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Showing papers on "Bessel function published in 2007"


Journal ArticleDOI
TL;DR: A novel family of paraxial laser beams forming an overcomplete yet nonorthogonal set of modes that have a singular phase profile and are eigenfunctions of the photon orbital angular momentum are studied.
Abstract: We studied a novel family of paraxial laser beams forming an overcomplete yet nonorthogonal set of modes. These modes have a singular phase profile and are eigenfunctions of the photon orbital angular momentum. The intensity profile is characterized by a single brilliant ring with the singularity at its center, where the field amplitude vanishes. The complex amplitude is proportional to the degenerate (confluent) hypergeometric function, and therefore we term such beams hypergeometric-Gaussian (HyGG) modes. Unlike the recently introduced hypergeometric modes [Opt. Lett. 32, 742 (2007)], the HyGG modes carry a finite power and have been generated in this work with a liquid-crystal spatial light modulator. We briefly consider some subfamilies of the HyGG modes as the modified Bessel Gaussian modes, the modified exponential Gaussian modes, and the modified Laguerre-Gaussian modes.

240 citations


Posted Content
TL;DR: In this article, the authors generalized the Abel-Plana formula and derived the divergent parts from the vacuum expectation values for the local physical observables in a manifestly cutoff independent way and presented them in the form of strongly convergent integrals.
Abstract: One of the most efficient methods to obtain the vacuum expectation values for the physical observables in the Casimir effect is based on using the Abel-Plana summation formula. This allows us to derive the regularized quantities in a manifestly cutoff independent way and present them in the form of strongly convergent integrals. However, the application of AbelPlana formula, in its usual form, is restricted by simple geometries when the eigenmodes have a simple dependence on quantum numbers. The author generalized the Abel-Plana formula which essentially enlarges its application range. Based on this generalization, formulae have been obtained for various types of series over the zeros of some combinations of Bessel functions and for integrals involving these functions. It has been shown that these results generalize the special cases existing in literature. Further, the derived summation formulae have been used to summarize series arising in the mode summation approach to the Casimir effect for spherically and cylindrically symmetric boundaries. This allows us to extract the divergent parts from the vacuum expectation values for the local physical observables in a manifestly cutoff independent way. The present paper reviews these results. Some new considerations are also added.

207 citations


Journal ArticleDOI
TL;DR: In this paper, a relatively simple and general method for calculating the mutual inductance and self-inductance of both coaxial and non-coaxial cylindrical coils is given.
Abstract: A relatively simple and general method for calculating the mutual inductance and self-inductance of both coaxial and noncoaxial cylindrical coils is given. For combinations of cylindrical coils, thin solenoids, pancake coils, and simple circular loops, the mutual inductance can be reduced to a one-dimensional integral of closed form expressions involving Bessel and related functions. Coaxial and noncoaxial cases differ only by the presence of an extra Bessel factor J 0(sp) in the noncoaxial integral, where p is the perpendicular distance separating the coil axes and s is the variable of integration. The method is related to a recently given noncoaxial generalization of Ruby's formula for a nuclear radiation source and detector system, the analogy being close but not exact. In many cases, the Bessel function integral for the inductance can be easily evaluated directly using Maple or Mathematica. In other cases, it is better to transform the integral to a more numerically friendly form. A general analytical solution is presented for the inductance of two circular loops which lie in the same plane

162 citations


Journal ArticleDOI
TL;DR: In this paper, the concept of Bessel multipliers is introduced, which unifies the approach used for Gabor multipliers for arbitrary analysis/synthesis systems, which form Bessel sequences, like wavelet or irregular Gabor frames.

145 citations


Journal ArticleDOI
TL;DR: The exact scattering by a sphere centered on a Bessel beam is expressed as a partial wave series involving the scattering angle relative to the beam axis and the conical angle of the wave vector components of the Besselbeam.
Abstract: The exact scattering by a sphere centered on a Bessel beam is expressed as a partial wave series involving the scattering angle relative to the beam axis and the conical angle of the wave vector components of the Bessel beam. The sphere is assumed to have isotropic material properties so that the nth partial wave amplitude for plane wave scattering is proportional to a known partial-wave coefficient. The scattered partial waves in the Bessel beam case are also proportional to the same partial-wave coefficient but now the weighting factor depends on the properties of the Bessel beam. When the wavenumber-radius product ka is large, for rigid or soft spheres the scattering is peaked in the backward and forward directions along the beam axis as well as in the direction of the conical angle. These properties are geometrically explained and some symmetry properties are noted. The formulation is also suitable for elastic and fluid spheres. A partial wave expansion of the Bessel beam is noted.

