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Showing papers on "Bessel function published in 1982"



Journal ArticleDOI
TL;DR: In this article, the q-Lommel polynomials are shown to be orthogonal with respect to a purely discrete measure with bounded support, which is then used to prove that the prositive zeros of x−νJ(2)v(x; q) are real simple and interlace with the zero of x −ν − 1J( 2)v + 1(x, q), when ν > −1.

192 citations


Journal ArticleDOI
TL;DR: In this article, the authors established a link between the canonical measure of a hitting time and the spectral measure of the differential generator of the diffusion, and showed that the derivative of this canonical measure with respect to natural scale (as a function of the point being hit) equals the spectral measures of the generator on a restricted interval.
Abstract: All first hitting times for a one-dimensional diffusion belong to the Bondesson class of infinitely divisible distributions on $\lbrack 0, \infty\rbrack$. A distribution in this class can be conveniently represented in terms of its canonical measure. In this paper we establish a link between the canonical measure of a hitting time and the spectral measure of the differential generator of the diffusion. In particular, it is shown that the derivative of the canonical measure with respect to natural scale (as a function of the point being hit) equals the spectral measure of the differential generator on a restricted interval. The canonical measure is then calculated for several examples arising from the Bessel diffusion process, including the inverse of a gamma variate and the Hartman-Watson mixing distribution.

58 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered the sum of products of Bessel functions of the form J∞n = −∞(n/jJ2n)/ (n+μ).
Abstract: In our investigations of the linear theory of the stability of relativistic beam‐plasma systems immersed in a magnetic field we have been led to consider sum rules for an infinite series of products of Bessel functions of the form J∞n = −∞(n jJ2n)/ (n+μ). In this work we report on the sum of this series treated as a special case of a more general infinite series. We also mention the extension of the results beyond the range of the parameters for which formulae are explicitly given and indicate how intermediate results obtained may be useful in their own right. Finally, an additional application of our result is indicated.

47 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the Rayleigh-ritz method of the calculus of variations can be used to obtain an upper bound of (,+ l)1/2((, + 2)l/2+2+l), and that for large v,jr = v + 0(v]/2).
Abstract: It is shown, using the Rayleigh-Ritz method of the calculus of variations, that an upper bound for the first zeroy„, of z'v Jv(z), v > -1, is given by (,+ l)l/2{(„ + 2)l/2+l}, and that for large v,jr = v + 0(v]/2). 1. The following upper bound is given by Watson [4] for the first zero jv of Jv(x) (v>0) ¡A 1 '/2 (1) J. <{ f(\" +!)(\" + 5) j . It may be shown that a better bound may be obtained, valid for v > -1, namely (2) (,+ l)1/2((, + 2)1/2+l). 2. Consider the function (3) U(z) = T{v+\\)(2/(yz))'j,(yz). The differential equation satisfied by u(z) is given by Watson [3] to be (4) z2u\" + (2v + \\)zu' + y2z2u = 0 with the boundary condition u(0) = 1, and if y is a zero of J„, u(\\) — 0. Equation (4) can be written in Sturm-Liouville form (5) ¿(.-'fJ+TV-H.-* Multiplying Eq. (5) by u and integrating over 0 < z < 1, it follows that (6) 2=:/¿'2'+l\"'2 -{, and u(\\) vanishes. Thus the relation (6) provides a variational formulation, as indicated by Irving and Mullineux [1], for y2 which is an eigenvalue for the differential equation (5). The first eigenvalue will be7,2. The functional fl ,2v+l,,>2 j„ )q z u> az (7) A(«) = &z2>+Wdz Received February 19, 1981; revised July 13, 1981. 1980 Mathematics Subject Classification. Primary 33A40, 49G10. ©1982 American Mathematical Society 0025-5718/81/0000-1069/$01.50 589 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

33 citations


Journal ArticleDOI
TL;DR: In this paper, a general formula for angular integrations in many-dimensional spaces (derived in a previous paper) is applied to several problems connected with solution of the Schrodinger equation for many-particle systems.
Abstract: A general formula for angular integrations in many-dimensional spaces (derived in a previous paper) is applied to several problems connected with solution of the Schrodinger equation for many-particle systems. Matrix elements of the Hamiltonian are derived for cases where the potential can be expressed in terms of functions of the generalized radius multiplied by polynomials in the m coordinates. The theory of hyperspherical harmonics is reviewed, and a sum rule is derived relating the sum over all the harmonics belonging to a particular eigenvalue of angular momentum to the Gegenbauer polynomial corresponding to that eigenvalue. A formula is derived for projecting out the component of an arbitrary function corresponding to a particular eigenvalue of generalized angular momentum, and the formula is applied to polynomials in the m coordinates. An expansion is derived for expressing a many-dimensional plane wave in terms of hyperspherical harmonics and functions which might be called “hyperspherical Bessel functions.” It is shown how this expansion may be used to calculate many-dimensional Fourier transforms. A formula is derived expressing the effect of a group-theoretical projection operator acting on a many-dimensional plane wave. Finally, the techniques mentioned above are used to expand the Coulomb potential of a many-particle system in terms of Gegenbauer polynomials.

