On the Square of the First Zero of the Bessel Function Jv(z)

1999 ◽  
Vol 42 (1) ◽  
pp. 56-67 ◽  
Author(s):  
Árpád Elbert ◽  
Panayiotis D. Siafarikas
Keyword(s):  
Convex Function ◽  
Bessel Function ◽  
Series Expansion ◽  
Positive Zero ◽  
Monotonicity Properties

AbstractLet Jv,1 be the smallest (first) positive zero of the Bessel function Jv(z), v > −1, which becomes zero when v approaches −1. Then can be continued analytically to −2 < v < −1, where it takes on negative values. We show that is a convex function of v in the interval −2 < v ≤ 0, as an addition to an old result [Á. Elbert and A. Laforgia, SIAM J. Math. Anal. 15(1984), 206–212], stating this convexity for v > 0. Also the monotonicity properties of the functions are determined. Our approach is based on the series expansion of Bessel function Jv(z) and it turned out to be effective, especially when −2 < v < −1.

Higher monotonicity properties of normalized Bessel functions

2014 ◽  
Vol 12 (05) ◽  
pp. 1461007
Author(s):  
Yongping Liu
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Positive Zero ◽  
Local Maxima ◽  
Monotonicity Properties

Denote by Jν the Bessel function of the first kind of order ν and μν,k is its kth positive zero. For ν > ½, a theorem of Lorch, Muldoon and Szegö states that the sequence [Formula: see text] is decreasing, another theorem of theirs states that the sequence [Formula: see text] has higher monotonicity properties. In the present paper, we proved that when ν > ½ the sequence [Formula: see text] has higher monotonicity properties and the properties imply those of the sequence of the local maxima of the function x-ν+1|Jν-1(x)|, x ∈ (0, ∞), i.e. the sequence [Formula: see text] has higher monotonicity properties.

Higher monotonicity properties of certain Sturm-Liouville functions. V

1977 ◽  
Vol 77 (1-2) ◽  
pp. 23-37 ◽  
Author(s):  
M. E. Muldoon
Keyword(s):  
Bessel Function ◽  
Positive Zero ◽  
Monotonicity Properties ◽  
Sturm Liouville ◽  
Special Solutions ◽  
Image Position

SynopsisThe principal concern here is with conditions on f or on special solutions of the equationwhich ensure that the higher differences of the zeros and related quantities of solutions of (1) are regular in sign. In particular, by choosing f(x)= 2v−2x1/v−2, it is shown that if ⅓ ≦|v|<½, thenwhere cvk denotes the kth positive zero of a Bessel function of order v and Δµk = Δk+1 − µk. Lorch and Szego [15] conjectured that (2) should hold for the larger range | v | < ½ but the methods used here do not apply to the range | v <| ⅓.

Inflection Points of Bessel Functions of Negative Order

1991 ◽  
Vol 43 (6) ◽  
pp. 1309-1322 ◽  
Author(s):  
Lee Lorch ◽  
Martin E. Muldoon ◽  
Peter Szego
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Positive Zero ◽  
Second Derivative ◽  
Monotonicity Properties ◽  
Negative Order ◽  
Inflection Points

AbstractWe consider the positive zeros j″vk, k = 1, 2,…, of the second derivative of the Bessel function Jν(x). We are interested first in how many zeros there are on the interval (0,jν1), where jν1 is the smallest positive zero of Jν(x). We show that there exists a number ƛ = —0.19937078… such that and . Moreover, j″v1 decreases to 0 and j″ν2 increases to j″01 as ν increases from ƛ to 0. Further, j″vk increases in —1 < ν< ∞, for k = 3,4,… Monotonicity properties are established also for ordinates, and the slopes at the ordinates, of the points of inflection when — 1 < ν < 0.

scholarly journals The zeros ofaz2J″ν(z)+bzJ′ν(z)+cJν(z)as functions of order

1992 ◽  
Vol 15 (2) ◽  
pp. 319-322 ◽  
Author(s):  
A. McD. Mercer
Keyword(s):  
Bessel Function ◽  
General Result ◽  
Positive Zero ◽  
Special Case

Ifj″νkdenotes thekthpositive zero of the Bessel functionJ″ν(x), it has been shown recently by Lorch and Szego [2] thatj″ν1increases withνinν>0and that (withkfixed in2,3,…)j″νkincreases in0<ν≤3838. Furthermore, Wong and Lang have now extended the latter result, as well, to the rangeν>0. The present paper, by using a different kind of analysis, re-obtains these conclusions as a special case of a more general result concerning the positive zeros of the functionaz2J″ν(z)+bzJ′ν(z)+cJν(z). Here, the constantsa,bandcare subject to certain mild restrictions.

