Showing papers on "Bessel function published in 2003"
TL;DR: In this article, the first hitting times of squared Bessel processes and radial Ornstein-Uhlenbeck processes with negative dimensions or negative starting points are studied. But the authors focus on the first time a Bessel process hits a given barrier.
Abstract: Bessel processes play an important role in financial mathematics because of their strong relation to financial models such as geometric Brownian motion or Cox-Ingersoll-Ross processes. We are interested in the first time Bessel processes and, more generally, radial Ornstein-Uhlenbeck processes hit a given barrier. We give explicit expressions of the Laplace transforms of first hitting times by (squared) radial Ornstein-Uhlenbeck processes, that is, Cox-Ingersoll-Ross processes. As a natural extension we study squared Bessel processes and squared Ornstein-Uhlenbeck processes with negative dimensions or negative starting points and derive their properties.
336 citations
TL;DR: An improved recurrence algorithm to calculate the scattering field of a multilayered sphere is developed that is efficient, numerically stable, and accurate for a large range of size parameters andrefractive indices.
Abstract: An improved recurrence algorithm to calculate the scattering field of a multilayered sphere is developed. The internal and external electromagnetic fields are expressed as a superposition of inward and outward waves. The alternative yet equivalent expansions of fields are proposed by use of the first kind of Bessel function and the first kind of Hankel function instead of the first and the second kinds of Bessel function. The final recursive expressions are similar in form to those of Mie theory for a homogeneous sphere and are proved to be more concise and convenient than earlier forms. The new algorithm avoids the numerical difficulties, which give rise to significant errors encountered in practice by previous methods, especially for large, highly absorbing thin shells. Various calculations and tests show that this algorithm is efficient, numerically stable, and accurate for a large range of size parameters and refractive indices.
201 citations
TL;DR: In this article, a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel, is proven, where one of the factors in the product formula is replaced by its mean over a sphere.
Abstract: It is an open conjecture that generalized Bessel functions associated with root systems have a positive product formula for nonnegative multiplicity parameters of the associated Dunkl operators. In this paper, a partial result towards this conjecture is proven, namely a positive radial product formula for the non-symmetric counterpart of the generalized Bessel function, the Dunkl kernel. Radial here means that one of the factors in the product formula is replaced by its mean over a sphere. The key to this product formula is a positivity result for the Dunkl-type spherical mean operator. It can also be interpreted in the sense that the Dunkl-type generalized translation of radial functions is positivity-preserving. As an application, we construct Dunkl-type homogeneous Markov processes associated with radial probability distributions.
181 citations
TL;DR: The infinitely divisible distributions of random variables with Laplace transforms are characterized for various in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their Levy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transform.
Abstract: The infinitely divisible distributions on of random variables with Laplace transforms respectively are characterized for various in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their Levy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a one-dimensional Brownian motion. The distributions of are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and the Dirichlet -function associated with the quadratic character modulo 4. Related families of infinitely divisible laws, including the gamma, logistic and generalized hyperbolic secant distributions, are derived from by operations such as Brownian subordination, exponential tilting, and weak limits, and characterized in various ways.
111 citations
TL;DR: This work demonstrates for the first time to their knowledge a branch from the family of higher-order Bessel beams that has fractional azimuthal variation at its beam axis and has the ability to perform dynamic optical manipulation with dynamic control of a spatial light modulator.
Abstract: Higher-order Bessel beams have been demonstrated to have the ability to trap and rotate low- and high-index particles simultaneously [Phys. Rev. A 66, 063402 (2002)]. The rotation and trapping is caused by the presence of orbital angular momentum arising from its azimuthal phase variation (that changes at integer multiples of π) and the concentric rings of the Bessel mode. We demonstrate for the first time to our knowledge a branch from the family of higher-order Bessel beams that has fractional azimuthal variation at its beam axis. This new family of laser beams has the ability to perform dynamic optical manipulation with dynamic control of a spatial light modulator. Furthermore, we take the opportunity to explore the propagation characteristics of higher-order Bessel beams for which the azimuthal phase changes at noninteger multiples of 2π .
94 citations
TL;DR: It is shown that stationary, spatiotemporal localized Bessel X-wave transmission is also possible in the anomalous dispersion regime and all previously reported cases of suppression of normal material group velocity dispersion by using angular dispersion in tilted pulses, pulsed Bessel beams, and BesselX waves are compared and presented in a unified way.
Abstract: We study Bessel X waves with cone dispersion propagating in free space and dispersive media. Their propagation features find simple explanation when viewed as cylindrically symmetric versions of the so-called tilted pulses. All previously reported cases of suppression of normal material group velocity dispersion by using angular dispersion in tilted pulses, pulsed Bessel beams, and Bessel X waves are compared and presented in a unified way. We show that stationary, spatiotemporal localized Bessel X-wave transmission is also possible in the anomalous dispersion regime.
