Showing papers on "Bessel function published in 1993"
TL;DR: It is demonstrated that under certain circumstances, Bessel beams can be viewed as light beams with tunable wavelength.
Abstract: We investigated the frequency-doubling properties of light beams whose transverse profile is given by the zero-order Bessel function ${\mathit{J}}_{0}$(r), Bessel beams. Phase-matched second-harmonic generation in a KDP crystal was observed at angles usually not suited for phase matching. It is thereby demonstrated that under certain circumstances, Bessel beams can be viewed as light beams with tunable wavelength. A variety of applications in the field of nonlinear optics is expected.
161 citations
108 citations
TL;DR: In this paper, a method of solving diffraction integrals is presented, based on Gegenbauer polynomials of the first kind, which can be used to obtain a uniformly convergent series of orthogonal functions.
Abstract: Vector diffraction theory for optical systems has been of interest for a long time. Ignatovsky and Wolf have formulated these problems in terms of diffraction integrals and Wolf has presented very interesting results. Usually, the quadrature of diffraction integrals is numerically intensive, therefore these problems have remained of interest and many authors have worked on the Ignatovsky-Wolf formulation or some variation thereof. This paper presents yet another method of solving diffraction integrals. Since a certain part of the kernel of these integrals is Riemann integrable in the interval [0, π], the Weierstrass theorem says that it can be approximated by a uniformly convergent series of orthogonal functions. Thus it is possible to expand these functions into a series of Gegenbauer polynomials of the first kind. Once these expansions are substituted in the diffraction integrals, the resulting integrals are readily evaluated, over the surface of unit sphere, in terms of the spherical Bessel fu...
78 citations
TL;DR: In this paper, the authors derive the expansion of a cylindrical wave into plane waves and give some examples of applications, such as the application of diffraction theory to plane wave propagation, where the product of a Hankel function of integer order times a sinusoidal angular factor is considered.
72 citations
TL;DR: In this paper, the authors consider bounds on the Neumann eigenvalues of the Laplacian on domains in $I\mathbb{R}^n $ in the light of their recent results on Dirichlet eigen values, in particular, their proof of t...
Abstract: The authors consider bounds on the Neumann eigenvalues of the Laplacian on domains in $I\mathbb{R}^n $ in the light of their recent results on Dirichlet eigenvalues, in particular, their proof of t...
70 citations
TL;DR: In this paper, the authors formulate several criteria on square-integrable functions in terms of certain smoothness and rate of decay that guarantee that these functions generate Bessel sequences, and show that one can obtain affine frames by arbitrarily oversampling any of the well-known wavelets.
67 citations
TL;DR: In this article, the sum zeta H(s)= Sigma jEj-s over the eigenvalues Ej of the Schrodinger equation in a spherical domain with Dirichlet walls, threaded by a line of magnetic flux.
Abstract: The authors study the sum zeta H(s)= Sigma jEj-s over the eigenvalues Ej of the Schrodinger equation in a spherical domain with Dirichlet walls, threaded by a line of magnetic flux. Rather than using Green function techniques, they tackle the mathematically non-trivial problem of finding exact sum rules for the zeros of Bessel functions Jv, which are extremely helpful when seeking numerical approximations to ground state energies. These results are particularly valuable if v is neither an integer nor half an odd integer.
61 citations
TL;DR: It is shown that the generating function for skew Ferrers diagrams according to their width and area is the quotient of new basic Bessel functions.
59 citations
TL;DR: The optical-phonon modes of a circular wire are studied by means of the generalized Born-Huang equation and its eigensolutions are found to be confined bulk and interface modes for almost all wave vectors parallel to the wire axis.
Abstract: The optical-phonon modes of a circular wire are studied by means of the generalized Born-Huang equation. Its eigensolutions are found to be confined bulk and interface modes for almost all wave vectors parallel to the wire axis. At particular wave vectors hybridization of the two mode branches occurs. The eigenfrequencies and displacement eigenfields are derived for confined and interface modes. Three branches of confined modes exist, two TO and one LO. For modes confined to the wire the radial wave numbers are quantized, and determined by the zeros of Bessel functions of the first kind and of their derivatives. The twofold TO degeneracy is removed from reasons of symmetry. The displacement fields may be understood in terms of superpositions of linearly polarized plane waves with wave vectors forming a cone around the wire axis. For modes confined to the surroundings of the wire radial wave vectors remain continuous. Two types of interface modes are found. The dispersion of their frequencies with wave-vector components parallel to the wire is given in terms of modified Bessel functions. The same dispersion relations are obtained from electrostatic matching conditions. The Fr\"ohlich interaction is studied for confined LO and interface modes. The corresponding interaction Hamiltonians are given in explicitly quantized form.