BESSEL FUNCTIONS IN A QUANTUM-BILLIARD CONFIGURATION PROBLEM

2003 ◽  
Vol 01 (04) ◽  
pp. 421-428
Author(s):  
ÁRPÁD ELBERT ◽  
LEE LORCH ◽  
PETER SZEGO
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Positive Zero ◽  
General Function ◽  
Cylinder Function ◽  
Quantum Billiard ◽  
Configuration Problem ◽  
Math Phys

In studying various quantum-billiard configurations, R. L. Liboff (J. Math. Phys.35 (1994) 2218), was led to investigate the vanishing of f(ν)=j2ν,1 - jν2, where jμk is the kth positive zero of the Bessel function Jμ(x). Here we show that the even more general function fα(ν)=cαν,k - cν,k+l is increasing and vanishes once (and only once) in 0<ν<∞, provided α≥π/2 and [Formula: see text], k, l=1,2,3,…. As usual, cμn is the nth positive zero of the cylinder function Cμ(x)=Jμ (x) cos θ - Yμ(x) sin θ. Specialized to Liboff's case, f(ν), this yields not only the existence of a zero of f(ν) but also its uniqueness.

scholarly journals Monotonicity properties of the zeros of Bessel functions

1982 ◽  
Vol 24 (1) ◽  
pp. 67-85 ◽  
Author(s):  
Roger C. McCann ◽  
E. R. Love
Keyword(s):  
Bessel Functions ◽  
Positive Zero ◽  
Zeros Of Bessel Functions ◽  
Monotonicity Properties

AbstractLet jν, denote the first positive zero of Jν. It is shown that jν/(ν + α) is a strictly decreasing function of ν for each positive α provided ν is sufficiently large. For each α lowe bounds on ν are given to assure the monotonicity of jν/(ν + α). From this it is shown that jν > ν + j0 for all ν > 0, which is both simpler and an improvement on the well known inequality Jν ≥ (ν (ν + 2))1/2.

scholarly journals Bounds for the generalized Marcum function of the second kind

2021 ◽  
Author(s):  
Árpád Baricz ◽  
Nitin Bisht ◽  
Sanjeev Singh ◽  
V. Antony Vijesh
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Modified Bessel Function ◽  
Monotonicity Properties

AbstractIn this paper, we focus on the generalized Marcum function of the second kind of order $$\nu >0$$ ν > 0 , defined by $$\begin{aligned} R_{\nu }(a,b)=\frac{c_{a,\nu }}{a^{\nu -1}} \int _b ^ {\infty } t^{\nu } e^{-\frac{t^2+a^2}{2}}K_{\nu -1}(at)\mathrm{d}t, \end{aligned}$$ R ν ( a , b ) = c a , ν a ν - 1 ∫ b ∞ t ν e - t 2 + a 2 2 K ν - 1 ( a t ) d t , where $$a>0, b\ge 0,$$ a > 0 , b ≥ 0 , $$K_{\nu }$$ K ν stands for the modified Bessel function of the second kind, and $$c_{a,\nu }$$ c a , ν is a constant depending on a and $$\nu $$ ν such that $$R_{\nu }(a,0)=1.$$ R ν ( a , 0 ) = 1 . Our aim is to find some new tight bounds for the generalized Marcum function of the second kind and compare them with the existing bounds. In order to deduce these bounds, we include the monotonicity properties of various functions containing modified Bessel functions of the second kind as our main tools. Moreover, we demonstrate that our bounds in some sense are the best possible ones.

A Bessel function inequality connected with stability of least square smoothing, II

1968 ◽  
Vol 9 (2) ◽  
pp. 119-122 ◽  
Author(s):  
Lee Lorch ◽  
Peter Szego
Keyword(s):  
Bessel Function ◽  
Least Square ◽  
Positive Zero ◽  
Image Position

In accordance with customary notation, Jv(t) denotes the Bessel function of the first kind and order v, jvk its kth positive zero and jv, 0 = 0.The object of this note is to prove that

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