80 citations
TL;DR: In this article, the effect of the Kerr nonlinearity on linear non-diffractive Bessel beams is investigated analytically and numerically using the nonlinear Schrodinger equation.
76 citations
TL;DR: In this paper, the authors presented the solutions for displacements and stresses subjected to a vertical point load in a continuously inhomogeneous transversely isotropic half-space with Young's and shear moduli varying exponentially with depth.
71 citations
TL;DR: In this article, the generalized shift operator, generated by Bessel differ-ential operator B, is considered, by means of which maximal functions (B-maximal functions) and fractional integrals are investigated.
Abstract: In this paper we consider the generalized shift operator, generated by Bessel differ- ential operator B, by means of which maximal functions (B-maximal functions) and frac- tional integrals (B-fractional integrals) are investigated. The Lp(B)-boundedness result for the B-maximal function and (Lp(B),Lq(B))-boundedness result for the B-fractional integral are obtained.
67 citations
TL;DR: In this article, a general analytical formulae for the generation and propagation of non-fracting Bessel beams is proposed, where the width of the slit is compared with the ideal case represented by a Dirac d transmittance function.
54 citations
TL;DR: In this article, the authors derived analytic bounds for the zeros of the third Jackson -Bessel function J v ( 3 ) ( z ; q ) for the case where q is the number of zeros.
Abstract: We derive analytic bounds for the zeros of the third Jackson
q -Bessel function J v ( 3 ) ( z ; q ) .
TL;DR: A detailed study of the axicon-based Bessel-Gauss resonator with concave output coupler is presented, using a technique to convert the Huygens-Fresnel integral self-consistency equation into a matrix equation and then finding the eigenvalues and the Eigenfields of the resonator at one time.
Abstract: A detailed study of the axicon-based Bessel-Gauss resonator with concave output coupler is presented. We employ a technique to convert the Huygens-Fresnel integral self-consistency equation into a matrix equation and then find the eigenvalues and the eigenfields of the resonator at one time. A paraxial ray analysis is performed to find the self-consistency condition to have stable periodic ray trajectories after one or two round trips. The fast-Fourier-transform-based Fox and Li algorithm is applied to describe the three-dimensional intracavity field distribution. Special attention was directed to the dependence of the output transverse profiles, the losses, and the modal-frequency changes on the curvature of the output coupler and the cavity length. The propagation of the output beam is discussed.
TL;DR: In this paper, the authors studied the canonical solution of a family of classical n-vector spin models on a generic d-dimensional lattice; the couplings between two spins decay as the inverse of their distance raised to the power α, with α < d.
Abstract: We study the canonical solution of a family of classical n-vector spin models on a generic d-dimensional lattice; the couplings between two spins decay as the inverse of their distance raised to the power α, with α < d. The control of the thermodynamic limit requires the introduction of a rescaling factor in the potential energy, which makes the model extensive but not additive. A detailed analysis of the asymptotic spectral properties of the matrix of couplings was necessary to justify the saddle point method applied to the integration of functions depending on a diverging number of variables. The properties of a class of functions related to the modified Bessel functions had to be investigated. For given n, and for any α, d and lattice geometry, the solution is equivalent to that of the α = 0 model, where the dimensionality d and the geometry of the lattice are irrelevant.
TL;DR: In this article, a generalized version of the Gordeyev-integral for electrostatic waves in magnetized plasma with a kappa or generalized Lorentzian velocity distribution is derived.
Abstract: A Gordeyev-type integral for the investigation of electrostatic waves in magnetized plasma having a kappa or generalized Lorentzian velocity distribution is derived. The integral readily reduces, in the unmagnetized and parallel propagation limits, to simple expressions involving the Zκ function. For propagation perpendicular to the magnetic field, it is shown that the Gordeyev integral can be written in closed form as a sum of two generalized hypergeometric functions, which permits easy analysis of the dispersion relation for electrostatic waves. Employing the same analytical techniques used for the kappa distribution, it is further shown that the well-known Gordeyev integral for a Maxwellian distribution can be written very concisely as a generalized hypergeometric function in the limit of perpendicular propagation. This expression, in addition to its mathematical conciseness, has other advantages over the traditional sum over modified Bessel functions form. Examples of the utility of these generalized ...
TL;DR: In this article, the authors derived Laplace approximations to three functions of matrix argument which arise in statistics and elsewhere: matrix Bessel Av, matrix bessel Bv, and the type II confluent hypergeometric function, Ψ.