54 citations
TL;DR: In this paper, particular solutions of the discrete Painlev\'e III (d-P$\rm_{III}$) equation of rational and special function (Bessel) type are presented.
Abstract: We present particular solutions of the discrete Painlev\'e III (d-P$\rm_{III}$) equation of rational and special function (Bessel) type. These solutions allow us to establish a close parallel between this discrete equation and its continuous counterpart. Moreover, we propose an alternate form for d-P$\rm_{III}$ and confirm its integrability by explicitly deriving its Lax pair.
53 citations
TL;DR: A new more efficient algorithm for the evaluation of the fast rotation function coefficients is derived and its implementation in standard programs is straightforward.
Abstract: A new more efficient algorithm for the evaluation of the fast rotation function coefficients is derived. Its implementation in standard programs is straightforward.
TL;DR: In this article, the authors prove an equiconvergence theorem between Fourier-Bessel expansions of functions in certain weighted Lebesgue spaces and the classical cosine Fourier expansions of suitable related functions.
Abstract: We shall prove an equiconvergence theorem between Fourier-Bessel expansions of functions in certain weighted Lebesgue spaces and the classical cosine Fourier expansions of suitable related functions. These weighted Lebesgue spaces arise naturally in the harmonic analysis of radial functions on euclidean spaces and we shall use the equiconvergence result to deduce shag results for the pointwise almost everywhere convergence of Fourier integrals of radial functions in the Lorentz spaces L p,q (R n ). Also we shall briefly apply the above approach to the study of the harmonic analysis of radial functions on noneuclidean hyperbolic spaces
TL;DR: In this paper, a general procedure is outlined by which forms of density profile can be generated permitting analytical solutions for plane wave propagation in a stratified medium, and the same procedure is applicable to calculate forms of sound speed profile permitting exact solutions in the presence of a given density stratification.
Abstract: The Helmholtz equation for a variable density medium is considered and solutions are sought for the case of horizontal stratification, both sound speed and density varying continuously with depth. It is shown that the nonlinear density dependence in the Helmholtz equation can be reduced to a linear form by means of appropriate transformations. A general procedure is then outlined by which forms of density profile can be generated permitting analytical solutions for plane wave propagation in a stratified medium. The same procedure is applicable to calculate forms of sound‐speed profile permitting exact solutions in the presence of a given density stratification. Some specific examples are considered, for which representative density profiles are calculated in terms of well‐known special functions such as Bessel and Airy functions. Comparison with measured data shows that the analytical density and sound‐speed profiles derived are capable of giving good agreement with real density and speed profiles in marine sediments.
TL;DR: In this paper, an exact quadrature formula for entire functions of exponential type is obtained, where the nodes of the formula are zeros of the Bessel function of the first kind of order a.
Abstract: An exact quadrature formula for entire functions of exponential type is obtained. The nodes of the formula are zeros of the Bessel function of the first kind of order a . It generalizes and refines a known quadrature formula related to the sampling theorem. The uniqueness of the nodes is studied.
TL;DR: In this article, a class of second-order linear differential equations with a large parameter u is considered, and it is shown that Liouville-Green type expansions for solutions can be expressed using factorial series in the parameter, and such expansions converge for Re (u) > 0, uniformly for the independent variable lying in a certain subdomain of the domain of asymptotic validity.
Abstract: A class of second-order linear differential equations with a large parameter u is considered. It is shown that Liouville-Green type expansions for solutions can be expressed using factorial series in the parameter, and that such expansions converge for Re (u) > 0, uniformly for the independent variable lying in a certain subdomain of the domain of asymptotic validity. The theory is then applied to obtain convergent expansions for modified Bessel functions of large order.
TL;DR: In this paper, the authors considered the time domain scattering problem of a stratified dissipative half-space excited by a point source and treated both the direct and inverse scattering problem.
Abstract: In this paper the time domain scattering problem of a stratified dissipative half-space excited by a point source is considered. Both the direct and inverse scattering problem are treated. The dissipative wave equation is reduced to an equation in one spatial dimension by spatial Fourier series and a Hankel transform. The transformed wave equation is then solved by a combination of a time domain wave-splitting technique and a Green function technique. The Green functions represent the intemal fields inside the stratified medium. The splitting is expressed in terms of a Neumann operator K which has an explicit expression as a convolution integral with a Bessel function kernel
TL;DR: In this paper, a new series formula of Sharma and its connection with traditional Bessel function series is established, and an alternative new series is developed which isolates the steady-state component for all values of traffic intensity and which turns out to be computationally superior.
