Integrals Involving Bessel Functions Research Articles

Domain wall dynamics of the Ising chain in a transverse field

We show that the dynamics of an Ising spin chain in a transverse field conserves the number of domains (strings of down spins in an up-spin background) at discrete times. This enables the determination of the eigenfunctions of the time-evolution operator, and the dynamics of initial states with domains. The transverse magnetization is shown to be identically zero in all sectors with a fixed number of domains. For an initial state with a single string of down spins, the local magnetization, the equal-time and double-time spin-spin correlation functions, are calculated analytically as functions of time and the initial string size. The domain size distribution function can be expressed as a simple integral involving Bessel functions.

Hypergeometric reproducing kernels and analytic continuation from a half-line

Indefinite inner product spaces of entire functions and functions analytic inside a disk are considered and their completeness studied. Spaces induced by the rotation invariant reproducing kernels in the form of the generalized hypergeometric function are completely characterized. A particular space generated by the modified Bessel function kernel is utilized to derive an analytic continuation formula for functions on ℝ+ using the best approximation theorem of Saitoh. As a by-product several new area integrals involving Bessel functions are explicitly evaluated.

Elementary evaluation of certain infinite integrals involving Bessel functions

Although it is known theoretically that certain infinite integrals of Bessel functions can be expressed in terms of elementary functions, the practical evaluation of such integrals was quite difficult due to the algebraic complexity of the expressions involved. A simple and elegant algebra is introduced here which allows these integrals to be calculated in an elementary way in terms of elementary functions. Some relationships are shown between the integrals involving Bessel functions and two-dimensional integrals over a circle of elementary functions involving distances between points. A comparison is made with existing results, and some of them were found in error (or were misprints).

Casimir energy in the axial gauge

The zero-point energy of a conducting spherical shell is studied by imposing the axial gauge via path-integral methods, with boundary conditions on the electromagnetic potential and ghost fields. The coupled modes are then found to be the temporal and longitudinal modes for the Maxwell field. The resulting system can be decoupled by studying a fourth-order differential equation with boundary conditions on longitudinal modes and their second derivatives. The exact solution of such an equation is found by using a Green-function method, and is obtained from Bessel functions and definite integrals involving Bessel functions. Complete agreement with a previous path-integral analysis in the Lorenz gauge, and with Boyer's value, is proved in detail.

On the gamma factor of the triple L-function, I

Introduction. The study of the triple L-function began with Garrett [2]. From a representation-theoretic point of view, Piatetski-Shapiro and Rallis [9] and the author [6] defined the archimedean or nonarchimedean L-factor as the greatest common divisor (GCD) of local integrals. The purpose of this paper is to prove that the archimedean GCD L-factor agrees with the expected L-factor up to invertible functions if the local representation comes from a cuspidal automorphic representation. This theorem follows from the local functional equation by formal argument except when the representation is unramified. To treat the unramified case, we carry out the explicit calculation of the unramified local integral. Stimulated by Stade [12], we make use of a hypergeometric function 3F2 and its two-term relations and three-term relation. We also need a theorem on some infinite integral involving Bessel functions, proved by Bailey [1] in 1936. The key lemma is the following (see Lemma 2.4).

Analytical solution for the steady-state diffusion towards an inlaid disc microelectrode in a multi-layered medium

The rigorous analytical expression for the steady-state diffusion-limited current produced by an electroactive species diffusing towards an inlaid disc electrode through a multi-layered medium (such as the membrane covered sensor) is presented. This theoretical approach reformulates the problem in terms of a dual integral equation and takes full account of the edge effect that one-dimensional formulations cannot include. The solution is found by solving a linear system of equations where the entries of the matrix are integrals involving Bessel functions, and can be easily and quickly computed with standard software such as Mathematica. Comparison of this analytical solution (corresponding to the axisymmetric geometry) with previous models (linear, spherical and mixed) for the membrane-covered gas sensor allows us to demonstrate the restricted ranges of applicability of these previous models. Through a variety of approximations, we show that it is possible to obtain accurate estimates of the current (also valid for liquid samples) using very simple expressions, which in turn allows procedures such as the fast estimation of a medium permeability. The model can be easily extended to any number of parallel layers with a variety of boundary conditions in the interface or layer furthest away from the insulator plane; thus some interesting limiting cases are analysed (such as the validity of ignoring a very thin layer next to the electrode or the unattainability of steady-state just by horizontal radial diffusion).

Computation of infinite integrals involving Bessel functions of arbitrary order by the [formula omitted

The D- transformation due to the author is an effective extrapolation method for computing infinite oscillatory integrals of various kinds. In this work two new variants of this transformation are designed for computing integrals of the form ∫ a ∞ g(t) C v(t) dt , where g( x) is a nonoscillatory function and C v(x) may be an arbitrary linear combination of the Bessel functions of the first and second kinds J v ( x) and Y v ( x), of arbitrary real order v. When applied to such integrals, the D- transformation and its new variants are observed to produce very accurate results. It is also seen that their performance is very similar to that of the modified W-transformation due to the author, as extended in a recent work by Lucas and Stone with C v(x) = J v(x) . The present paper is concluded by stating the relevant convergence and stability results and by appending a numerical example.