105 citations


Proceedings ArticleDOI
07 Sep 2007
TL;DR: In this article, a special class of homogeneous monogenic polynomials constructed in the framework of hypercomplex function theory is presented, in order to be an Appell set of polynomial functions.
Abstract: In this paper we present applications of a special class of homogeneous monogenic polynomials constructed, in the framework of hypercomplex function theory, in order to be an Appell set of polynomials. In particular, we derive important properties of an associated exponential function from R3 to R3 and propose a generalization to Rn+1.

98 citations


Journal ArticleDOI
TL;DR: In this article, the authors studied the second-order Bessel differential equation in Liouville form, where the parameter ν represents the order of the associated Bessel functions and λ is the complex spectral parameter involved in considering properties of the equation in the Hilbert function space L2(0,∞).

87 citations


Journal ArticleDOI
TL;DR: Energy characteristics of the superposition of TE- and TM-polarized electromagnetic Bessel beams are studied and the following phenomena are predicted, which should confirm negative beam propagation: reflection of the beam from a circular aperture and unusual movement of microparticles in the beam.
Abstract: Energy characteristics of the superposition of TE- and TM-polarized electromagnetic Bessel beams are studied. For some phase differences between TE and TM waves the components of the Poynting vector vary in sign. We call this situation "negative propagation," because locally the beam may behave like a wave propagating in the direction opposite to the conventional one. We predict the following phenomena, which should confirm negative beam propagation: reflection of the beam from a circular aperture and unusual movement of microparticles in the beam.

85 citations


Journal ArticleDOI
TL;DR: In this article, the conditional characteristic function of the position of a particle after n changes of direction was obtained from this characteristic function and the conditional distributions in terms of (n+1)−fold integrals of products of Bessel functions.
Abstract: We consider in this paper random flights in ℝd performed by a particle changing direction of motion at Poisson times. Directions are uniformly distributed on hyperspheres S1d. We obtain the conditional characteristic function of the position of the particle after n changes of direction. From this characteristic function we extract the conditional distributions in terms of (n+1)−fold integrals of products of Bessel functions. These integrals can be worked out in simple terms for spaces of dimension d=2 and d=4. In these two cases also the unconditional distribution is determined in explicit form. Some distributions connected with random flights in ℝ3 are discussed and in some special cases are analyzed in full detail. We point out that a strict connection between these types of motions with infinite directions and the equation of damped waves holds only for d=2.

82 citations


Journal ArticleDOI
TL;DR: In this article, a model of non-intersecting squared Bessel processes in the confluent case is studied, where all paths start at time $t = 0$ at the same positive value, remain positive, and are conditioned to end at time$t = T$ at $x = 0$.
Abstract: We study a model of $n$ non-intersecting squared Bessel processes in the confluent case: all paths start at time $t = 0$ at the same positive value $x = a$, remain positive, and are conditioned to end at time $t = T$ at $x = 0$. In the limit $n \to \infty$, after appropriate rescaling, the paths fill out a region in the $tx$-plane that we describe explicitly. In particular, the paths initially stay away from the hard edge at $x = 0$, but at a certain critical time $t^*$ the smallest paths hit the hard edge and from then on are stuck to it. For $t eq t^*$ we obtain the usual scaling limits from random matrix theory, namely the sine, Airy, and Bessel kernels. A key fact is that the positions of the paths at any time $t$ constitute a multiple orthogonal polynomial ensemble, corresponding to a system of two modified Bessel-type weights. As a consequence, there is a $3 \times 3$ matrix valued Riemann-Hilbert problem characterizing this model, that we analyze in the large $n$ limit using the Deift-Zhou steepest descent method. There are some novel ingredients in the Riemann-Hilbert analysis that are of independent interest.