33 citations


01 Jul 1982
TL;DR: In this article, a five-parameter gamma distribution with two shape parameters, two location parameters, and a correlation parameter is investigated for wind gust modeling for the ascent flight of the space shuttle.
Abstract: A five-parameter gamma distribution (BGD) having two shape parameters, two location parameters, and a correlation parameter is investigated. This general BGD is expressed as a double series and as a single series of the modified Bessel function. It reduces to the known special case for equal shape parameters. Practical functions for computer evaluations for the general BGD and for special cases are presented. Applications to wind gust modeling for the ascent flight of the space shuttle are illustrated.

32 citations


01 Oct 1982
TL;DR: In this paper, the electric field inside high gain micro-channel plate multipliers was studied using the solution of the Maxwell equations applied to the microchannel plate (MCP) rather than on the conventional lumped RC model.
Abstract: Electric field inside high gain microchannel plate multipliers was studied The calculations were based directly on the solution of the Maxwell equations applied to the microchannel plate (MCP) rather than on the conventional lumped RC model The results are important to explain the performance of MCP's, i) under a pulsed bias tension and, ii) at high rate conditions The results were tested experimentally and a new method of MCP operation free from the positive ion feedback was demonstrated

28 citations


Journal ArticleDOI
TL;DR: In this article, quadrature formulas for the Fourier and Bessel transforms were obtained for the Laplace transform, which correspond to the Gauss-Laguerre formula for the Bessel transform.
Abstract: Quadrature formulas are obtained for the Fourier and Bessel transforms which correspond to the well-known Gauss-Laguerre formula for the Laplace transform. These formulas provide effective asymptotic approximations, complete with error bounds. Comparison is also made between the quadrature formulas and the asymptotic expansions of these transforms.

26 citations


Journal ArticleDOI
TL;DR: In this article, a simple and accurate algorithm for determining the propagation constants of cylindrical dielectric waveguides of arbitrary refractive index was proposed, which eliminates the use of Bessel or other complicated functions.
Abstract: From Maxwell's equations we derive a simple and accurate algorithm which can determine the propagation constants of the modes of cylindrical dielectric waveguides of arbitrary refractive index. It eliminates the use of Bessel or other complicated functions, and could be implemented in small microcomputers.

25 citations


Journal ArticleDOI
TL;DR: In this paper, efficient stratagems are developed for numerically evaluating one and two-dimensional integrals over x, y with integrand exp(x y)IO(2p-xy).
Abstract: Efficient stratagems are developed for numerically evaluating oneand two-dimensional integrals over x, y with integrand exp(-x y)IO(2p-xy). The integrals are expressed in terms of convergent series, which exhibit the correct limiting behavior, and which can be evaluated recursively. The performances of these stratagems are compared with numerical integration.

Journal ArticleDOI
TL;DR: In this paper, the trajectories followed in the complex plane by all the zeros of the Hankel function and those of its derivative, when the order varies continuously along real values, are discussed.
Abstract: The trajectories followed in the complex plane by all the zeros of the Hankel function and those of its derivative, when the order varies continuously along real values, are discussed.

Journal ArticleDOI
01 Jan 1982
TL;DR: In this paper, it was shown that the Wronskian W(c′vm, y′vk) v > yǫv1, where c′vm is the kth zero of the derivative C′v(x) with respect to x of the general Bessel function.
Abstract: Let y(x) be a non-trivial solution of the differential equation yn+p(x)y=0. In this paper we extend to the zeros of the derivative y′(x) some known results on the zeros of y(x). In addition let c′vk be the kth zero of the derivative C′v(x) with respect to x of the general Bessel function Cv(x)=AJv(x)+BYv(x) where Jv(x) and Yv(x) are the Bessel functions of first and second kind, respectively. We show that c′v,k + m/c′vk decreases to 1 as k increases, m = 1, 2, …. Finally we show that the Wronskian W(c′vm, y′vk) v > y′v1); here c′vm and y′vk and zeros of the derivatives of two Bessel functions of the same order, not necessarily linearly independent.