Abstract: Past work relating to the computation of time-dependent state probabilities in M/M/1 queueing systems is reviewed, with emphasis on methods that avoid Bessel functions. A new series formula of Sharma [13] is discussed and its connection with traditional Bessel function series is established. An alternative new series is developed which isolates the steady-state component for all values of traffic intensity and which turns out to be computationally superior. A brief comparison of our formula, Sharma's formula, and a classical Bessel function formula is given for the computation time of the probability that an initially empty system is empty at time t later.
TL;DR: In this article, the standard Bessel function formula for the potential of a thin axisymmetric disc is extended to include arbitrary vertical structure, yielding a formula which reduces the potential to a single quadrature in the important cases of a disc with constant scale-height and either exponential or Gaussian vertical density profiles.
Abstract: The standard Bessel function formula for the potential of a thin axisymmetric disc is extended to include arbitrary vertical structure, yielding a formula which reduces the potential to a single quadrature in the important cases of a disc with constant scale-height and either exponential or Gaussian vertical density profiles. The general solution of Poisson's equation for an axisymmetric body is also given as a double integral over a Legendre function. A new formulation for the potential of thin axisymmetric discs is also given. The potential is expressed as a double integral over elementary functions in the most general case, and can usually be reduced to a single quadrature
TL;DR: In this paper, a path integral representation of the conformations of polymers with different degrees of bond correlation is presented, and an expression for the distribution function of the chains' end-to-end distance is derived by several independent methods, including direct evaluation of the discrete limit of the path integral, decomposition into normal modes, and solution of a partial differential equation.
Abstract: The statistical properties of fractional Brownian walks are used to construct a path integral representation of the conformations of polymers with different degrees of bond correlation. We specifically derive an expression for the distribution function of the chains’ end‐to‐end distance, and evaluate it by several independent methods, including direct evaluation of the discrete limit of the path integral, decomposition into normal modes, and solution of a partial differential equation. The distribution function is found to be Gaussian in the spatial coordinates of the monomer positions, as in the random walk description of the chain, but the contour variables, which specify the location of the monomer along the chain backbone, now depend on an index h, the degree of correlation of the fractional Brownian walk. The special case of h=1/2 corresponds to the random walk. In constructing the normal mode picture of the chain, we conjecture the existence of a theorem regarding the zeros of the Bessel function.
TL;DR: A two-element refracting system is designed to transform a Gaussian laser beam into a diffraction-free Bessel beam because the resulting input and output surfaces are almost spherical.
Abstract: A two-element refracting system is designed to transform a Gaussian laser beam into a diffraction-free Bessel beam. The resulting input and output surfaces are almost spherical, which makes for easy implementation of the system.
TL;DR: In this article, the authors describe an algorithm for the evaluation of Bessel functions J ν ( x ), Y ν( x ) and H ( j ) ν, x 1/3 for arbitrary positive orders and arguments at a constant CPU time.
TL;DR: In this article, a Symmetrie cone in the real w-dimensional inner product space (/, ) is considered, and it is shown that the connected group A of automorphisms of Ω acts transitively on Ω.
Abstract: Let Ω be a Symmetrie cone in the real w-dimensional inner product space (/, by which one means that Ω is non-empty, open in /, convex, and self-dual, and that the connected group A of automorphisms of Ω acts transitively on Ω. Then / carries the structure of a formally real Jordan algebra, relative to which Ω is the interior of the set of squares in /. Let J be the complexification of /, and H = /+ i Ω the Siegel upper halfplane in J. For convenience assume that Ω is irreducible. Let r denote the rank of Ω, set m = n/r, and let t be the identity in /.
TL;DR: In this article, exact formulas for the longitudinal and transverse dielectric correlation times and complex polarizability tensor, of a single axis rotator with two equivalent sites were found by writing the Laplace transforms of the dipole autocorrelation functions as three term recurrence relations and solving them in terms of continued fractions.
Abstract: It is shown how exact formulas for the longitudinal and transverse dielectric correlation times and complex polarizability tensor, of a single axis rotator with two equivalent sites may be found. This is accomplished by writing the Laplace transforms of the dipole autocorrelation functions as three term recurrence relations and solving them in terms of continued fractions. The solution of these recurrence relations, in the zero frequency limit, yields the correlation times in terms of modified Bessel functions of the first kind. The previous result of Lauritzen and Zwanzig for the longitudinal relaxation time, based on an asymptotic expansion of the Sturm–Liouville equation, is regained in the limit of high potential barriers.
