Showing papers on "Bessel function published in 1986"

Dynamic localization of a charged particle moving under the influence of an electric field.

Abstract: The motion of a charged particle on a discrete lattice under the action of an electric field is studied with the help of explicit calculations of probability propagators and mean-square displacements. Exact results are presented for arbitrary time dependence of the electric field on a one-dimensional lattice. Existing results for the limiting cases of zero frequency and zero field are recovered. A new phenomenon involving the dynamic localization of the moving particle is shown to result in the case of a sinusoidally varying field: The particle is generally delocalized except for the cases when the ratio of the field magnitude and the field frequency is a root of the ordinary Bessel function of order 0. For these special cases it is found to be localized. This localization could be used, in principle, for inducing anisotropy in the transport properties of an ordinarily isotropic material.

Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two‐center problem in molecular quantum mechanics

Abstract: The differential equation, x(x−x0)(d2y/dx2)+(B1+B2x) (dy/dx)+[ω2x(x−x0) −[2ηω(x−x0)+B3]y=0, arises both in the quantum scattering theory of nonrelativistic electrons from polar molecules and ions, and, in the guise of Teukolsky’s equations, in the theory of radiation processes involving black holes. This article discusses analytic representations of solutions to this equation. Previous results of Hylleraas [E. Hylleraas, Z. Phys. 71, 739 (1931)], Jaffe [G. Jaffe, Z. Phys. 87, 535 (1934)], Baber and Hasse [W. G. Baber and H. R. Hasse, Proc. Cambridge Philos. Soc. 25, 564 (1935)], and Chu and Stratton [L. J. Chu and J. A. Stratton, J. Math. Phys. (Cambridge, Mass.) 20, 3 (1941)] are reviewed, and a rigorous proof is given for the convergence of Stratton’s spherical Bessel function expansion for the ordinary spheroidal wave functions. An integral is derived that relates the eigensolutions of Hylleraas to those of Jaffe. The integral relation is shown to give an integral equation for the scalar field quasinormal modes of black holes, and to lead to irregular second solutions to the equation. New representations of the general solutions are presented as series of Coulomb wave functions and confluent hypergeometric functions. The Coulomb wave‐function expansion may be regarded as a generalization of Stratton’s representation for ordinary spheroidal wave functions, and has been fully implemented and tested on a digital computer. Both solutions given by the new algorithms are analytic in the variable x and the parameters B1, B2, B3, ω, x0, and η, and are uniformly convergent on any interval bounded away from x0. They are the first representations for generalized spheroidal wave functions that allow the direct evaluation of asymptotic magnitude and phase.

Transform Analysis of Generalized Functions

Abstract: Preliminaries. Finite Parts of Integrals. Base Spaces. Definition of Distributions. Properties of Generalized Functions and Distributions. Operations on Generalized Functions and Distributions. Other Operations on Distributions. The Fourier Transformation. The Laplace Transformation. Applications of the Laplace Transformation. The Stieltjes Transformation. The Mellin Transformation. Hankel Transformation and Bessel Series. Bibliography. Author Index.

Asymptotic solutions for the scattered field of plane wave by a cylindrical obstacle buried in a dielectric half-space

Abstract: Two-dimensional scattering of a plane wave by an embedded conducting strip is formulated rigorously using the concept of Kobayashi potential, in which potential or wave function is expressed in terms of infinite integrals including the Bessel function in the integrand. By imposition of the required boundary conditions at the interface of the dielectric half-space and on the strip, the problem is reduced to a dual integral equation (DIE). Using the discontinuous properties of Weber-Schafheitlin's integrals, DIE is transformed into a matrix equation with infinite unknowns whose elements are expressed by infinite integrals. Asymptotic solutions for the matrix elements are derived when the separation between the interface of the different media and the obstacle is large compared to the wavelength. Using these results, the expression for the scattered field is derived in a general form which can be applied to an arbitrary cylindrical obstacle. Some numerical results are given for conducting strip and circular cylinder to see the effect of inhomogeneity on the surrounding medium, size of the obstacle, and the angle of incidence on the scattered field.

Trigonometric Approximations for Bessel Functions

Abstract: Bessel functions, used extensively in mathematical physics, electromagnetics, and communication-system theory, must often be approximated by appropriate formulas. The functions J?(x) and I?(x) for integer values of p are well approximated by the sum of a small number of circular and hyperbolic sines or cosines, respectively, when x is not too large, e.g., less than 10, and convenient expressions are obtained for their errors. These approximations are especially convenient for use on personal computers (PCs) and pocket calculators and in analytically evaluating integrals.

