Showing papers on "Bessel function published in 1990"

Multidimensional digital image representations using generalized Kaiser–Bessel window functions

Abstract: Inverse problems that require the solution of integral equations are inherent in a number of indirect imaging applications, such as computerized tomography. Numerical solutions based on discretization of the mathematical model of the imaging process, or on discretization of analytic formulas for iterative inversion of the integral equations, require a discrete representation of an underlying continuous image. This paper describes discrete image representations, in n-dimensional space, that are constructed by the superposition of shifted copies of a rotationally symmetric basis function. The basis function is constructed using a generalization of the Kaiser-Bessel window function of digital signal processing. The generalization of the window function involves going from one dimension to a rotationally symmetric function in n dimensions and going from the zero-order modified Bessel function of the standard window to a function involving the modified Bessel function of order m. Three methods are given for the construction, in n-dimensional space, of basis functions having a specified (finite) number of continuous derivatives, and formulas are derived for the Fourier transform, the x-ray transform, the gradient, and the Laplacian of these basis functions. Properties of the new image representations using these basis functions are discussed, primarily in the context of two-dimensional and three-dimensional image reconstruction from line-integral data by iterative inversion of the x-ray transform. Potential applications to three-dimensional image display are also mentioned.

Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter

Abstract: Bessel functions of purely imaginary order are examined. Solutions of both the modified and unmodified Bessel equations are defined which, when their order is purely imaginary and their argument is real and positive, are pairs of real numerically satisfactory functions. Recurrence relations, analytic continuation formulas, power series representations, Wronskian relations, integral representations, behavior at singularities, and asymptotic forms of the zeros are derived for these numerically satisfactory functions. Also, asymptotic expansions in terms of elementary and Airy functions are derived for the Bessel functions when their order is purely imaginary and of large absolute value.Second-order linear ordinary differential equations having a large parameter and a simple pole are then examined, for the case where the exponent of the pole is complex. Asymptotic expansions are derived for the solutions in terms of the numerically satisfactory Bessel functions of purely imaginary order.

Analytic continuation considerations when using generalized formulations for scattering problems

Abstract: A generalized operator equation has been developed for a simple scattering problem for which a closed-form solution exists. The operator equation has been solved numerically via the method of moments using spatially impulsive fictitious sources as expansion functions together with a simple point-matching testing procedure. The study focused on the convergence and accuracy of the solution and examined how they are dependent on the location and distance of the fictitious sources relative to the area containing the singularities of the actual field simulated by these sources. As expected, it was found that when the actual field simulated by the fictitious impulsive sources has singularities lying between the physical boundary and the closed surface over which the sources are placed, the impulsive expansion does not yield a uniformly convergent solution. In this case, instabilities are encountered as the number of sources increases, in the sense that a small improvement of the boundary error requires a considerable change in the currents. The moment matrix is then difficult to invert and easily susceptible to large round-off errors. Conversely, if the actual field has no singularities lying between the physical boundary and the closed surface over which the sources are placed, the impulsive expansion does yield a uniformly convergent solution to any degree of precision. >

