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:

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.

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 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.

Asymptotic analysis of the Cooke-Noble integral equation

1982 ◽  
Vol 92 (2) ◽  
pp. 293-306 ◽  
Author(s):  
L. R. F. Rose
Keyword(s):  
Integral Equation ◽  
Asymptotic Analysis ◽  
Boundary Value Problems ◽  
Bessel Function ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Mathematical Problem ◽  
Hankel Transform ◽  
Simple Function ◽  
Image Position

Those mixed boundary-value problems which can usefully be treated analytically often lead to the following mathematical problem. Two functions u(x), σ(x), defined over the interval ([0, ∞), take prescribed values over complementary portions of that interval; specifically, letwhere p(x) is usually a simple function, for example a constant or a power of x. There exists a relation between u(x) and σ(x) which can be most simply expressed as a relation between their Hankel transforms. Using a circumflex to denote the Hankel transform, for example withwhere Jv denotes as usual the Bessel function of the first kind of order v, we can state that relation between u and σ as follows:where A(ξ) is a known function, determined at an earlier stage of the analysis. The problem is to derive u(x) for (xє [ 0, a), or σ(x) for x є (a, ∞).

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 The inversion of Hankel transforms of order zero and unity

1969 ◽  
Vol 10 (2) ◽  
pp. 156-161 ◽  
Author(s):  
Ian N. Sneddon
Keyword(s):  
Bessel Functions ◽  
Hankel Transform ◽  
Science Students ◽  
Order Zero ◽  
Inversion Theorem ◽  
Hankel Transforms ◽  
Image Position ◽  
Valid Proof

In teaching the elements of transform theory to students of physics and engineering it is very useful to have available, as early as possible, the inversion theorem for the Hankel transformThe difficulty is that a valid proof for general values of v (cf. [1], p. 456) is complicated and involves a greater familiarity with the processes of analysis and the properties of Bessel functions than is possessed by most science students.

Self reciprocal properties of certain functions

1961 ◽  
Vol 57 (3) ◽  
pp. 561-567 ◽  
Author(s):  
V. V. L. N. Rao
Keyword(s):  
Bessel Function ◽  
Hankel Transform ◽  
Hankel Transforms ◽  
Image Position

The object of this note is to study the properties of some functions self reciprocal in the Hankel transform. I denote a function f(x) as Rμ, if it is self reciprocal for Hankel transforms of order μ so that it is given bywhere Jμ(x) is a Bessel function of order μ. If μ = ½, f(x) is denoted by Rs while f(x) is written as Rc when μ = − ½.

A METHOD FOR THE DIRECT INTERPRETATION OF ELECTRICAL SOUNDINGS MADE OVER A FAULT OR DIKE

Geophysics ◽  
1973 ◽  
Vol 38 (4) ◽  
pp. 762-770 ◽  
Author(s):  
Terry Lee ◽  
Ronald Green
Keyword(s):  
Bessel Functions ◽  
Direct Analysis ◽  
Hankel Transform ◽  
Approximation Technique ◽  
Polarization Data ◽  
Resistivity Data ◽  
Vertical Fault ◽  
Direct Interpretation ◽  
Electrical Soundings ◽  
Infinite Integral

The potential function for a point electrode in the vicinity of a vertical fault or dike may be expressed as an infinite integral involving Bessel functions. Beginning with such an expression, two methods are presented for the direct analysis of resistivity data measured both normal and parallel to dikes or faults. The first method is based on the asymptotic expansion of the Hankel transform of the field data and is suitable for surveys done parallel to the strike of the dike or fault. The second method is based on a successive approximation technique which starts from an initial approximate solution and iterates until a solution with prescribed accuracy is found. Both methods are suitable for programming on a digital computer and some illustrative numerical results are presented. These examples show the limitations of the methods. In addition, the application of resistivity data to the interpretation of induced‐polarization data is pointed out.

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