Showing papers on "Bessel function published in 1988"

Comparison of Bessel and Gaussian beams

Abstract: A comparison of beam divergence and power-transport efficiency is made between Gaussian and Bessel beams when both beams have the same initial total power and the same initial full width at half-maximum.

The Morse oscillator in position space, momentum space, and phase space

Abstract: We present a unified description of the position‐space wave functions, the momentum‐space wave functions, and the phase‐space Wigner functions for the bound states of a Morse oscillator. By comparing with the functions for the harmonic oscillator the effects of anharmonicity are visualized. Analytical expressions for the wave functions and the phase space functions are given, and it is demonstrated how a numerical problem arising from the summation of an alternating series in evaluating Laguerre functions can be circumvented. The method is applicable also for other problems where Laguerre functions are to be calculated. The wave and phase space functions are displayed in a series of curves and contour diagrams. An Appendix discusses the calculation of the modified Bessel functions of real, positive argument and complex order, which is required for calculating the phase space functions for the Morse oscillator.

On the Scattering of Sound by Two Semi-Infinite Parallel Staggered Plates. I. Explicit Matrix Wiener-Hopf Factorization

Abstract: This paper discusses the two-dimensional scattering of sound waves by two semi-infinite rigid parallel plates. The plates are staggered, so that a line in the plane of the motion passing through both edges is not in general perpendicular to the plane of either plate. The problem is formulated as a matrix Wiener-Hopf functional equation, which exhibits the difficulty of a kernel containing exponentially growing elements. We show how this difficulty may be overcome by constructing an explicit product decomposition of the matrix kernel with both factors having algebraic behaviour at infinity. This factorization is written in terms of a single entire auxiliary function that has a simple infinite series representation. The Wiener-Hopf equation is solved for arbitrary incident wave fields and we derive an asymptotic expression for the field scattered to infinity; the latter includes the possibility of propagating modes in the region between the plates. In part II of this work we will evaluate our solution numerically and obtain some analytical estimates in a number of physically interesting limits.

A unified theory of polynomial expansions and their applications involving Clebsch-Gordan type linearization relations and Neumann series

Abstract: Motivated by their potential for applications in several diverse fields of physical, astrophysical, and engineering sciences, this paper aims at presenting a unified study of various classes of polynomial expansions and multiplication theorems associated with the general multivariable hypergeometric function (studied recently by A. W. Niukkanen and H. M. Srivastava), which provides an interesting and useful unifiation of numerous families of special functions in one and more variables, encoutered naturally (and rather frequently) in many physical, quantum chemical, and quantum mechanical situations. Several interesting applications of these general polynomial expansions are considered, not only in the derivations of various Clebsch-Gordan type linearization relations involving products of several Jacobi or Laguerre polynomials, but also to associated Neumann expansions in series of the Bessel functionsJ v (z) andI v (z) (and of their suitable products).

Exact Cylindrical Soliton Solutions of the Sine-Gordon Equation, the Sinh-Gordon Equation and the Periodic Toda Equation

Abstract: We consider the sine-Gordon equation, the sinh-Gordon equation and the periodic Toda equation which are written respectively as Δ u +sin u =0, Δ u +sinh u =0 and Δ u n -exp (- u n + u n -1 )+exp (- u n +1 + u n )=0, ( u n = u n + N ' ), where Δ ≡∂ 2 /∂ x 2 +∂ 2 /∂ y 2 . The exact analytic solutions of the above equations corresponding to the cylindrical solitons have been obtained. The solutions are expressed by the series expansions of the Bessel functions.

Minkowski Bessel modes.

Abstract: The global Minkowski Bessel (MB) modes, whose explicit form allows the identification and description of the condensed vacuum state resulting from the operation of a pair of accelerated refrigerators, are introduced. They span the representation space of a unitary representation of the Poincar\'e group on two-dimensional Lorentz space-time. Their three essential properties are (1) they are unitarily related to the familiar Minkowski plane waves, (2) they form a unitary representation of the translation group on two-dimensional Minkowski space-time, and (3) they are eigenfunctions of Lorentz boosts around a given reference event. In addition the global Minkowski Mellin modes are introduced. They are the singular limit of the MB modes. This limit corresponds to the zero-transverse-momentum solutions to the zero-rest-mass wave equation. Also introduced are the four Rindler coordinate representatives of each global mode. Their normalization and density of states are exhibited in a (semi-infinite) accelerated frame with a finite bottom. In addition we exhibit the asymptotic limit as this bottom approaches the event horizon and thereby show how a mode sum approaches a mode integral as the frame becomes bottomless. This is the infinite Regge-Wheeler volume limit.