Diffusion of impurities in a dense fluid near the glass transition

Abstract: The diffusion of impurities of low concentration in a surrounding dense medium is studied near the liquid-glass transition point. Within a simplified model we obtain an analytical solution in the long-time or small-frequency regime for a generalized diffusion constant, which is expressed in terms of Bessel functions. At the glass transition of the medium the impurities can either become localized or still be delocalized. In the former case, there exists an Einstein-Stokes relation between the diffusion constant and the viscosity of the medium. In the latter case, the situation is similar to a superionic glass and reduces essentially to the Lorentz model. Explicit results for the generalized diffusion constant in the various regimes are obtained from asymptotic expansions of the Bessel functions.

Uniform asymptotic solutions of a class of second order linear differential equations having a turning point and a regular singularity, with an application to Lengendre functions

Abstract: The asymptotic behaviour, as a parameter $u \to \infty $, of solutions of second-order linear differential equations with a turning point and a regular (double pole) singularity is considered. It is shown that the solutions can be approximated by expressions involving Bessel functions in a region which includes both the turning point and the singularity. Explicit error bounds for the difference between the approximations and the exact solutions are established. The theory is applied to find uniform asymptotic expansions for Legendre functions.

A double integral containing the modified Bessel function: Asymptotics and computation

Abstract: On considere une integrale bidimensionnelle comportant exp(−u−t) Io(2√ut) Io(z) est la fonction de Bessel modifiee et l'integrale est prise sur le rectangle o≤u≤x, o≤t≤y. On presente des algorithmes efficaces de calcul

Some integrals involving three modified Bessel functions. II

Abstract: The integrals ∫∞0Zμ(at)Kν(bt) iKρ(ct)dt, where Zμ=Iμ, Kμ, are calculated, with the help of the factorization properties of the function F4. Results are given for real parameters a, b, c both when they are and are not in a triangle configuration. Some generalizations using derivation with respect to the parameters are considered.

Analytical solutions to the general problem of oblique wave growth and damping

Abstract: Analytical solutions are presented for the linear growth or damping rate caused by resonant particle interactions with plasma waves propagating in any arbitrary direction relative to the ambient magnetic field. For the case of three realistic and widely adopted representations for the particle distribution function, namely for a bi‐Maxwellian, a bi‐Lorenzian, and a loss‐cone distribution, the angular dependence of the net growth rate can be expressed in terms of simple, smoothly varying functions involving modified Bessel functions of the first and second kind. Furthermore, in the limits of both quasilongitudinal and quasitransverse wave propagation, the analytical results reduce to simple algebraic expressions that may readily be used to compare the contributions from any specific harmonic resonance. These analytical solutions eliminate the need for costly and time‐consuming numerical integration that hitherto was the standard procedure for obtaining growth rates for oblique waves.

Poisson integrals of regular functions

Abstract: Tangential convergence of Poisson integrals is proved for certain spaces of regular functions which contain the spaces of Bessel potentials of Lp functions, 1 1.

The use of Bessel functions to extend the range of optical diffraction techniques for in-process surface finish measurements of high precision turned parts

Abstract: To extend the useful range of optical diffraction techniques, algorithms are derived to extract the roughness parameter from the diffraction pattern when the ratio Rq/ lambda <0.03. It is shown that when Rq/ lambda <0.03 the optical Fourier transform of a surface is directly related to the mathematical Fourier transform of the surface profile. These algorithms, closely related to Bessel functions, will extend the range to Rq/ lambda =0.13. They are applied to experimental data and produce a good correlation with values obtained by conventional stylus techniques.

Titius-bode and the helicity connection: a quantized field theory of protostar formation

Abstract: The Titius-Bode relation is discussed in terms of the basic physics of the formation of the solar system. It is demonstrated that the Bode numbers are the eigenvalues of the Euler-Lagrange equation resulting from the variation of the free energy of the generic plasma that formed the sun and planets. It is postulated that a major disturbance in the plane of the galaxy resulted in the formation of a plasma structure that proceeded to lose energy to it surroundings until it reached its lowest possible free-energy state. The free energy of the plasmoid is varied subject to appropriate side conditions. The resulting morphology is described by first-and second-order Bessel functions. There is close agreement between the Bode numbers and the zeros of the Bessel functions of the first kind of order one. The result is a minimum action field theory of planetary formation that does not invoke the concept of "action at a distance" forces.

Complex temperature plane zeros in the mean-field approximation

Abstract: The authors derive asymptotic expressions for the complex temperature plane zeros of the infinite-range Ising model in the scaling regime. The results also apply to high-dimensional, short-range Ising systems. For the nth zero in a system of N spins, the leading asymptotic result is t/sub n/ varies as (n/N)/sup 1/2/(-1+/-i).