Theory of generalized Bessel functions@@@Теория обобщенных функций Бесселя

Abstract: SummaryIn this paper we discuss the theory of generalized Bessel functions which are of noticeable importance in the analysis of scattering processes for which the dipole approximation cannot be used. We introduce these functions in their standard form and their modified version. We state the relevant generating functions and Graf-type addition theorems. The usefulness of the results to construct a fast algorithm for their quantitative computation is also devised. We comment on the possibility of getting two-index generalized Bessel functions ine.g. the study of sum rules of the type $$\sum\limits_{n = - \infty }^\infty {t^n J_n^3 (x)} $$ , whereJn is the cylindrical Bessel function of the first kind. The usefulness of the results for problems of practical interest is finally commented on. It is shown that a modified Anger function can be advantageously introduced to get an almost straightforward computation of the Bernstein sum rule in the theory of ion waves.RiassuntoIn questo lavoro si discute la teoria delle funzioni di Bessel generalizzate, che assumono notevole rilevanza nell’analisi dei processi di scattering in cui non può essere utilizzata l’approssimazione di dipolo. Si introducono tali funzioni nelle loro forme standard e modificate. Vengono derivate le relative funzioni generatrici e teoremi di addizione di tipo Graf. Si commenta inoltre l’importanza dei risultati ottenuti per costruire un algoritmo numerico per il loro calcolo esplicito. Si discute inoltre la possibilità di ottenere funzioni di Bessel a due indici utili nello studio di regole di somma del tipo $$\sum\limits_{n = - \infty }^\infty {t^n J_n^3 (x)} $$ , whereJn è la funzione di Bessel cilindrica di primo tipo. Si analizza l’utilità dei risultati ottenuti per lo studio di problemi fisici concreti. Si dimostra infine che è possibile introdurre una funzione di Anger modificata per riottenere una derivazione immediata della regola di somma di Bernstein relativa alla teoria delle onde ioniche.РезюмеВ этой статье мы обсуждаем теорию обобщенных функций Бесселя, которые играют важную роль при анализе процессов рассеяния, для которых невозможно использовать дипольное приближение. Мы вводим эти функции в их стандартной форме и их модифицированный вариант. Мы формулируем соответствующие производящие функции и теоремы сложения типа Графа. Отмечается полезность полученных результатов для конструирования быстрого алгоритма для количественных вычислений. Мы отмечаем возможность получения обобщенных функций Бесселя с двумя индексами, например, при исследовании правия сумм типа $$\sum\limits_{n = - \infty }^\infty {t^n J_n^3 (x)} $$ , whereJn, чдеJn есть цилиндрическая функция Бесселя первого рода. В заключение указывается полезность этих результатов для проблем, представляющих практический интерес. Показывается, что модифицированная функция Ангера может быть с успехом использована для непосредственного вычисления правила сумм Бернейнстейна в теории ионных волн.

The numerical computation of Bessel functions of the first and second kind for integer orders and complex arguments

Abstract: A rigorous algorithm for computing integer order Bessel functions of the first and second kind with complex arguments is discussed. The algorithm makes use of backward recurrence for computing Bessel functions of the first kind where applicable, and of Hankel's asymptotic expansion for large arguments. The argument is limited only by the ability of the computer to handle large numbers. Orders up to a few thousand have been computed to 13-digit accuracy. >

Non-overlapping partitions, continued fractions, bessel functions and divergent series

Abstract: The counting sequence of a special class of set partitions leads to special numbers called Bessel numbers. The corresponding ordinary generating function has a simple continued fraction expansion related to Bessel functions. We determine here the asymptotic form of Bessel numbers and discuss their relation to Bell numbers. The estimation problem is of some methodological interest as it is necessary to find the asymptotic form of coefficients in an asymptotic but divergent expansion.

Stieltjes transforms and the Stokes phenomenon

Abstract: Recently, Berry, Olver and Jones have found uniform asymptotic expansions for the exponentially small remainder terms that result when asymptotic expansions are optimally truncated. These uniform expansions describe the rapid change in the behaviour of the remainders as a Stokes line is crossed. We show how such uniform expansions may be found when a function can be expressed as a Stieltjes transform. Such an approach has the following advantages: the uniform expansion is calculated directly, non-uniform expansions away from the Stokes line are readily found, and explicit error bounds may be established. We illustrate the method by application to the modified Bessel function K v ( z ).

On the zeta‐function regularization of a two‐dimensional series of Epstein–Hurwitz type

Abstract: As a further step in the general program of zeta‐function regularization of multiseries expressions, some original formulas are provided for the analytic continuation, to any value of s, of two‐dimensional series of Epstein–Hurwitz type, namely, ∑∞n1,n2=0[a1(n1+c1)2 +a2(n2+c2)2]−s, where the aj are positive reals and the cj are not simultaneously nonpositive integers. They come out from a generalization to Hurwitz functions of the zeta‐function regularization theorem of the author and Romeo [Phys. Rev. D 40, 436 (1989)] for ordinary zeta functions. For s=−k,0,2, with k=1,2,3,..., the final results are, in fact, expressed in terms of Hurwitz zeta functions only. For general s they also involve Bessel functions. A partial numerical investigation of the different terms of the exact, algebraic equations is also carried out. As a by‐product, the series ∑∞n=0exp[−a(n+c)2], a,c>0, is conveniently calculated in terms of them.