A Fourier-Bessel expansion for solving radial Schr�dinger equation in two dimensions

The spectrum of the two-dimensional Schrodinger equation for polynomial oscillators bounded by infinitely high potentials, where the eigenvalue problem is defined on a finite interval r ∈ [0, L), is variationally studied. The wave function is expanded into a Fourier-Bessel series, and matrix elements in terms of integrals involving Bessel functions are evaluated analytically. Numerical results presented accurate to thirty digits show that, by the time L approaches a critical value, the low-lying state energies behave almost as if the potentials were unbounded. The method is applicable to multi-well oscillators as well.

Evaluating infinite integrals involving Bessel functions of arbitrary order

The evaluation of intergrals of the form I n = ∫ 0 ∞ ƒ(x)J n(x) dx is considered. In the past, the method of dividing an oscillatory integral at its zeros, forming a sequence of partial sums, and using extrapolation to accelerate convergence has been found to be the most efficient technique available where the oscillation is due to a trigonometric function or a Bessel function of order n = 0, 1. Here, we compare various extrapolation techniques as well as choices of endpoints in dividing the integral, and establish the most efficient method for evaluating infinite integrals involving Bessel functions of any order n, not just zero or one. We also outline a simple but very effective technique for calculating Bessel function zeros.

Comparative study of acceleration techniques for integrals and series in electromagnetic problems

Most electromagnetic problems can be reduced to either integrating oscillatory integrals or summing up complex series. However, limits of the integrals and the series usually extend to infinity, and, in addition, they may be slowly convergent. Therefore numerically efficient techniques for evaluating the integrals or for calculating the sum of an infinite series have to be used to make the numerical solution feasible and attractive. In the literature there are a wide range of applications of such methods to various EM problems. In this paper our main aim is to critically examine the popular series transformation (acceleration) methods which are used in electromagnetic problems and to introduce a new acceleration technique for integrals involving Bessel functions and sinusoidal functions.

Some integrals involving Bessel functions

A certain class of integrals involving Bessel functions is considered and the general theory to develop them as a series is discussed. In order to obtain an effective series approximation, it is also shown as the convergence of the expansion can be accelerated. A physical example and a mathematical one are completely developed.

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

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.

A short note on some integrals involving bessel functions

A new expansion for a certain class of integrals involving Bessel functions is considered. An example is completely developed showing as an efficient series expansion can be obtained.

Some Integrals Involving Bessel Functions

A number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized hypergeometric function with subsequent reduction to special cases. Connection is made with Weber′s second exponential integral and Laplace transforms of products of three Bessel functions.

Magnetic field of a direct current in a cylindrical-shaped conductor imbedded in a resistive half-space beneath a conductive surface layer

New applications of Fourier series and integral equation methods have yielded results of significance for electrical prospecting. This paper presents the first model calculation in three dimensions of the magnetic field due to a direct current flow in a conductor imbedded in a half-space beneath a conductive surface layer. The conductor is in the shape of a cylinder of semi-infinite length, and the current flow is from a point source in the surface layer. The vertical component of the magnetic field is given here by a Fourier series integral over a function which satisfies a Cauchy singular integral equation. For the two special cases, namely, an outcropping cylinder with no surface layer and a nearly invisible (in the sense of negligible conductivity contrast with its surrounding) cylinder, this integral equation can be solved exactly to yield closed-form expressions for the vertical component of the magnetic field. The exact solution in the first special case is based on a new generalization of the Hankel integral transform (published separately), and it extends the class of exactly solvable singular integral equations. This exact solution has also yielded as useful byproducts two new standard integrals involving Bessel functions. The integral equation approach used here, in contrast to the well-established surface integral equation approach, is suitable for numerical calculations, and typical magnetic field profiles are presented. The magnetic field results given here would be valuable for interpretation in electrical prospecting (based on the measurement of magnetic fields of direct currents), and the mathematical techniques used here would be useful for future calculations in electromagnetism.

An Asymptotic Approximation for a Class of Oscillatory Infinite Integrals

The computation of a class of strongly oscillating integrals involving Bessel functions depending on a parameter is considered. Using an expansion in terms of Laguerre polynomials, which has been recently proposed for evaluating these integrals for small values of the parameter, an asymptotic formula effective for large values of the parameter is established. Five examples are given.

A class of discontinuous integrals involving Bessel functions

A general theorem, which appears to be newly discovered although it is of a very classical sort, gives simple evaluations for a large class of infinite integrals containing Bessel functions in product with other suitably constrained analytic functions.

Hypervirial calculation of integrals involving Bessel functions

A general and simple procedure is presented for evaluating matrix elements that involve Bessel functions. The method is based upon hypervirial relationships for systems subjected to Dirichlet boundary conditions.

On High Precision Methods for Computing Integrals Involving Bessel Functions

The technique of Bakhvalov and Vasil’eva for evaluating Fourier integrals is generalized to integrals involving exponential and Bessel functions.

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

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.