Mathematical Society
Volume	351
Issue number	7
DOIs	https://doi.org/10.1090/s0002-9947-99-02165-0
Publication status	Published - 1999
Externally published	Yes

Research Keywords

Asymptotic expansions
Bcssel functions
Inequalities
Zeros

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10.1090/s0002-9947-99-02165-0

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@article{3b1a43a3b7ae4e6c98df31285b016b56,

title = "{"}Best possible{"} upper and lower bounds for the zeros of the bessel function Jv(x)",

abstract = "Let jvk denote the k-th positive zero of the Bessel function Jv(x). In this paper, we prove that for v > 0 and k = 1, 2, 3, ..., v-ak/21/3v1/3 v,k k/21/3v1/3 + 3/20ak 221/3/v1/3. These bounds coincide with the first few terms of the well-known asymptotic expansion jv,k ∼ v - ak/21/3v1/3 + 3/20ak 221/3/v1/3 as v → ∞, k being fixed, where ak is the k-th negative zero of the Airy function Ai(i), and so are {"}best possible{"}. {\textcopyright}1999 American Mathematical Society.",

keywords = "Asymptotic expansions, Bcssel functions, Inequalities, Zeros",

author = "Qu, \{C. K.\} and R. Wong",

year = "1999",

doi = "10.1090/s0002-9947-99-02165-0",

language = "English",

volume = "351",

pages = "2833--2859",

journal = "Transactions of the American Mathematical Society",

issn = "0002-9947",

publisher = "American Mathematical Society",

number = "7",

}

TY - JOUR

T1 - "Best possible" upper and lower bounds for the zeros of the bessel function Jv(x)

AU - Qu, C. K.

AU - Wong, R.

PY - 1999

Y1 - 1999

N2 - Let jvk denote the k-th positive zero of the Bessel function Jv(x). In this paper, we prove that for v > 0 and k = 1, 2, 3, ..., v-ak/21/3v1/3 v,k k/21/3v1/3 + 3/20ak 221/3/v1/3. These bounds coincide with the first few terms of the well-known asymptotic expansion jv,k ∼ v - ak/21/3v1/3 + 3/20ak 221/3/v1/3 as v → ∞, k being fixed, where ak is the k-th negative zero of the Airy function Ai(i), and so are "best possible". ©1999 American Mathematical Society.

AB - Let jvk denote the k-th positive zero of the Bessel function Jv(x). In this paper, we prove that for v > 0 and k = 1, 2, 3, ..., v-ak/21/3v1/3 v,k k/21/3v1/3 + 3/20ak 221/3/v1/3. These bounds coincide with the first few terms of the well-known asymptotic expansion jv,k ∼ v - ak/21/3v1/3 + 3/20ak 221/3/v1/3 as v → ∞, k being fixed, where ak is the k-th negative zero of the Airy function Ai(i), and so are "best possible". ©1999 American Mathematical Society.

KW - Asymptotic expansions

KW - Bcssel functions

KW - Inequalities

KW - Zeros

UR - https://www.scopus.com/pages/publications/22644451810

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-22644451810&origin=recordpage

U2 - 10.1090/s0002-9947-99-02165-0

DO - 10.1090/s0002-9947-99-02165-0

M3 - RGC 21 - Publication in refereed journal

SN - 0002-9947

VL - 351

SP - 2833

EP - 2859

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

IS - 7

ER -