Asymptotic estimates for Laguerre polynomials

Abstract: We give a brief summary of recent results concerning the asymptotic behaviour of the Laguerre polynomialsL () (x). First we summarize the results of a paper of Frenzen and Wong in whichn→∞ and α>−1 is fixed. Two different expansions are needed in that case, one with aJ-Bessel function and one with an Airy function as main approximant. Second, three other forms are given in which α is not necessarily fixed. These results follow from papers of Dunster and Olver, who considered the expansion of Whittaker functions. Again Bessel and Airy functions are used, and in another form the comparison function is a Hermite polynomial. A numerical verification of the new expansion in terms of the Hermite polynomial is given by comparing the zeros of the approximant with the related zeros of the Laguerre polynomial.

The operational matrix of integration for Bessel functions

Abstract: A general expression for the operational matrix of integration P for the case of Bessel functions is derived. Using this P, several problems such as identification, analysis and optimal control may be studied. Examples are included to illustrate the theoretical results.

Optimized Gaussian basis sets for representation of continuum wavefunctions

Abstract: Exponents of Gaussian-type basis functions have been optimized for electron-molecule scattering purposes. The criterion for optimization was to obtain the best least-squares fit to six Bessel functions jl(kh(l)*r) representing the continuum functions. The values for the radial momentum kh(l) are defined by the boundary conditions for the Bessel functions to have vanishing radial derivatives at r=20 au. For each l=0, 1 and 2, Gaussian basis sets of eight functions have been optimized. The results are of excellent quality. It is therefore concluded that usual atomic Gaussian basis sets, augmented by these functions, can be sufficient in electron-molecule scattering calculations, such as R-matrix calculations, for example.

A uniform expansion for the eigenfunction of a singular second-order differential operator

Abstract: In a recent work, Frenzen and Wong [Canad. J. Math., 37 (1985), pp. 979–1007] have obtained a uniform asymptotic expansion for the Jacobi polynomials in terms of Bessel functions. An analogous expansion for the Jacobi functions had been given earlier by Stanton and Tomas [Acta Math., 140 (1978), pp. 251–276]. The common starting point of these papers is an integral representation.In this paper it is shown that in general such expansions can be obtained directly from an eigenfunction of a singular second-order differential operator, and with additional assumptions they converge in some interval. This leads to an expansion for the eigenfunction of an integral representation of Mehler type with good information on the kernel.

Evaluation of a nondiffracting transducer for tissue characterization

Abstract: The authors describe a method for estimating backscatter coefficients of excised human liver samples and the RMI413A tissue equivalent phantom, using the J/sub 0/ Bessel nondiffracting transducer. A formula for the calculation of backscatter coefficients that requires only two one-dimensional integrations in addition to one-dimensional Fourier transforms is presented. This equation in combination with backscatter from a J/sub 0/ Bessel nondiffracting beam results in the ability to calculate backscatter coefficients in real time. In addition, estimations of the backscatter coefficients are more distance-independent because of the nonspreading nature of the J/sub 0/ Bessel nondiffracting beam. The present results compare well to those obtained by the conventional focused Gaussian beam transducer. >

The Initial Phase of Transient Thermal Stresses Due to General Boundary Thermal Loads in Orthotropic Hollow Cylinders

Abstract: The stresses and displacements are obtained using the Hankel asymptotic expansions for the Bessel functions of the first and second kind. The material properties are assumed to be independent of temperature. A constant applied temperature at the one surface and convection into a medium at a different temperature at the other surface is studied

