Showing papers on "Bessel function published in 1984"

Resonant spectra of dielectric spheres

Abstract: The natural resonant frequencies and poles associated with the electromagnetic modes of a dielectric sphere with a relative index of refraction of 1.4 have been calculated for size parameters ranging from 1 to 50. Determining pole locations in the complex plane entailed the computation of spherical Bessel functions for large complex arguments. The symbolic programming language reduce was used to provide independent verifications of the convergence and accuracy of the numerical Bessel function routines required in these computations. To determine pole locations, we used a standard zero-finding routine to find the zeros of the scattering coefficient denominators. In addition, we used a separate zero-counting routine in conjunction with the search routine to ensure that all poles within a given region of the complex plane were found. The real parts of the calculated poles agree with the location of peaks in the resonance spectrum (calculated for real frequency excitation), whereas the imaginary parts are related to the widths of these peaks. The intensity inside the sphere, averaged over all spherical angles, was computed as a function of radius. When the particle is excited at resonance, the internal intensity exhibits a sharp peak near, but not on, the surface. The intensity was found to be the strongest when the particle is driven at resonant frequencies whose poles have small imaginary components in the complex plane.

Linear augmented-Slater-type-orbital method for electronic-structure calculations

Abstract: The purpose of this paper is to explore the possibility of using augmented Slater-type orbitals (STO) as basis functions for electronic-structure calculations. STO's have a radial dependence given by ${r}^{n\ensuremath{-}1}\mathrm{exp}(\ensuremath{-}\ensuremath{\zeta}r)$ and as a result have a number of important advantages. They are localized about sites and have the same asymptotic form as actual atomic orbitals. They are regular at the origin and possess analytic Fourier transforms. The Fourier transform can be manipulated to yield an addition theorem, that is, a reexpansion formula for an STO about another site which is similar to the one used for spherical Bessel functions. Augmenting the STO's with numerical solutions of the Schr\"odinger equation within touching spheres leads to a small secular matrix since the numerical functions are orthogonal to all the core states and the STO's are only used in the interstitial region. The method has been applied to copper, silver, and palladium using Chodorow-type potentials and accounting for all relativistic effects except spin-orbit coupling. The results on copper are in good agreement with previous calculations and with experiments. The results on Pd and Ag are in better agreement with photoemission experiments than fully self-consistent local-density calculations.

Gyromagnetic emission and absorption - Approximate formulas of wide validity

Abstract: Approximate formulas of wide validity are derived for gyro magnetic emission and absorption of gyromagnetic waves by mildly relativistic electrons. An averaged emissivity is defined by replacing the sum over harmonic number by an integral and averaging over the pitch angle distribution of the radiating particles. A method for performing the average over pitch angle without approximation to the Bessel functions is developed and the resulting expressions are then approximated using Wild-Hill formulas which interpolate between the non-relativistic and ultra-relativistic limits.

Generalized Fock Spaces and Associated Operators

Abstract: A class of generalized Fock spaces associated with Bessel functions is studied. The generalized Fock space is a Hilbert space of even entire functions weighted by a modified Bessel function of the third kind, whereas ordinary Fock space is a Hilbert space of entire functions of several complex variables weighted by a Gaussian kernel. The generalized Fock space has a reproducing kernel which is a modified Bessel function of the first kind.Commutator relations between the Schrodinger radial kinetic energy operator and multiplication by $z^2 $ lead to a generalized class of Weyl relations for the Bessel functions.

Surface-wave computations and the synthesis of theoretical seismograms at high frequencies