Journal ArticleDOI
TL;DR: In this paper, the probability density function of the intensity derivative is given by an infinite series of modified Bessel functions of the second kind and the effect of integrating over space and time has also been discussed.
Abstract: The statistics of the derivative of intensity for partially developed speckle patterns have been investigated. The probability density function of the intensity derivative is given by an infinite series of modified Bessel functions of the second kind. The effect of integrating over space and time has also been discussed. The approximate form for the probability density function of the integrated intensity of the differentiated partially developed speckle pattern is also given by an infinite series of modified Bessel functions of the second kind. In the special case of fully developed speckle patterns, the probability density functions derived in this paper reduce to those given by Ebeling [ Opt. Commun.35, 323 ( 1980)].

Journal ArticleDOI
TL;DR: In this article, rational approximations of the Bessel functions Jv(x), v=0,1,…,10 were presented, which can be used to simplify the Hankel transform to the computation of two Fourier transforms.
Abstract: We present rational approximations of the Bessel functions Jv(x), v=0,1,…,10, which can be used to simplify the computation of the Hankel transform to the computation of two Fourier transforms.

Journal ArticleDOI
01 Feb 1982
TL;DR: In this article, it was shown that the function (yb h` K/(bx 1/2)K,(axl/ 2) Va Kj, (ax 1 /2) K,(bx1/2),(ax 1 2 )K, (bx 2 /2 )I,(a 1/3 )I,(ax 2 /3)I,
Abstract: We prove that the function (yb h` K/(bx 1/2)K,(axl/2) Va Kj,(ax1/2)K,,(bx1/2) is the Laplace transform of an infinitely divisible probability distribution when v > IA > 0 and b > a > 0. This implies the complete monotonic ity of the function. We also establish a representation as a Stieltjes transform, which implies in particular that the function has positive real part when x lies in the right half-plane. We conjecture that b I-'I,z(ax 112)Iv(bxl /2) Va IjA(bx1/2)I,,(ax1/2) also is the Laplace transform of an infinitely divisible probability distribution. It is known that in the limit as v -_ oo, the infinite divisibility property holds for both functions.

Journal ArticleDOI
TL;DR: In this article, 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.
Abstract: 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.

Journal ArticleDOI
O. Shimbo1, L. Nguyen
TL;DR: In this analysis, it is shown that the suitably chosen values for α cannot be large (roughly, \alpha \leq 1.6 ).
Abstract: The Bessel function expansion has been used [1] to approximate the TWTA nonlinear characteristics to assist in the evaluation of intelligible crosstalk between FDM/FM carriers using the Shimbo-Pontano method [2]. In particular, the arbitrary constant α was chosen to be in the range of 100-1000. This correspondence presents a further theoretical clarification on the selection of suitable values for α, which was briefly explained in [3]. In this analysis, it is shown that the suitably chosen values for α (as given in the main text) cannot be large (roughly, \alpha \leq 1.6 ).

Proceedings ArticleDOI
01 May 1982
TL;DR: Three basic techniques for calculating Fourier-Bessel transforms have been recently developed and a brief review of these methods and a discussion of their accuracy are provided.
Abstract: The Fourier-Bessel transforms (also designated as Hankel transforms) arise in numerous signal processing problems. They are particularly important in image analysis, reconstruction of unknown media from projections, synthesis of waves from their spatial spectrum. Three basic techniques (and a number of possible variations) for calculating Fourier-Bessel transforms have been recently developed by us. The present paper provides a brief review of these methods and a discussion of their accuracy. Test calculations illustrate this analysis and yield further indications for proper application of the proposed methods.

Journal ArticleDOI
TL;DR: In this paper, Steed's method for the calculation of both the regular and irregular Coulomb functions for positive energy, F λ ( η,x),G λ( η,x), and their derivatives, is extended into the region of very high precision (∼ 30S).

Journal ArticleDOI
TL;DR: In this article, a simple technique was developed for calculating the correlation function directly from small-angle Xray scattering curves obtained with an finfinitely long' primary beam profile, based on expanding the correlation functions in a series of zero-order Bessel functions of the first kind, where the coefficients of the series are proportional to the intensities of the measured curve.
Abstract: A simple technique has been developed for calculating the correlation function directly from small-angle Xray scattering curves obtained with an finfinitely long' primary beam profile. "ihe method is based on expanding the correlation function in a series of zeroorder Bessel functions of the first kind, where the coefficients of the series are proportional to the intensities of the measured curve. The correlation function thus is represented by an analytical expression and can be calculated easily.