Bessel beam radar system using sequential spatial modulation

Abstract: A spatially modulated Bessel beam radar system for enhancing the resolution with which a range and an azimuth of a plurality of closely spaced targets is determined. In a Bessel beam radar system, a Bessel beam is generated by sequential spatial modulation of the radar signal while maintaining a constant spatial polarization, and the return signal from one or more targets is processed to determine its Bessel function content. To spatially modulate the radar beam, the point at which the radar signal is transmitted is moved around a circular orbit. In a first embodiment of the spatially modulated Bessel beam radar system (80), a radar dome (86) mounted on the distal end of a mast (84) is pivoted around an orbit (90). The radar signal is transmitted in a predefined direction, along a Poynting vector that is generally aligned with the plane of the orbit. In a second embodiment (110), a plurality of parabolic antennas (116) are arranged in a spaced-apart circular array around a common center. The radar signal is sequentially spatially modulated as it is transmitted from each of the parabolic antennas in sequence around the circular array, and the Poynting vector of the spatially modulated Bessel beam radar signal is generally transverse to a plane in which the parabolic antennas are disposed. The signal reflected back from plural targets comprises a complex phase history. To determine a range and azimuth for each target, a controller/processor (180), processes this signal to develop closed form Bessel functions from which target azimuth and range are determined. Alternatively, target azimuth is determined from a convolution of the complex phase history using a dot product detector (202).

Continued-fraction solutions to the Riccati equation and integrable lattice systems

Abstract: Continued-fraction solutions to the matrix Riccati equation are discussed which are constructed by using the concept of form invariance It is demonstrated that this technique is related to the AKNS method of deriving integrable nonlinear lattice systems This gives an explanation why continued-fraction solutions related to the Toda lattice were obtained in a previous work Continued fractions corresponding to Kac-Van Moerbeke, discrete nonlinear Schrodinger and discrete modified KdV lattice equations are constructed A method for linearising the Kac-Van Moerbeke lattice equations is rederived and particular solutions are generated The authors approach demonstrates the crucial role played by the boundary condition at the finite end of the lattice for the existence of this method These results are extended to the other two lattice systems above in the semi-infinite case and corresponding particular solutions generated in terms of Bessel functions

Effective Performance of Bessel Arrays

Abstract: The Bessel array is a configuration of five, seven, or nine identical loudspeakers in an equal-spaced line array that provides the same overall polar pattern as a single loudspeaker of the array. The results of a computer simulation are described. The various Bessel configurations are compared to one-, two-, and five-source equal-spaced equal-level equal-polarity line arrays

Eigenmodes for electromagnetic waves propagating in a toroidal cavity

Abstract: A solution has been attempted by means of the Helmholtz equation for an electromagnetic wave propagating in an empty torus in a system of toroidal coordinates. The electromagnetic fields are expressed in terms of the Hertz vector to obtain a scalar Helmholtz equation. The latter has been solved by making use of an inverse aspect ratio expansion of the solution. Unlike most previous workers, the authors have obtained their solutions in terms of hypergeometric functions whose static limit is the toroidal harmonics. The cylindrical solutions in terms of Bessel functions can also be recovered by taking the appropriate large aspect ratio limit. The eigenmodes, with arbitrary toroidal and poloidal mode numbers, have been obtained by applying the boundary conditions on the metallic walls of infinite conductivity, and they cannot be distinguished as TE or TM modes. Eigenfrequencies for various toroidal and poloidal mode numbers are plotted against the inverse aspect ratio. First-order approximations to the fields in the toroidal cavity have also been derived. >

Time-dependent treatment of scattering. II, Novel integral equation approach to quantum wave packets

Abstract: The novel wave-packet propagation scheme presented is based on the time-dependent form of the Lippman-Schwinger integral equation and does not require extensive matrix inversions, thereby facilitating application to systems in which some degrees of freedom express the potential in a basis expansion. The matrix to be inverted is a function of the kinetic energy operator, and is accordingly diagonal in a Bessel function basis set. Transition amplitudes for various orbital angular momentum quantum numbers are obtainable via either Fourier transform of the amplitude density from the time to the energy domain, or the direct analysis of the scattered wave packet.