Complex-plane methods for evaluating highly oscillatory integrals in nuclear physics. II

Abstract: For pt.I see ibid., vol.14, p.973(1988). Products of Green functions for pion-nucleon interactions can most naturally be evaluated by transformation to a momentum-space representation. Such transformations often involve an integral whose integrand is highly oscillatory. By deforming the contour of integration in the complex plane, such an integral can be made to converge rapidly. All oscillatory terms are converted into pure damping terms which decrease exponentially. Examples are given for simple integrals, for the integral of three spherical Bessel functions, and for the Fourier and spherical Bessel function transforms of the Woods-Saxon potential (Fermi-Dirac distribution function). They authors also show that oscillatory principal-value integrals can usually be accurately evaluated using this method. Convergence with respect to the number of Gaussian points is studied. It is shown that the method normally converges much better than ordinary integration along the real axis, especially as the frequency of the oscillations increases relative to the damping. A comparison is made with the method of steepest descent.

An addition theorem and some product formulas for q-Bessel functions

Abstract: The most familiar series representation of the Bessel function is 1.1 Jackson [12] gave the following q-analogues: 1.2 1.3 where 0 < q < 1, the q-shifted factorials are defined by 1.4 and the q-gamma function is given by 1.5

The distribution of zeros of the polynomial eigenfunctions of ordinary differential operators of arbitrary order

Abstract: The distribution of zeros of the polynomial eigenfunctions of ordinary differential operators of arbitrary order with polynomial coefficients is calculated via its moments directly in terms of the parameters which characterize the operators. Some results of K.M. Case and the authors are extended. In particular, the restriction for the degree of the polynomial coefficient of the ith-derivative to be not greater than i is relaxed. Applications to the Heine polynomials, the Generalized Hermite polynomials and the Bessel type orthogonal polynomials (which no trivial properties of zeros were known of) are shown.

Solitons in Higher Dimensions

Abstract: We first review the recent developments of studies on solitons in higher dimensions. Next we extract two characteristics about solitons in higher dimensions: (i) these solitons are written by the special functions such as the Bessel function, (ii) existence of the transformation which connects 1+1d soliton equation and its cylindrical or spherical equation. We check that to what extent these two characteristics hold in the recently found examples of the various higher dimensional solitons

The soliton number of optical soliton bound states for two special families of input pulses

Abstract: To gain a better understanding of the generation of optical solitons the author investigates the linear eigenvalue problem associated with the non-linear Schrodinger equation. Two families of initial envelope functions are discussed. It is found that, for a purely imaginary initial envelope function of width a and height b, and its Galilei transforms, the soliton number of soliton bound states is the integer smaller than 1/2+ab/ pi . For the initial envelope function i beta exp(- alpha mod x mod ) and its Galilei transforms, the soliton number of soliton bound states is equal to the number of intersections of the Bessel functions J-1/2 and +or-J1/2 below beta / alpha , which is the integer smaller than 1/2+2 beta / alpha pi .

Exact solutions for buckling of some divergence-type nonconservative systems in terms of Bessel and Lommel functions

Abstract: Closed-form solutions for a divergence-type nonconservative system, that of the column under follower distributed forces, is given for three types of boundary conditions. The solution generalizes the previous classical studies by Pfluger, who found the solution for the column simply supported at its ends in terms of Bessel functions, as well as by Leipholz and Madan, who formulated the series solutions for the column clamped at one end, and simply supported or clamped at the other. In the present work the solution is formulated in terms of Bessel and Lommel functions, yielding exact characteristics equations, with attendant buckling loads found within any desired numerical accuracy.

A theorem concerning differentiability of eigenvectors and eigenvales with some applications

Abstract: Let H(ν) be a family of self-adjoint operators, bounded or unbounded, in a separable Hilbert space H. It is assumed that H (ν) as a function of the real parameter ν is differentiable, with respect to the operator norm, in some open interval l 0 and the derivative H′(ν) is uniformly bounded on compact subsets of l 0. Under the additional assumption that an isolated eigenvalue λ(ν) exists and is simple for every v in L it is proved that the corresponding eigenvector x(ν) is strongly differentiable in l 0. The differential equation λ′ (ν) = (H′ (ν)x(ν), x(v)), ‖ x(ν) ‖ =1 that follows immediately generalizes a diferential equation found for the zeros of Bessel functions in ref [2] and can be applied to many other eigenvalue problems. Techniques, known from the theory of Bessel functions, are used to give bounds and qualitative properties of the eigenvalues of several eigenvalue problems for which both the existence and the simplicity of eigenvalues is well-known

An orthonormal Laguerre expansion yielding Rice's envelope density function for two sine waves in noise

Abstract: Through an orthonormal Laguerre expansion, expressions are derived for a lesser known Rician probability distribution-the probability density function (PDF) of the envelope of two fixed-amplitude randomly phased sine waves in narrowband Gaussian noise-and for the integral of the density, the cumulative distribution function (CDF). The principal formula derived has been checked analytically, numerically, and (approximately) graphically. Analytically, the moment-generating function for the PDF of the square of the envelope has been found to be a three-term product of elementary functions times an I/sub 0/ Bessel function (and thus to be in closed form); in confirmation, the same result has been secured via another, more direct route. >