Abstract: Multi-mode, surface-wave dispersion—eigenvalue—computations are studied in the range of frequencies from 1 to 10,000 Hz, for structures composed of a sequence of homogeneous layers. When the original version of the Thomson-Haskell technique is applied to such structures, a loss-of-precision problem is encountered with Rayleigh-wave computations. Into the range of high frequencies, we now want to know whether the Love-wave formulation and the alternate Rayleigh-wave formulations are completely free of this problem. At low, intermediate, high, and very high frequencies, our numerical testing shows no evidence whatsoever of loss-of-precision problems; not for Love waves, for Rayleigh waves when Knopoff's method is used, nor for Rayleigh waves when the deltamatrix extension of the original Thomson-Haskell formulation is employed. In these three cases, when P digits are used in the computations, the resulting phase velocities can be obtained to full. P digit accuracy. All overflow/underflow problems can be controlled by simple modifications of the usual computational methods; these modifications do not affect the accuracy of the calculated dispersion. The computation of displacement-depth and stress-depth functions—eigenfunctions—is also studied up to extremely high frequencies. Once again, the original Rayleigh-wave formulation encounters loss-of-precision difficulties in these eigenfunction evaluations. The use of Knopoff's method completely solves this aspect of the problem with precision loss. As with dispersion calculations, all overflow/underflow problems are easily controlled by means that do not affect the accuracy of the computed eigenfunctions. In addition to precision loss and overflow/underflow problems, three further details affect efficiency and accuracy when synthesizing high-frequency seismograms by multi-mode summation of dispersed surface waves. Avoid branch-line integrals and spherical Bessel functions of nonintegral order by using spherical structures combined with transformations permitting replacement of Bessel functions with circular or exponential functions. The number of higher modes treated explicitly, at each of the higher frequencies, should be limited by selective sampling combined with summation procedures approximating inclusion of all modes. As frequency increases, the structural specification must be monitored to ensure the homogeneous layers used to approximate the true structure do not become much thicker than the vertical extent of the eigenfunction lobes.

An Inequality for the Bessel Function $J_\nu (\nu x)$

Abstract: An inequality for the Bessel function $J_ u ( u x)$, $ u > 0$, $0 < x \leq 1$ involving both upper and lower bounds is derived. Inequalities for the modified Bessel functions are also obtained.

Numerical calculation of four-center Coulomb integrals

Abstract: A numerical method for the calculation of an arbitrary four‐center Coulomb integral for angular momentum wave functions is developed. The method involves expanding the product of two angular momentum wave functions at two different centers as a linear combination of angular momentum wave functions centered at a point on the line segment joining the two centers. A formalism is developed that reduces the storage requirements for the functions defining the expansion of the product. The integral is therefore expressed as a sum of two‐center integrals that are calculated in momentum space. A recently developed method for calculating spherical Bessel transforms is valuable in the final step. Sample results for a number of calculations are given and the possible application of the method in molecular structure calculations is considered.

A series expansion for the first positive zero of the Bessel functions

Abstract: It is shown that the first positive zerojv X of the Bessel function J,(x) is given by lp. = 2(v + 1) 1/2 1 + (v+ 1) 7(v-+ 1)2 + 49(v + 1)3 8363(v + 1) 4+ ] = ? + ~ ~ ~ 9 6 1152 276480 + for 1 1, the Bessel function J,(x) has an infinite number of zeros and that all zeros are real (Watson [9]). We denote the sth positive zero of Jp(x) byj s. Several approximations, asymptotic expansions or bounds for the zeros of Bessel functions exist (see [1], [2], [4], [6], [7], [9]). Especially McMahon's expansion for large zeros (see Abramowitz and Stegun [1]), Olver's asymptotic expansion for large orders and Olver's uniform asymptotic expansions (see Olver [6]) are interesting formulas, but, unfortunately, they are not applicable when s and v are small. The purpose of this note is to give a series expansion forj'1,I when 1 < v < 0. 2. Cayley [3] noticed that Graeffe's method for solving a polynomial equation can be applied for the efficient computation of (1) 2r (r) r = 1,2.

Propagation on a Sheath Helix in a Coaxially Layered Lossy Dielectric Medium (Short Paper)

Abstract: Radial and axial dependence of the azimuthally symmetric fields in each coaxial layer may be expressed in terms of modified Bessel functions and complex exponentials, respectively

Exact solutions for spectra and Green's functions in random one-dimensional systems

Abstract: For chains of harmonic oscillators with random masses a set of equations is derived, which determine the spatial Fourier components of the average one-particle Green's function. These equations are valid for complex values of the frequency. A relation between the spectral density and functions introduced by Schmidt is discussed. Exact solutions for this Green's function and the less complicated characteristics function-the analytic continuation into the complex frequency plane of the accumulated spectral density and the inverse localization length of the eigenfunctions-are derived for exponential distributions of the masses. For some cases the characteristic function is calculated numerically. For gamma distributions the equations are cast in the form of ordinary, higher order differential equations; these have been solved numerically for determining the characteristic function. For arbitrary mass distributions a cumulant expansion and a peculiar symmetry of the Green's function are discussed. The method is also applied to chains where the spring constants and/or the masses have random values. Also for these systems exact solutions are discussed; for exponential distributions, e.g., of both masses and spring constants the characteristic function is expressed in Bessel functions. The relation with certain random relaxation models is shown. Finally, X-Y Hamiltonians with random exchange constants and/or magnetic fields-or, equivalently, tight-binding electron models with diagonal and/or off-diagonal disorder-are considered. Here the Green's function does not depend on the wave number if the distribution of exchange constants is symmetric around the origin. New solutions for the characteristic function and Green's function are derived for a number of cases, including exponentially distributed magnetic fields and power law distributed exchange constants.

