A theorem on the roots of a certain equation involving Bessel functions

1970 ◽  
Vol 67 (1) ◽  
pp. 93-96 ◽  
Author(s):  
R. Steinitz
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Real Constant ◽  
Image Position

It is well known (Watson, A treatise on Bessel functions) that if Jν(z) is a Bessel function of the first kind, and A is a real constant, then all roots of the equation are either real or purely imaginary.

On a singular eigenvalue problem

1968 ◽  
Vol 64 (2) ◽  
pp. 439-446 ◽  
Author(s):  
D. Naylor ◽  
S. C. R. Dennis
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Bessel Functions ◽  
Asymptotic Condition ◽  
Real Constant ◽  
Limit Circle ◽  
Expansion In Eigenfunctions ◽  
Image Position

Sears and Titchmarsh (1) have formulated an expansion in eigenfunctions which requires a knowledge of the s-zeros of the equationHere ka > 0 is supposed given and β is a real constant such that 0 ≤ β < π. The above equation is encountered when one seeks the eigenfunctions of the differential equationon the interval 0 < α ≤ r < ∞ subject to the condition of vanishing at r = α. Solutions of (2) are the Bessel functions J±is(kr) and every solution w of (2) is such that r−½w(r) belongs to L2 (α, ∞). Since the problem is of the limit circle type at infinity it is necessary to prescribe a suitable asymptotic condition there to make the eigenfunctions determinate. In the present instance this condition is

scholarly journals On some infinite series involving the zeros of Bessel functions of the first kind

1960 ◽  
Vol 4 (3) ◽  
pp. 144-156 ◽  
Author(s):  
Ian N. Sneddon
Keyword(s):  
Positive Integer ◽  
Bessel Function ◽  
Rational Function ◽  
Recurrence Relation ◽  
Infinite Series ◽  
Bessel Functions ◽  
Zeros Of Bessel Functions ◽  
Image Position

In this paper we shall be concerned with the derivation of simple expressions for the sums of some infinite series involving the zeros of Bessel functions of the first kind. For instance, if we denote by γv, n (n = l, 2, 3,…) the positive zeros of Jv(z), then, in certain physical applications, we are interested in finding the values of the sumsandwhere m is a positive integer. In § 4 of this paper we shall derive a simple recurrence relation for S2m,v which enables the value of any sum to be calculated as a rational function of the order vof the Bessel function. Similar results are given in § 5 for the sum T2m,v.

The analyticity of cross-product Bessel function zeros

1966 ◽  
Vol 62 (2) ◽  
pp. 215-226 ◽  
Author(s):  
James Alan Cochran
Keyword(s):  
Bessel Function ◽  
Bessel Functions ◽  
Cross Product ◽  
Image Position ◽  
Function Zeros

Introduction. In this paper we consider the two cross-product combinations of Bessel functionswhere δ = (k− 1)zand (') denotes differentiation with respect to the argument. HereJνandYνdesignate respectively the Bessel functions of the first and second kind of order ν.

scholarly journals Theorems relating Hankel and Meijer's Bessel transforms

1963 ◽  
Vol 6 (2) ◽  
pp. 107-112 ◽  
Author(s):  
K. C. Sharma
Keyword(s):  
Bessel Function ◽  
Hypergeometric Function ◽  
Bessel Functions ◽  
Hankel Transform ◽  
Bessel Transforms ◽  
Image Position

In this note a theorem, giving a relation between the Hankel transform of f(x) and Meijer's Bessel function transform of f(x)g(x), is proved. Some corollaries, obtained by specializing the function g(x), are stated as theorems. These theorems are further illustrated by certain suitable examples in which certain integrals involving products of Bessel functions or of Gauss's hypergeometric function and Appell's hypergeometric function are evaluated. Throughout this note we use the following notations:

scholarly journals The Modified Bessel Function Kn(z)

1919 ◽  
Vol 38 ◽  
pp. 10-19
Author(s):  
T. M. MacRobert
Keyword(s):  
Asymptotic Expansion ◽  
Bessel Function ◽  
Bessel Functions ◽  
Modified Bessel Function ◽  
Definition Of ◽  
Image Position

Gray and Mathews, in their treatise on Bessel Functions, define the function Kn(z) to beWe shall denote this function by Vn(z). This definition only holds when z is real, and R(n)≧0. The asymptotic expansion of the function is also given; but the proof, which is said to be troublesome and not very satisfactory, is omitted. Basset (Proc. Camb. Phil. Soc., Vol. 6) gives a similar definition of the function.

scholarly journals Some Bessel Function Integrals

1956 ◽  
Vol 2 (4) ◽  
pp. 183-184 ◽  
Author(s):  
T. M. MacRobert
Keyword(s):  
Bessel Function ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Basic Formula ◽  
Confluent Hypergeometric Functions ◽  
Image Position

The basic formula to be proved iswhere p≧q + 1, z ≠0; | amp z | < π, R(n)>0, r = 1, 2,…,p. For other values of pand qthe result holds if the integral converges. From this formula some results, involving Bessel functions and Confluent Hypergeometric functions, will be deduced.

An Addition Theorem and Some Product Formulas for q-Bessel Functions

1988 ◽  
Vol 40 (5) ◽  
pp. 1203-1221 ◽  
Author(s):  
Mizan Rahman
Keyword(s):  
Bessel Function ◽  
Gamma Function ◽  
Bessel Functions ◽  
Series Representation ◽  
Addition Theorem ◽  
Image Position ◽  
Product Formulas

The most familiar series representation of the Bessel function is1.1Jackson [12] gave the following q-analogues:1.21.3where 0 < q < 1, the q-shifted factorials are defined by1.4and the q-gamma function is given by1.5

scholarly journals Integrals involving Products of Bessel Functions

1956 ◽  
Vol 2 (4) ◽  
pp. 180-182 ◽  
Author(s):  
F. M. Ragab
Keyword(s):  
Bessel Functions ◽  
Image Position

The first formula to be proved iswhere R(z)>0.

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.

scholarly journals An Integral involving a Modified Bessel Function of the Second Kind and an E-function

1953 ◽  
Vol 1 (3) ◽  
pp. 119-120 ◽  
Author(s):  
Fouad M. Ragab
Keyword(s):  
Positive Integer ◽  
Bessel Function ◽  
Modified Bessel Function ◽  
Image Position

§ 1. Introductory. The formula to be established iswhere m is a positive integer,and the constants are such that the integral converges.

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