TL;DR: The purpose of the article is to study how to apply a basic Borel-Laplace transformation to q-difference equations satisfied by the q-Bessel functions of F.H. Jackson.
TL;DR: In this paper, a simple physical model that allows the analysis of high-order harmonic generation in gases when the pumping laser beam has an intensity profile that is not Gaussian but truncated Bessel is presented.
Abstract: We describe a very simple physical model that allows the analysis of high-order harmonic generation in gases when the pumping laser beam has an intensity profile that is not Gaussian but truncated Bessel. This is the typical experimental condition when sub-10-fs pump-laser pulses, generated by the hollow fiber compression technique, are used. This model is based on the analysis of the phase-matching conditions for the harmonic generation process revisited in view of the new spatial mode of the fundamental beam. In particular, the role of the atomic dipole phase and the geometric phase terms are evidenced both for harmonics generated in the plateau and in the cutoff spectral regions. The influence of dispersion introduced by free electrons produced by laser ionization has also been discussed in some detail. Spatial patterns of far-field harmonics are then obtained by means of a simplified algorithm which allows one to avoid the numerical integration of the harmonic beam propagation equation. Experimental spatial distributions and divergence angles of high-order harmonics generated in Ne with 7-fs titanium-sapphire pulses are compared with numerical simulations in various experimental conditions. The agreement between measurements and calculated results is found to be very satisfactory.
TL;DR: In this paper, a new analytic solution of the mild-slope long wave equation is derived for studying the effects of bottom topography on combined refraction and diffraction, which is essentially of a series form involving the Bessel functions of real orders.
TL;DR: In this paper, the propagation of a combined beam consisting of two coaxial Bessel Gauss and Laguerre Gauss vortex beams is investigated and it is demonstrated that the vortical properties of light field under diffraction depend on propagation length as well as on amplitude ratio of individual beams.
TL;DR: In this paper, it was shown that the relative Bessel distribution attached to a distinguished (that is, self contragradient) unitary representation of GL 2 (k ) where k is a p -adic field is given by a locally integrable function, namely the relative bessel function.
Abstract: In this paper we develop the local theory for a Jacquet's relative trace formula. The local theory is essential to the application of the trace formula: it is an identity of Bessel and relative Bessel distributions. We show that the relative Bessel distribution attached to a distinguished (that is, self contragradient) unitary representation of GL 2 ( k ) where k is a p -adic field is given by a locally integrable function, namely the relative Bessel function. We compute the relative Bessel function for principal series, complementary series and special representations. We also show that the Bessel distributions associated to unitary irreducible admissible representations of the double covers of GL 2 ( k ) and SL 2 ( k ) are given by a locally integrable Bessel functions. We compute these Bessel functions for principal series, complementary series and special representations. Finally, we obtain the Waldspurger correspondence via Bessel identities between relative Bessel functions on GL 2 ( k ) and Bessel functions on the double cover of SL 2 ( k ). The Bessel identity also implies the identity between Bessel and relative Bessel distributions.
TL;DR: In this paper, the perturbation series of the two-level system driven by a sinusoidal field till second order was obtained, and the third order correction to the energy levels was obtained by proving how this correction does not modify at all the localization condition for a strong field as arising from the zeros of the zeroth Bessel function of integer order.
Abstract: Using perturbation theory in the strong coupling regime, that is, the dual Dyson series, and renormalization-group techniques to resum secular terms, we obtain the perturbation series of the two-level system driven by a sinusoidal field till second order. The third order correction to the energy levels is obtained by proving how this correction does not modify at all the localization condition for a strong field as arising from the zeros of the zeroth Bessel function of integer order. A comparison with weak coupling perturbation theory is done showing how the latter is contained in the strong coupling expansion in the proper limits. The strong coupling expansion we obtain proves to be accurate in the regime of high-frequency driving field. This computation gives an explicit analytical form to Floquet eigenstates and quasienergies for this problem, for high-frequency driving fields, supporting recent theoretical and experimental findings for quantum devices expected to give a representation for qubits in quantum computation.
TL;DR: An asymptotic expression of the incomplete gamma matrix function and integral expressions of Bessel matrix functions are given and results are applied to study the asymPTotic behavior of the modified Bessel function.
TL;DR: The paper investigates the analogies between Stirling numbers and Bessel numbers and a generating function for the BesselNumbers is obtained, and a proof of their unimodality is given.
TL;DR: In this paper, two new approximations for the generalized Bessel function were developed for the tunnelling environment in atomic ionization, and they are analytically much simpler than the currently used low-frequency asymptotic approximation.