Laplace transforms, Mellin transforms and mixed poisson processes

Abstract: 1. Introduction While trying to generalize results of McNolty (1964), the present author discovered the usefulness of Laplace and Mellin transforms for the computation of the counting distribution Pn (t)=P(N(t)=n) of a mixed Poisson process {N(t); t⩾0}. For various mixing distributions U we present the corresponding expressions for Pn (t), expressions which have not been contained in literature so far, but also neater forms or easier derivations of existing results. Mixing distributions which are considered are: χ 2, χ, Maxwell, Rayleigh, Weibull, generalized gamma, Pearson Type I, III, V, VI, F, Pareto, Bessel and truncated normal.

Scattering by a cylinder: A fast exact numerical solution

Abstract: The known exact analytic solutions to the problem of (acoustic and electromagnetic) scattering by a two‐dimensional infinite right circular cylinder are not in closed form, but consist of infinite series, each term of which contains Hankel and Bessel functions of increasing order, and the convergence rate of which decreases rapidly with increasing (ka) numbers. For a given numerical solution, it is thus necessary to compute many Hankel and Bessel functions in high order for each spatial datum point for which the scattered field need be calculated. Presented is an exact numerical method of solution, which is in closed form, and requires the computation of only one Hankel function of order unity per spatial datum point for which the fields need be calculated. The method consists of a closed form numerical deconvolution solution of the scattering integral equation, executed efficiently with the aid of the fast Fourier transform algorithm, thus requiring only ( (3)/(2) N log2 N) complex arithmetic multiply–a...

Series Representations for Rice's Ie Function

Abstract: Two new series representations for the Rice function Ie (k, x) are presented. One of the series involves the modified Struve functions and the other involves the modified Bessel functions. These two series complement each other in their convergence speeds as functions of the values of k and x . The truncation error bounds are derived for both series. Therefore, they can be used alternatively with high efficiency and known precision.

Aero-optical foundations and applications

Abstract: Nomenclature A = coefficient defined in Eq. (A40) A Q = area B = magnetic field c = speed of light C = correlation function Cn = index of refraction structure function constant Cp = specific heat D = diameter of aperture D = electric displacement E = spectrum E = electric field H = magnetic induction / = intensity // = Bessel function of order / k = wave number / = turbulence scale size L = path length

Variational R-matrix calculations of e-+H2 scattering using numerical asymptotic basis functions

Abstract: Variational R-matrix calculations of elastic electron scattering from the ground state of H2 are reported for the 2 Sigma u+ resonant scattering state at R=1.402a0. The orbital basis set includes exponential functions centred on each nucleus, spherical Bessel functions and numerical functions obtained by integrating asymptotic coupled differential equations. Polarisation response is treated in terms of polarisation pseudo-states.

Product Formulas for Bessel, Whittaker, and Jacobi Functions via the Solution of an Associated Cauchy Problem

Abstract: Delsarte’s approach to generalized translation operators via the solution of an associated Cauchy problem is used to derive the product formulas for the Bessel, Whittaker, and Jacobi functions in kernel form. As essential prerequisites, explicit representations of the corresponding Riemann functions are given for three cases. The main part of the paper deals with the Jacobi case, for which the derivation of the translation kernel is carried out explicitly. In the Bessel case, the results of Delsarte are covered, and in the Whittaker case, a generalization of previous results of the author on the Laguerre polynomial product formula is obtained.

Technique for Evaluating Indefinite Integrals Involving Products of Certain Special Functions

Abstract: A new technique is described for evaluating a general class of indefinite integrals involving products of many of the special functions of physics such as Bessel functions, Legendre functions, Hermite functions, etc. The technique is a generalization of the method used by Sonine to evaluate certain indefinite integrals of Bessel functions. It involves replacing the integral to be evaluated by a coupled set of linear, inhomogeneous differential equations. A particular solution of the set of differential equations is then sufficient to express the result of integration. Several examples are given to illustrate the technique.