Electromagnetic scattering by rough conduction circular cylinders. I. Angular corrugation

Abstract: The coherent and incoherent responses of perfectly conducting, angularly corrugated surfaces (which are infinite circular cylinders on the average) are calculated using a second-order perturbation method. The general approach used is to apply the extinction theorem to write an integral equation for the surface current density, and then to use perturbation theory to solve it. These results are recast by using the only nontrivial Pade approximant for this approximation. This recasting yield a considerable qualitative improvement which, at least in one special case, is shown to be quantitative as well. >

Zeros of combinations of bessel functions and their derivatives

Abstract: We use the Mittag-Leffler partial fractions expansion of and certain monotonicity properties of this and related functions to study the zeros of where α, β, τ and δ are real. Thus we simplify and, in roost cases, strengthen results of E. K Ifantis and P. D. Siafarikas, Applicable Anal. 23 (1986), 85-110. We derive further properties of these zeros in certain special cases

Accurate recursive generation of spherical Bessel and Neumann functions for a large range of indices

Abstract: Partial wave expansion is encountered in a wide range of problems in theoretical physics. Typically, spherical Bessel functions are encountered in scattering of nuclei, atoms, and molecules. A routine is described for computing spherical Bessel and Neumann functions jl( x) and nl( x), respectively, by backward and forward recurrence relations. The method is fast, very accurate, not code intensive, and easily adopted for a personal computer. These features make the code well suited for the physics student research environment.

Invariant expansion. IV. The exponentials of tensorial expressions

Abstract: The expansion of the exponential of a tensorial expression such as the interaction or pair correlation function between two nonspherical molecules 1, 2 is of the form ∑mnl λmnlΦmnl(12), where Φmnl(12) are invariant tensorial expressions that depend only on the orientation of 1 and 2. The generating function e−∑mnl λmnlΦmnl =∑pqt ipqt(λ) Φpqt defines a generalized Bessel function (GBF). We discuss integral representations and recurrence relations for the GBF. The first GBFs for dipolar and linear quadrupolar exponents, which are of interest in the theory of ionic solutions are computed explicitly.

Distributions of quadratic forms

Abstract: : For independent chi-square variables x squared sub m and x squared sub n with m and n degrees of freedom, respectively, we consider the quadratic form where the positive ci are distinct. This paper gives exact finite expressions for the distribution of Q in terms of available functions such as the distribution function of chi-square random variables, modified Bessel Functions, Dawson's integral. These formulas are useful for checking the accuracy of approximations and tables of the distribution of Q and provide a simple alternative in their absence. For large m and n, reasonable approximations to the distribution of Q are available. For the general quadratic form Williams (1984) compares algorithms for truncations of infinite series expansions of the distribution.

The imaginary zeros of a mixed Bessel function

Abstract: In the present work we study the existence and monotonicity properties of the imaginary zeros of the mixed Bessel functionMv(z)=(βz2+α)Jv(z)+zJ′v(z). Such a function includes as particular cases the functionsJ′v(z)(α=β=0), J″v(z)(α=−v2,β=1)x andHv(z)=αJv(z)+zJ′v(z), whereJv(z) is the Bessel function of the first kind and of orderv>−1 andJ′v(z), J″v(z) are the first two derivatives ofJv(z). Upper and lower bounds found for the imaginary zeros of the functionsJ′v(z), J″v(z) andHv(z) improve previously known bounds.

A note on certain integrals of Bessel functions

Abstract: In a book on Bessel functions by Watson (1980), there appear to be minor errors in one particular integral of Bessel functions. This integral is given a series expansion which may be useful in certain physical problems. The asymptotic nature of this series is very briefly discussed.

Dynamics and symmetries of a field partitioned by an accelerated frame.

Abstract: The canonical evolution and symmetry generators are exhibited for a Klein-Gordon (KG) system which has been partitioned by an accelerated coordinate frame into a pair of subsystems. This partitioning of the KG system is conveyed to the canonical generators by the eigenfunction property of the Minkowski Bessel (MB) modes. In terms of the MB degrees of freedom, which are unitarily related to those of the Minkowski plane waves, a near complete diagonalization of these generators can be realized.

Eigenfunctions of the magnetospheric radial-diffusion operator

Abstract: The applicability of eigenfunction methods to the theory of magnetospheric radial diffusion is investigated analytically, with a focus on the operator (Lambda) for radiation-belt and ring-current particles. It is shown that the eigenfunctions of Lambda can be expressed in terms of a linear combination of ordinary Bessel functions of the first and second kinds, and that the solution for the drift-averaged phase-space density can be expressed as the sum of a quasi-static solution and a time-dependent superposition of the eigenfunctions of Lambda. Results for illustrative and practical sample problems are presented in extensive tables and graphs and discussed in detail.