Infinite Integrals Involving Bessel Functions Research Articles

On the evaluation of infinite integrals involving Bessel functions

In this paper we consider the numerical method for computing the infinite highly oscillatory Bessel integrals of the form ∫a∞f(x)Cv(ωx)dx, where Cv(ωx) denotes Bessel function Jv(ωx) of the first kind, Yv(ωx) of the second kind, Hv(1)(ωx) and Hv(2)(ωx) of the third kind, f is a smooth function on [a,∞),limx→∞f(k)(x)=0(k=0,1,2,…),ω is large and a⩾1ωk with k≤1. We construct the method based on approximating f by a combination of the shifted Chebyshev polynomial so that the generalized moments can be evaluated efficiently by the truncated formula of Whittaker W function. The method is very efficient in obtaining very high precision approximations if ω is sufficiently large. Furthermore, we give the error which depends on the endpoint “a”. Numerical examples are provided to confirm our results.

Infinite integrals involving Bessel functions by an improved approach of contour integration and the residue theorem

Two lemmas are established to evaluate integrals involving Bessel functions or products of two Bessel functions on a large semicircle. These lemmas are as useful as Jordan’s lemma. With the help of them, the approach of contour integration and residue theorem to infinite integrals becomes much more powerful than ever before, yet is still simple and easy. A general formula is derived for infinite integrals involving rational functions, powers and Bessel functions, the index of the power and the order of the Bessel function being independent, and the result is expressed in terms of residues and coefficients of the Laurent series of the rational function. Similar formulae are presented for infinite integrals involving Neumann or Struve functions. Another general formula is derived for infinite integrals involving rational functions, powers and two Bessel functions, the index of the power and the orders of the Bessel functions being independent. The formulation presented may be extended to infinite integrals of similar types. Typical examples are worked out for each type of integral, and many of the results obtained seem not available in the literature.

Infinite integrals involving Bessel functions by contour integration

The approach of contour integration and residue theorem is systematically developed to evaluate infinite integrals involving Bessel functions or closely related ones. Three independent formulations applicable to different types of integrals are presented. The conventional formulation using Hankel functions as auxiliary functions, whose previous development leave considerable room for improvement, is fully developed. The second and third formulations, which appear unnoticed before, uses as auxiliary functions linear combinations of Bessel and Struve functions, and linear combinations of Anger and Weber functions, respectively. These functions play a crucial role as the Hankel functions do in the conventional formulation. Several general formulae are derived, each expressing an infinite integral in terms of residues. The integrands are products of rational functions, powers and Bessel (Neumann, Struve, Anger or Weber) functions, with some relation satisfied between the index of the power and the order of the Bessel or related function. Typical examples are worked out for each type of integral.

Eddy currents induced in a conducting rod of finite length by a coaxial encircling coil

This paper describes the calculation of eddy currents in a cylindrical conductive rod of finite length due to a coaxial circular coil carrying an alternating current. The coil impedance variation with frequency is determined from the field for an arbitrary coaxial location of the coil. Expressions for electromagnetic field and impedance of a coil encircling an infinite cylindrical rod are well known, the results being expressed as infinite integrals involving Bessel functions. For a finite length rod, additional boundary conditions must be satisfied at the ends. The extra boundary conditions are accommodated here by recasting the problem in a domain of finite extent in the axial direction. The axial length of the truncated domain is arbitrary and can be large compared with the coil length or the length of the rod. Therefore, the truncated domain solution can yield results that are numerically as close to the infinite domain solution as desired. The results provide a simple means of calculating the impedance of an inductor with a lossy core and can be used to investigate the linear properties of the core material.

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).

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.

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.

A Method for Computing Bessel Function Integrals

Infinite integrals involving Bessel functions are recast, by means of an Abel transform, in terms of Fourier integrals. As there are many efficient numerical methods for computing Fourier integrals, this leads to a convenient way of approximating Bessel function integrals.

A method for computing Bessel function integrals

Infinite integrals involving Bessel functions are recast, by means of an Abel transform, in terms of Fourier integrals. As there are many efficient numerical methods for computing Fourier integrals, this leads to a convenient way of approximating Bessel function integrals.

Tables for the Rapid and Accurate Numerical Evaluation of Certain Infinite Integrals Involving Bessel Functions

Tables for the Rapid and Accurate Numerical Evaluation of Certain Infinite Integrals Involving Bessel Functions

Tables for the rapid and accurate numerical evaluation of certain infinite integrals involving Bessel functions

Tables for the rapid and accurate numerical evaluation of certain infinite integrals involving Bessel functions

Some relations between Bessel and Legendre Functions

Certain infinite integrals involving Bessel Functions are expressed in terms of Associated Legendre Functions in oblate and prolate spheroidal coordinates. Some expansions of Bessel Functions in series involving Legendre Functions are also given.

The generalized radially symmetric wave equation

In this paper a detailed study is made of solutions of the differential equation ∂ 2 ϕ /∂ R 2 + k/R ∂ ϕ /∂ R – ∂ 2 ϕ /∂ T 2 = 0 in the quarter plane R ≽ 0, T ≽ 0. The boundary value problem considered is that of finding a solution which satisfies Cauchy data on T = 0. The contour integral solutions developed for an equation occurring in gas dynamics, shown to be equivalent to that considered here, are the main aid in the investigation. The solution is obtained first for values of T ≼ R but is continued into the whole quarter plane. This continuation follows from a fundamental uniqueness result that knowledge of ϕ on the characteristic R = T specifies its value in the domain R ≼ T . A point emphasized is that the continuation is not in general the analytic continuation from the domain T ≼ R , even for analytic initial data. Different interpretations of the solutions found are examined. When k is a positive integer the equation is that satisfied by radially symmetric solutions of the wave equation in k + 1 space dimensions, and this leads to the solution of the full wave equation for given initial conditions on T = 0. The Huygens principle is clearly illustrated. For general positive values of k the discussion clarifies a problem in gas dynamics in the study of which the original contour integral solutions were first devised. The general solution is also compared with a solution by separation of variables, and some conclusions are drawn regarding certain infinite integrals involving Bessel functions. In the final section negative values of k are considered. The contour integral representation solves in a concise form the singular initial value problem of finding a solution which takes prescribed values on R = 0, thus generalizing a result well known for positive value of k .

On Certain Infinite Integrals Involving Bessel Functions

Journal of Mathematics and PhysicsVolume 34, Issue 1-4 p. 179-181 Article On Certain Infinite Integrals Involving Bessel Functions† G. M. Muller, G. M. Muller Genaral Electric Company Richland, WashingtonSearch for more papers by this author G. M. Muller, G. M. Muller Genaral Electric Company Richland, WashingtonSearch for more papers by this author First published: April 1955 https://doi.org/10.1002/sapm1955341179Citations: 2 †This work was performed under contract W-31-109-Eng-52 with the Atomic Energy Commission. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume34, Issue1-4April 1955Pages 179-181 RelatedInformation

Some Infinite Integrals Involving Bessel Functions

Proceedings of the London Mathematical SocietyVolume s2-40, Issue 1 p. 37-48 Papers Some Infinite Integrals Involving Bessel Functions W. N. Bailey, W. N. Bailey The University, ManchesterSearch for more papers by this author W. N. Bailey, W. N. Bailey The University, ManchesterSearch for more papers by this author First published: 1936 https://doi.org/10.1112/plms/s2-40.1.37Citations: 73AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes2-40, Issue11936Pages 37-48 RelatedInformation