Point charge passing a resonator with beam tubes

Abstract: Charged particle beams interact with the surroundings because of the co-travelling electromagnetic fields. This interaction limits the beam. One of the most interesting cases is a point charge travelling with arbitrary velocity along the axis of a cylindrical resonator with two semi-infinite tubes. Without the cylindrical wall of the resonator, the structure consists of two tubes connected by an infinite radial line. In the tube region the fields are given by the source fields plus a continuous spectrum of waveguide modes. In the resonator region a discrete set of modes is used. Matching the fields at the common interface yields a system of linear equations for the expansion coefficients. The occurring integrals contain fractions of the Bessel function and its derivative, which were expanded into algebraic series of Bessel function zeros, and are solved by means of the residuum calculus. Numerical results are given for the coupling impedance between charge and structure and for the energy loss of a Gaussian charge distribution.

Ordering relations between the zeros of miscellaneous bessel functions

Abstract: A method applied in previous work [l,10,11] to the study of the zeros of the ordinary Bessel function Je (z) is here extended and also applied to the zeros of the function , is the derivative of Jν (z). It is proved that in the case where ν is real and ν>−1, the zeros of F ν(z) are the same with the zeros of the function , where T (x), in the case of real positive zeros, meromorphic with poles the positive zeros of Jν(z). Moreover the function T(x) is real for x real and increases as x increases in each of the intervals . This result unifies generalizes and improves, many known results for the zeros of the interesting function .

The analytic structure of Oseen flow past a sphere as a function of Reynolds number

Abstract: The feasibility of Van Dyke’s [28] proposal that one go from zero to infinite Reynolds number R by computer extension of a Stokes series in powers of R depends on the analytic structure of the flow as a function of R. This structure is investigated in detail for Oseen flow past a sphere. We find that there is a doubly infinite array of simple pole singularities in the left half complex R-plane, each of which lies close to one of the n zeros of the modified Bessel functions $K_{n + (1/2)} (R)$, for some positive integer n. Consequently, $R = \infty $ is an accumulation point of poles and hence a nonisolated singularity.We have extended the Stokes series for the drag coefficient $C_D (R)$ to 66 terms in double precision, and have sought the asymptotic behavior of $C_D (R)$ as $R\to \infty$. Neither Pade approximants nor an Euler transformation formed from our long series give good convergence. We suggest that the asymptotic behavior of $C_D (R)$ is primarily as an expansion in powers of $R^{ - 2/3} $ though...

Dynamical problems of thermoelastic solids

Abstract: The dynamical problem for thermal stresses in an infinite isotropic elastic cylinder of radius a with its axis along the z-axis, subject to fixed boundary conditions is studied. The Fourier heat conduction equation has been solved applying the Fourier transform and the theory of complex variable. The thermoelastic equation of motion has been separated into two wave equations which can be solved separately. The temperature, the displacement and the stress components have been obtained in analytical form as series involving Bessel function of first kind and of order zero.

A method to extend the spectral iteration technique

Abstract: A method has been developed to extend the spectral iteration technique to encompass a more general class of subdomain type basis functions. While retaining the advantages of the spectral iteration technique, this method enables the usage of basis functions other than piecewise constant basis functions in expressing the unknown current distribution. Advantages of this technique with the choice of piecewise sinusoidal basis functions as compared to piecewise constant basis functions have been demonstrated for both antenna and scattering problems. This choice of basis functions is found to result in a further reduction in the number of unknowns required to represent the current distribution. Numerical results of the input impedance for a cylindrical antenna and the scattering cross section of a thin fiat plate are presented and discussed.

Description and Philosophy of Spectral Methods

Abstract: We describe the use of spectral methods in computational fluid dynamics. Spectral methods are generally more accurate and often faster than finite-differences. For example, the ∇2 operator in 2 or 3 dimensions is easier to invert with spectral techniques because the spatial dependence of the operator separates in a more natural way. We warn against the use of some of the more common spectral expansions. Bessel series expansions of functions in cylindrical geometries converge poorly. However, other series expansions of the same functions converge quickly. We show how to choose basis functions that give fast convergence and outline the differences between Galerkin, tau, modal, collocation, and pseudo-spectral methods.

Numerical results on instabilities of ``top hat'' jets

Abstract: The Kelvin–Helhmoltz stability of the cylindrical interface between two compressible fluids is studied when the arguments of the Bessel and Hankel functions are large. The real and imaginary parts of the wave frequency, c, are derived through numerical computations.

Sturmian eigenvalue equations with a Bessel function basis

Abstract: A non‐self‐adjoint Sturmian eigenvalue equation of the form Av=f, encountered in quantum scattering theory, is solved as a complex general matrix eigenvalue problem. The matrix form is obtained on expansion of the solution in a discrete set of spherical Sturmian–Bessel functions of complex argument. This set of basis functions gives better convergence behavior for both the eigenvalues and eigenfunctions when compared to the results of a Chebyshev polynomial method reported previously.