75 citations


Journal ArticleDOI
TL;DR: This work has fabricated by photolithography a binary diffractive optical element able to produce in the focal plane of a spherical lens an optical vortex, which was then used to perform rotation of several polystyrene beads of diameter 5 microm.
Abstract: We derive what we believe to be new analytical relations to describe the Fraunhofer diffraction of the finite-radius plane wave by a helical axicon (HA) and a spiral phase plate (SPP). The solutions are deduced in the form of a series of the Bessel functions for the HA and a finite sum of the Bessel functions for the SPP. The solution for the HA changes to that for the SPP if the axicon parameter is set equal to zero. We also derive what we believe to be new analytical relations to describe the Fresnel and Fraunhofer diffraction of the Gaussian beam by a HA are derived. The solutions are deduced in the form of a series of the hypergeometric functions. We have fabricated by photolithography a binary diffractive optical element (a HA with number n=10) able to produce in the focal plane of a spherical lens an optical vortex, which was then used to perform rotation of several polystyrene beads of diameter 5 μm.

Journal ArticleDOI
TL;DR: In this paper, it was shown that among the multitude of rotating light beams whose complex amplitude can be represented as a linear superposition of the Laguerre Gaussian (LG) modes with definite numbers there are light beams with zero orbital angular angular momentum (OAM) and vice versa, multi-mode LG beams that show no rotation and are lacking the radially symmetric intensity distribution can possess the non-zero OAM.

Journal ArticleDOI
TL;DR: In this article, the closed series solution to scattering by an eccentric coated cylinder is realized in matrix form by boundary value analysis and the addition theorem of the Bessel's functions, and the results are compared to previously published works.

Journal ArticleDOI
TL;DR: The exact partial wave series for the scattering by a sphere centered on an ideal Bessel beam was recently given by Marston and is applied here to solid elastic spheres in water and to an empty spherical shell in water.
Abstract: The exact partial wave series for the scattering by a sphere centered on an ideal Bessel beam was recently given by Marston ["Scattering of a Bessel beam by a sphere," J. Acoust. Soc. Am. 121, 753-758 (2007)]. That series is applied here to solid elastic spheres in water and to an empty spherical shell in water. The examples are selected to illustrate the effect of varying the beam's conical angle so as to modify the coupling to specific resonances in the response of each type of sphere considered. The backscattering may be reduced or increased depending on properties of the resonance and of the specular contribution. Changing the conical angle is equivalent to changing the beamwidth. Some applications of the Van de Hulst localization principle to the interpretation of the partial wave series and to the interpretation of the scattering dependence on the beam's conical angle are discussed. Some potential applications to the analysis of the scattering by spheres of more general axisymmetric beams are noted.

Journal ArticleDOI
TL;DR: In this paper, it was shown that a variable exponent Bessel potential space coincides with the variable exponent Sobolev space if the Hardy-Littlewood maximal operator is bounded on the underlying variable exponent Lebesgue space.
Abstract: We show that a variable exponent Bessel potential space coincides with the variable exponent Sobolev space if the Hardy-Littlewood maximal operator is bounded on the underlying variable exponent Lebesgue space. Moreover, we study the Holder type quasi-continuity of Bessel potentials of the first order. Mathematics subject classification (2000): 46E35, 46E30, 26D10. Keywordsandphrases: Besselpotential space,Lebesguespacewithvariable exponent, quasi-continuity.

Journal ArticleDOI
TL;DR: In this paper, a new numerical analysis for the collocation method presented by Levin for √ √ f(x)S(rx)dx is presented, which gives more accurate error analysis about the integration of systems containing Bessel functions.
Abstract: The integration of systems containing Bessel functions is a central point in many practical problems in physics, chemistry and engineering. This paper presents a new numerical analysis for the collocation method presented by Levin for \(\int_a^b f(x)S(rx)dx\) and gives more accurate error analysis about the integration of systems containing Bessel functions. The effectiveness and accuracy of the quadrature is tested for Bessel functions with large arguments.

Journal ArticleDOI
Abstract: In this paper we introduce probability-preserving convolution algebras on cones of positive semidefinite matrices over one of the division algebras F = R, C or H which interpolate the convolution algebras of radial bounded Borel measures on a matrix space Mp,q(F )w ith p q. Radiality in this context means invariance under the action of the unitary group Up(F) from the left. We obtain a continuous series of commutative hypergroups whose characters are given by Bessel functions of matrix argument. Our results generalize wellknown structures in the rank-one case, namely the Bessel–Kingman hypergroups on the positive real line, to a higher rank setting. In a second part of the paper we study structures

Journal ArticleDOI
TL;DR: In this article, it was shown that the shape of a sound-soft/sound-hard ball in R 3 or a sound soft/soundhard disc in R 2 is uniquely determined by a single far-field datum measured at some fixed spot corresponding to a single incident plane wave.
Abstract: Some novel interlacing properties of the zeros for the Bessel and spherical Bessel functions are first presented and then applied to prove an interesting uniqueness result in inverse acoustic obstacle scattering. It is shown that in the resonance region, the shape of a sound-soft/sound-hard ball in R 3 or a sound-soft/sound-hard disc in R 2 is uniquely determined by a single far-field datum measured at some fixed spot corresponding to a single incident plane wave.

