On the Hankel transform of a generalized Laguerre polynomial and on the convolution involving special Bessel functions

1984 ◽  
Vol 25 (6) ◽  
pp. 1718-1720 ◽  
Author(s):  
H. Navelet
Keyword(s):  
Bessel Functions ◽  
Laguerre Polynomial ◽  
Hankel Transform

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.

An algorithm for the plane‐wave decomposition of point‐source seismograms

Geophysics ◽  
1990 ◽  
Vol 55 (10) ◽  
pp. 1380-1385 ◽  
Author(s):  
M. Dietrich
Keyword(s):  
Plane Wave ◽  
Point Source ◽  
Bessel Functions ◽  
Real Data ◽  
Hankel Transform ◽  
Considerable Reduction ◽  
Correct Formulation ◽  
Plane Wave Decomposition ◽  
Wave Decomposition ◽  
Discrete Integration

The correct formulation of the plane‐wave decomposition of point‐source seismograms involves a sequence of Fourier and Hankel transforms which can be evaluated in several ways. The procedure which is proposed here exploits the fact that the plane‐wave response is bandlimited along the horizontal slowness axis. This property permits to expand the Hankel transform into a Fourier‐Bessel series. In practice, this algorithm requires an interpolation in distance of the recorded dataset, but allows a considerable reduction of Bessel functions calculations. Numerical applications performed with synthetic and real data show that the Fourier‐Bessel summation technique yields results which are equivalent to a discrete integration of the Hankel transform.

scholarly journals Convolution of Hankel transform and its application to an integral involving Bessel functions of first kind

1995 ◽  
Vol 18 (3) ◽  
pp. 545-549 ◽  
Author(s):  
Vu Kim Tuan ◽  
Megumi Saigo
Keyword(s):  
Bessel Functions ◽  
Hankel Transform

In the paper a convolution of the Hankel transform is constructed. The convolution is used to the calculation of an integral containing Bessel functions of the first kind.

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 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.

Titchmarsh's theorem of Hankel transform

2019 ◽  
pp. 11-24
Author(s):  
Mohamed-Ahmed Boudref
Keyword(s):  
Number Theory ◽  
Data Analysis ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Lipschitz Condition ◽  
Analytic Number Theory ◽  
Hankel Transform ◽  
Partial Differential ◽  
Analytic Number ◽  
Bessel Transform

Hankel transform (or Fourier-Bessel transform) is a fundamental tool in many areas of mathematics and engineering, including analysis, partial differential equations, probability, analytic number theory, data analysis, etc. In this article, we prove an analog of Titchmarsh's theorem for the Hankel transform of functions satisfying the Hankel-Lipschitz condition.

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