Abstract: We develop two new approximations for the generalized Bessel function that frequently arises in the analytical treatment of strong-field processes, especially in non-perturbative multiphoton ionization theories. Both these new forms are applicable to the tunnelling environment in atomic ionization, and are analytically much simpler than the currently used low-frequency asymptotic approximation for the generalized Bessel function. The second of the new forms is an approximation to the first, and it is the second new form that exhibits the well-known tunnelling exponential.
TL;DR: In this paper, a polynomial approximation to Bessel functions arising from an electromagnetic scattering problem is examined, and the relationship between the approximation and the Taylor series is derived, and numerical calculations show that the approximation has similar accuracies.
Abstract: A polynomial approximation to Bessel functions that arises from an electromagnetic scattering problem is examined. The approximation is extended to Bessel functions of any integer order, and the relationship to the Taylor series is derived. Numerical calculations show that the polynomial approximation and the Taylor series truncated to the same order have similar accuracies.
TL;DR: In this article, an exact explicit analytical solution is provided, which allows the determination of the temperatures over the contact area with a very short computational time, regardless of the value of the Peclet number.
Abstract: So far in the literature, the distribution of stationary temperature over the surface of a half space subjected to a moving circular heat source has been reported in an integral or asymptotic form. In this paper an exact explicit analytical solution is provided, which allows the determination of the temperatures over the contact area with a very short computational time, regardless of the value of the Peclet number. The solution is based on special functions (Bessel and hypergeometric functions) that are preprogrammed under a formal calculation software (e.g., Maple). The results of the proposed solution are in agreement with the asymptotic models available in the literature.
TL;DR: In this paper, the Bessel-Gauss pulses and Bateman-Hillion relatively undistorted progressive waves are presented, and simple solutions describing localized wave propagation are found based on a kind of separation of variables.
Abstract: Tw on ew families of exact solutions of the wave equation uxx + uyy + uzz − c −2 utt = 0g eneralizing Bessel–Gauss pulses and Bateman–Hillion relatively undistorted progressive waves, respectively are presented. In each of these families new simple solutions describing localized wave propagation are found. The approach is based on a kind of separation of variables.
TL;DR: In this paper, the spreading of the quantum probability density for the highly-excited states of a single-particle system with an exponential-type potential on the positive semiaxis is quantitatively determined in both position and momentum spaces by means of the Boltzmann-Shannon information entropy.
Abstract: The spreading of the quantum probability density for the highly-excited states of a single-particle system with an exponential-type potential on the positive semiaxis is quantitatively determined in both position and momentum spaces by means of the Boltzmann–Shannon information entropy. This problem boils down to the calculation of the asymptotics of the entropy-like integrals of the modified Bessel function of the second kind (also called the Mcdonald function or Basset function). The dependence of the two physical entropies on the large quantum number n is given in detail. It is shown that the semiclassical (WKB) position–space entropy grows slower than the corresponding quantity of not only the harmonic oscillator but also the single-particle systems with any power-type potential of the form V(x)=x2k, x∈R and k∈N. The momentum–space entropy, calculated with a method based on the properties of the Mcdonald function, is rigorously found to have a behavior of the form −ln ln n, in strong contrast with the...
Book•
07 Aug 2003
TL;DR: In this paper, the authors present a model of a simple electromagnetic field with memory and a finite formulation of electromagnetic functions and problems in non-local electromagnetic fields.
Abstract: Preface 1. ELECTROMAGNETIC FIELDS 2. Green's functions and retarded potentials 3. Time-harmonic fields 4. Models of materials with memory 5. THERMODYNAMICS OF SIMPLE ELECTROMAGNETIC SYSTEMS 6. Thermoelectromagnetic systems 7. Existence and uniqueness 8. Wave propagation 9. Extremum principles 10. PROBLEMS IN NONLINEAR ELECTROMAGNETISM 11. Nonlocal electromagnetism and superconductivity 12. Magnetic hysteresis A. SOME PROPERTIES OF BESSEL FUNCTIONS B. Fourier transforms and Sobolev spaces C. Compact operators and eigenfunctions D. Differential operators in curvilinear coordinates E. Finite formulation of electromagnetism
TL;DR: In this article, the Taylor series expansion of the rth power of the modified Bessel function [Iν(z)]r is derived for arbitrary r. The result is expressed in terms of a recursive formula for a class of polynomials, which facilitates the systematic construction of the expansion.
Abstract: A formula for the Taylor series expansion of the rth power of the modified Bessel function [Iν(z)]r is derived for arbitrary r. The result is expressed in terms of a recursive formula for a class of polynomials, which facilitates the systematic construction of the expansion of [Iν(z)]r.