Journal ArticleDOI
TL;DR: In this paper, the authors presented a completely analytical method for determining the eddy currents in a cylindrical configuration, which consists of a permanent magnet rotating inside a conducting hollow cylinder (stator).
Abstract: The paper presents a completely analytical method for determining the eddy currents in a cylindrical configuration. The configuration consists of a cylindrical permanent magnet rotating inside a conducting hollow cylinder (stator). The solution is obtained by solving a generalized form of the diffusion equation and applying the modified Bessel functions. The determination of the magnetic field in the air and in the stator, and the losses generated by the eddy currents, is completely analytical. The results are verified by finite-element software.

Journal ArticleDOI
TL;DR: In this paper, a Bessel function method is proposed to obtain the exact solutions for the free-vibration analysis of rectangular thin plates with three edge conditions: (i) fully simply supported; (ii) fully clamped, and (iii) two opposite edges simply supported and the other two edges clamped.
Abstract: A novel Bessel function method is proposed to obtain the exact solutions for the free-vibration analysis of rectangular thin plates with three edge conditions: (i) fully simply supported; (ii) fully clamped, and (iii) two opposite edges simply supported and the other two edges clamped. Because Bessel functions satisfy the biharmonic differential equation of solid thin plate, the basic idea of the method is to superpose different Bessel functions to satisfy the edge conditions such that the governing differential equation and the boundary conditions of the thin plate are exactly satisfied. It is shown that the proposed method provides simple, direct, and highly accurate solutions for this family of problems. Examples are demonstrated by calculating the natural frequencies and the vibration modes for a square plate with all edges simply supported and clamped.

Journal ArticleDOI
TL;DR: In this article, a vector-valued version of the asymptotic expansion is constructed, which allows us to determine the order of a Levin-type method with highly oscillatory kernels, such as Airy functions or Bessel functions.
Abstract: We present a method for the efficient approximation of integrals with highly oscillatory vector-valued kernels, such as integrals involving Airy functions or Bessel functions. We construct a vector-valued version of the asymptotic expansion, which allows us to determine the asymptotic order of a Levin-type method. Levin-type methods are constructed using collocation, and choosing a basis based on the asymptotic expansion results in an approximation with significantly higher asymptotic order.

Journal ArticleDOI
TL;DR: In this article, the authors studied a family of paraxial laser beams forming an overcomplete yet nonorthogonal set of modes called hypergeometric gaussian (HyGG) modes.
Abstract: We studied a novel family of paraxial laser beams forming an overcomplete yet nonorthogonal set of modes. These modes have a singular phase profile and are eigenfunctions of the photon orbital angular momentum. The intensity profile is characterized by a single brilliant ring with the singularity at its center, where the field amplitude vanishes. The complex amplitude is proportional to the degenerate (confluent) hypergeometric function, and therefore we term such beams hypergeometric gaussian (HyGG) modes. Unlike the recently introduced hypergeometric modes (Opt. Lett. {\textbf 32}, 742 (2007)), the HyGG modes carry a finite power and have been generated in this work with a liquid-crystal spatial light modulator. We briefly consider some sub-families of the HyGG modes as the modified Bessel Gaussian modes, the modified exponential Gaussian modes and the modified Laguerre-Gaussian modes.

Journal ArticleDOI
TL;DR: Recent results of room acoustics analysis based on a spherical microphone array, which employs high spherical harmonics order for improved spatial resolution, and a dual-radius spherical measurement array to avoid ill-conditioning at the null frequencies of the spherical Bessel function are presented.
Abstract: The spatial and temporal distribution of early reflections in an auditorium is considered important for sound perception. Previous studies presented measurement and analysis methods based on spherical microphone arrays and plane-wave decomposition that could provide information on the direction and time of arrival of early reflections. This paper presents recent results of room acoustics analysis based on a spherical microphone array, which employs high spherical harmonics order for improved spatial resolution, and a dual-radius spherical measurement array to avoid ill-conditioning at the null frequencies of the spherical Bessel function. Spatial-temporal analysis is performed to produce directional impulse responses, while analysis based on the windowed Fourier transform is employed to detect direction of arrival of individual reflections at selected frequencies. Experimental results of sound-field analysis in a real auditorium are also presented.