Fake Airy functions and the asymptotics of reflectionlessness

Abstract: Two classes of analytic refractive-index profile P2(z, in ), whose reflection coefficients r are zero for all values of a parameter in , are studied as in to 0. The aim is to understand why r=0 rather than r varies as exp(-1/ in ) as for generic profiles. The authors find that reflectionlessness is a consequence of the fact that transition points of P2 (zeros or poles in the complex z plane) form tight clusters (whose size vanishes with in ) which can be regarded neither as coalesced nor well separated. Expansion near a cluster yields the local wave not as the usual Airy function, whose Stokes phenomenon generates reflection, but as Bessel functions of half-integer order (fake Airy functions) which are exactly trigonometric functions with no Stokes phenomenon and so no reflection.

Two-dimensional radiative transfer in a cylindrical layered medium with reflecting interfaces

Abstract: A system of exact linear integral equations for the source function, intensity, and flux is presented for a twodimensional cylindrical medium consisting of up to four layers with reflecting interfaces between the layers. Properties that may change from layer to layer are the single scattering albedo, optical thickness, and refractive index. The incident radiation is coUimated and has a Bessel function distribution. The Bessel function boundary condition reduces the two-dimensional problem to a one-dimensional problem. Superposition is then used to derive the solution for any other boundary condition that is Hankel transformable. A special case of a Gaussian distribution that models a laser beam is presented. Some one-dimensional numerical results are presented for the source function and intensity within one- and two-layer media. Two-dimensional results are presented for back-scattered intensity due to the laser-beam boundary condition. Only the conservative case, optical thicknesses of 2.0 and 5.0, and refractive indices of 1.00 and 1.33 are considered. Nomenclature A, = multiple reflection coefficient, defined in the Appendix B{J = function defined in the Appendix DN — function defined in the Appendix E = component of kernel function for source function integral equation Giy = function defined in the Appendix g = Hankel transformed function /, = intensity of radiation in layer i /+ = intensity in layer i in + TZ direction It~ = intensity in layer i in — rz direction I0 = magnitude of incident intensity outside the medium IT = intensity transmitted across an interface /, = magnitude of incident intensity J0 = zeroth order Bessel function of first kind Kfj = function defined in the Appendix W = total number of layers HIJ = ratio of layer i refractive index to that of layer ; P = scattering phase function Qij = function defined in the Appendix qzi = z direction flux in layer i Ri = function defined in the Appendix r = radial distance from center of medium r0 = laser beam radius 5, = source function in layer i Scio = source function lead term, defined in the Appendix T = transmission function, defined by Eq. (32) Ti = function defined in the Appendix tfj = transmission through the interface between layer i and / x = dummy integration variable

A derivation of the vector addition theorem

Abstract: An alternative derivation of the vector addition theorem is presented using the completeness of vector wave functions and integration by parts. The advantage of this derivation is that it leads directly to the simplified results of Bruning and Lo and of Stein. Moreover, the dichotomous results of the addition theorem when a spherical Hankel function is involved can be derived by contour integration.

Analysis of propagation through turbulence: evaluation of an integral involving the product of three Bessel functions

Abstract: Mellin-transform techniques are used to evaluate an integral involving the product of three Bessel functions and a power. This integral appears in a number of results concerning the statistics of propagation through turbulence. The solution takes the form of a generalized hypergeometric function, which is expressible as a series that converges rapidly for many cases of interest that pertain to atmospheric turbulence.

Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and coalescing turning point

Abstract: Second-order linear differential equations having a turning point and double pole with complex exponent are examined. The turning point is assumed to be a real continuous function of a parameter $\alpha $, and coalesces with the pole at the origin when $\alpha \to 0$. Asymptotic expansions for solutions, as a second parameter $u \to \infty$, are constructed in terms of Bessel functions of purely imaginary order. The asymptotic solutions are uniformly valid for the argument lying in both real and complex regions that include both the coalescing turning point and the pole. The theory is then applied to obtain uniform asymptotic expansions for Legendre functions of large real degree and purely imaginary order.