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

scholarly journals On the positive roots of an equation involving a Bessel function

1989 ◽  
Vol 32 (1) ◽  
pp. 157-164 ◽  
Author(s):  
Siegfried H. Lehnigk
Keyword(s):  
Bessel Function ◽  
Recurrence Relation ◽  
Modified Bessel Function ◽  
Positive Roots ◽  
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In this paper we shall discuss the positive roots of the equationwhere Iq is the modified Bessel function of the first kind. By means of a recurrence relation for Iq(r) [2, (5.7.9)], equation (1.1a) can also be written in the form

scholarly journals On some fractional integrals and their applications

1985 ◽  
Vol 28 (1) ◽  
pp. 97-105
Author(s):  
J. S. Lowndes
Keyword(s):  
Bessel Function ◽  
Fractional Integration ◽  
Fractional Integrals ◽  
Modified Bessel Function ◽  
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In previous papers [3, 4] the author has discussed the symmetric generalised Erdélyi–Kober operators of fractional integration defined bywhere α>0, γ≧0 and the operators ℑiγ(η,α) and defined as in equations (1) and (2) respectively but with Jα−1, the Bessel function of the first kind replaced by Iα−1, the modified Bessel function of the first kind.

scholarly journals Asymptotic Expressions for the Bessel Functions and the Fourier-Bessel Expansions

1920 ◽  
Vol 39 ◽  
pp. 13-20
Author(s):  
T. M. MacRobert
Keyword(s):  
Asymptotic Expansion ◽  
Bessel Functions ◽  
Asymptotic Expressions ◽  
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Asymptotic Expressions for the Bessel Functions.From the asymptotic expansion for Ku(z) it follows that, if − π < amp z < π,This theorem is also true if amp z = ± π; to prove this consider the formula

scholarly journals Integrals Involving a Modified Bessel Function of the Second Kind and an E-Function

1954 ◽  
Vol 2 (2) ◽  
pp. 93-96 ◽  
Author(s):  
T. M. Macrobert
Keyword(s):  
Bessel Function ◽  
Modified Bessel Function ◽  
Initial Value ◽  
Positive Direction ◽  
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The first formula to be proved iswhere p ≧ q + 1, | amp z | < л, R(k±n + αr)>0, r = l, 2, …, p. For other values of p and q the result is valid if the integral is convergent. A second formula is given in § 3.The following formulae are required in the proof:where R(z);>0, (1);where R(α)>0, | amp z | < л, (2);where the contour starts from -∞ on the ξ-axis, passes round the origin in the positive direction, and ends at -∞ on the ξ-axis, the initial value of amp ζ being - л, (3).

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 ◽  
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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.

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 ◽  
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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.

An inversion formula for the Varma transform

1966 ◽  
Vol 62 (3) ◽  
pp. 467-471 ◽  
Author(s):  
R. K. Saxena
Keyword(s):  
Integral Equation ◽  
Bessel Function ◽  
Inversion Formula ◽  
Fractional Integration ◽  
Integration Theory ◽  
Modified Bessel Function ◽  
Laplace Integral ◽  
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AbstractRecently Fox ((5)) has given an inversion formula for the transform whose kernel is xνKν(x), where Kν(x) is the modified Bessel function of the second kind, by the application of fractional integration theory. In the present paper it has been shown that the integral equationcan be thrown into the form of a Laplace integral, with the help of fractional integration, which can be solved by known methods.

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 ◽  
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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:

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