Journal ArticleDOI
A.J. Jerri1, I.A. Joslin2
TL;DR: In this article, the authors derived an upper bound for the truncation error of the generalized sampling expansion associated with the Hankel (Bessel) transform for optical systems with circular symmetry.
Abstract: Numerous upper bounds for the truncation error of the Shannon sampling expansion associated with the Fourier transforms are available in the literature. In this paper we derive an upper bound for the truncation error of the generalized sampling expansion associated with the Hankel (Bessel) transform. A clear example of the application of such series is in the analysis of optical systems with circular symmetry. A lower bound for the Bessel function Jm(jḿ,N+iy), which agrees with the known asymptotic one and is necessary to this analysis, is derived. The method, which employs complex contour integration, can be applied to other generalized sampling expansions.

Journal ArticleDOI
TL;DR: In this paper, a numerical modeling of partially buried vertical wire-antennas and scatterers is presented, where the problem of antenna grounding is formulated in terms of an integro-differential equation over the wire current across the air-ground interface, which is then converted into a numerically solvable form by means of Galerkin's method together with the piecewise linear functions.
Abstract: The purpose of the present work is the numerical modeling of partially buried vertical wire-antennas and scatterers. Our interest is in an efficient computation of their electromagnetic properties. Analytically the problem is formulated in terms of an integro-differential equation over the wire current across the air-ground interface. This equation is then converted into a numerically solvable form by means of Galerkin's method together with the piece-wise linear functions. The inverse Bessel transforms, commonly called Sommerfeld integrals, appearing in the Kernel of the integral equation have been evaluated by numerical integration over the real axis. The resulting computer code“STAKE,” which permits the numerical simulations of partially buried vertical wires, proved to be efficient and accurate in comparison with other computation methods. First the problem of antenna grounding has been investigated for the cases of soil and sea water. The current distributions, input admittances and radiatio...

Journal ArticleDOI
TL;DR: In this article, numerical, power-series and asymptotic solutions for the self-focusing problem are presented for the differential equation F'' + r^{ - 1} F'' - F' - F + F^3 = 0( 0 < r < ∞ )$ which arises in connection with self-focus.
Abstract: Numerical, power-series and asymptotic solutions are presented for the differential equation $F'' + r^{ - 1} F' - F + F^3 = 0( 0 < r < \infty )$ which arises in connection with self-focusing.

Journal ArticleDOI
TL;DR: In this article, the authors presented a method of evaluation of four integrals of Howland type, which involve a Bessel function in the integrands, with the aid of tabulated values, they are evaluated to 10D.
Abstract: This paper presents a method of evaluation of four integrals of Howland type, which involve a Bessel function in the integrands. With the aid of tabulated values, they are evaluated to 10D. Two of the four Howland integrals needed in the evaluation are evaluated anew to 20D in order to provide adequate accuracy. In a recent investigation of certain problems in elasticity concerning elliptic boundaries, four integrals of Howland type involving an additional Bessel function in the integrands were encountered. We believe that they deserve special consideration. The integrals are as follows: (1) F\"'k( \\ = tC mkj^ma\"> d (n + k> I), FZk{a) k\\ J0 sinh 2m ± 2m (n + k > 3), E\"'k( \\ = — (°° mkjn(ma) coth m dm (n + k > 2), E*,k k\\J0 sinh 2m ± 2m m (n + k > 4), where Jn is a Bessel function of the first kind of integral order n. n and k are nonnegative integers restricted as indicated above in order to render each integral convergent at the lower limit. The constant a may be real or complex. By using the usual series expression for Jn and integrating, the first integral


Journal ArticleDOI
TL;DR: In this paper, a solution for the seismic radiation from an arbitrarily growing spherical source in an inhomogeneously prestressed elastic medium is found for the general problem of the growing seismic source in a prestressed medium, formulated as a boundary value problem.
Abstract: Summary. A solution is found for the seismic radiation from an arbitrarily growing spherical source in an inhomogeneously prestressed elastic medium. The general problem of the growing seismic source in a prestressed medium is formulated as a boundary value problem. For the special case of the growing spherical source, an expansion in vector spherical harmonics reduces the problem to a set of one-dimensional Volterra integral equations. The equations can be easily formed through the use of Bessel function recursion relations. The integral equations for a growing spherical cavity are solved numerically. Waveforms are then computed for homogeneous and inhomogeneous stress fields for several growth histories. The resulting waveforms are similar to the waveforms of the corresponding instantaneous problem, but stretched out in time and reduced in amplitude. The effects of diffraction and the overshoot of equilibrium are reduced with a reduction in growth rate. The effects caused by inhomogeneity of the stress field are quite strong for the growing as well as for the instantaneous seismic source.