TL;DR: In this paper, the efficiency and accuracy of a few available algorithms for the computation of integer-order Bessel functions are considered, and it is shown that recurrence techniques are superior to the FFT technique, both in accuracy and speed.
Abstract: The efficiency and accuracy of a few available algorithms for the computation of integer-order Bessel functions are considered. First, the computation of integer-order Bessel functions of the first kind, using the fast Fourier transform (FFT) algorithm as opposed to recurrence techniques, is investigated. It is shown that recurrence techniques are superior to the FFT technique, both in accuracy and speed. An algorithm suggested in the literature and used in commercially available software, specifically MATLAB 3.5 and MATHEMATICA 1.2, for computing integer-order Bessel functions of the second kind is revealed to be erroneous by comparing these routines with an algorithm developed by the author. It is shown that catastrophic errors result from using the erroneous algorithm to compute high-order Bessel functions with nonreal arguments. >
TL;DR: A modification of Ambarzumian's method is used to develop the integro-differential equations for the source function, flux, and intensity at the boundary of a two-dimensional, semi-infinite cylindrical medium which scatters linearly.
Abstract: A modification of Ambarzumian's method is used to develop the integro-differential equations for the source function, flux, and intensity at the boundary of a two-dimensional, semi-infinite cylindrical medium which scatters linearly. The incident radiation is collimated, normal to the top surface of the medium, and is dependent only on the radial coordinate. The radial variation is assumed to be a Bessel function or a Gaussian distribution. The Gaussian boundary condition is used to simulate a laser beam. Numerical results are presented in graphical and tabular forms for both boundary conditions. Results for forward and backward scattering phase functions are compared with those for isotropic scattering. A method is presented for extending these results to the problem of a strongly anisotropic phase function which is made up of a spike in the forward direction superimposed on a linear phase function.
TL;DR: In this paper, a common generalization of both the Jackson and the Hahn-Exton q-Bessel function is studied, which yields q-analogues of an integral of Weber and Sonine and of the Hankel transform.
Abstract: A common generalization of both the Jackson and the Hahn-Exton q-Bessel function is studied. For this q-analogue of the Bessel function an analogue of Graf's addition formula is proved. From this result we obtain Hansen-Lommel type orthogonality relations and a q-analogue of Lommel's formula. Considering the results from a different point of view yields q-analogues of an integral of Weber and Sonine and of the Hankel (or Fourier-Bessel) transform.
TL;DR: Two Hansen-Lommel orthogonality relations for Jackson's q-Bessel functions were proved in this paper, where they were proved for all q-bessel functions.
TL;DR: In this paper, the dispersion relation for internal waves trapped in a layer and propagating linearly in a fluid of infinite depth with a rigid surface is estimated. But the main interest is in predicting the structure of internal wave wakes, but the results are applicable to any internal waves.
Abstract: An analysis is constructed in order to estimate the dispersion relation for internal waves trapped in a layer and propagating linearly in a fluid of infinite depth with a rigid surface. The main interest is in predicting the structure of internal wave wakes, but the results are applicable to any internal waves. It is demonstrated that, in general 1/cp = 1/CpO + k/ωmax + ∈(k) where cp is the wave phase speed for a particular mode, CpO is the phase speed at k = 0, ωmax is the maximum possible wave angular frequency and ωmax ≤ Nmax where Nmax is the maximum buoyancy frequency. Also, ∈(0) = 0, ∈(k) = o(k) for k large, and is bounded for finite k. In particular, when ∈(k) can be neglected, the dispersion relation for a lowest mode wave is approximately 1/cp ≈ (∫∞0N2(y)ydy)-½ + k/ωmax. The eigenvalue problem is analysed for a class of buoyancy frequency squared functions N2(x) which is taken to be a class of realvalued functions of a real variable x where O ≤ x ∞ such that N2(x) = O(e-βx) as x → ∞ and 1/β is an arbitrary length scale. It is demonstrated that N2(x) can be represented by a power series in e-βx. The eigenfunction equation is constructed for such a function and it is shown that there are two cases of the equation which have solutions in terms of known functions (Bessel functions and confluent hypergeometric functions). For these two cases it is shown that ∈(k) can be neglected and that, in addition, ωmax = Nmax. More generally, it is demonstrated that when k → ∞ it is possible to approximate the equation uniformly in such a way that it can be compared with the confluent hypergeometric equation. The eigenvalues are then, approximately, zeros of the Whittaker functions. The main result which follows from this approach is that if N2(x) is O(e-βx) as x → ∞ and has a maximum value N2max then a sufficient condition for 1/cp ∼ k/Nmax to hold for large k for the lowest mode is that N2(t)/t is convex for O ≤ t ≤ 1 where t = e-βx.