Journal ArticleDOI
TL;DR: In this paper, the authors present an analysis of the dynamics of conical waves partially obstructed by opaque objects, and show that the invariance of Bessel beams with finite transverse extension is no longer maintained under the mentioned conditions.
Abstract: We present an analysis of the dynamics of conical waves partially obstructed by opaque objects. The analysis yields the incoming and outgoing conical waves that form the Bessel beams (or any other propagation-invariant beams) when opaque obstructions are set on and off axis. The results show that the invariance of Bessel beams with finite transverse extension is no longer maintained under the mentioned conditions.

Journal ArticleDOI
TL;DR: Barbour et al. as discussed by the authors gave a shorter proof of Kanter's (J. Multivariate Anal. 6, 222-236, 1976) sharp Bessel function bound for concentrations of sums of independent symmetric random vectors.
Abstract: We give a shorter proof of Kanter’s (J. Multivariate Anal. 6, 222–236, 1976) sharp Bessel function bound for concentrations of sums of independent symmetric random vectors. We provide sharp upper bounds for the sum of modified Bessel functions I0(x) + I1(x), which might be of independent interest. Corollaries improve concentration or smoothness bounds for sums of independent random variables due to Cekanavicius & Roos (Lith. Math. J. 46, 54–91, 2006); Roos (Bernoulli, 11, 533–557, 2005), Barbour & Xia (ESAIM Probab. Stat. 3, 131–150, 1999), and Le Cam (Asymptotic Methods in Statistical Decision Theory. Springer, Berlin Heidelberg New York, 1986).

Journal ArticleDOI
TL;DR: In this paper, the authors derive representations for certain entire q-functions and apply their technique to the Ramanujan entire function (or q -Airy function) and q -Bessel functions.

Journal ArticleDOI
TL;DR: In this article, the spectrum of the lowest spin glueballs in pure Yang-Mills theory in 2+1$ dimensions was derived by solving the Schr\"odinger equation under certain assumptions.
Abstract: We present details of the analytic computation of the spectrum of lowest spin glueballs in pure Yang-Mills theory in $2+1$ dimensions. The new ingredient is provided by the conjectured new nontrivial expression for the (quasi)Gaussian part of the ground state wave functional. We show that this wave functional can be derived by solving the Schr\"odinger equation under certain assumptions. The mass spectrum of the theory is determined by the zeros of Bessel functions, and the agreement with available lattice data is excellent.

Journal ArticleDOI
TL;DR: In this paper, Bessel and Bessel-Gaussian beam propagation through an unapertured or apertured misaligned paraxial optical system is investigated, and analytical formulas are derived based on the generalized diffraction integral formula for treating the propagation of a laser beam through a misaligned POMO system in the cylindrical coordinate system.

Journal ArticleDOI
TL;DR: In this paper, the equivalence of the modulus of smoothness and the -functional is established for functions of the Nikol'skii-Besov type and function spaces of the type are defined.
Abstract: We study problems of approximation of functions on? in the metric of? with power weight using generalized Bessel shifts. We prove analogues of direct Jackson theorems for the modulus of smoothness of arbitrary order defined in terms of generalized Bessel shifts. We establish the equivalence of the modulus of smoothness and the -functional. We define function spaces of Nikol'skii-Besov type and describe them in terms?of best approximations. As a?tool for approximation, we use a?certain class of entire functions of exponential type. In this class, we prove analogues of Bernstein's inequality and others for the Bessel differential operator and its fractional powers. The main tool we use to solve these problems is Bessel harmonic analysis.

Journal Article
TL;DR: In this paper, the harmonic Green and Neumann functions are used to construct bi-harmonic Green, Neumann and particular Robin functions, which are then constructed via a convolution of the harmonic particular fundamental solutions.
Abstract: The harmonic Green and Neumann function and a particular Robin function are used to construct bi-harmonic Green, Neumann and particular Robin functions. Moreover hybrid bi-harmonic Green functions are given. They all are constructed via a convolution of the mentioned harmonic particular fundamental solutions. In case of the unit disc they are explicitly expressed. Besides these 9 bi-harmonic Green functions there is another bi-harmonic Green function in explicit form for the unit disc not defined by convolution. Related boundary value problems are not all well posed. In case they are, the unique solutions are given. For the other cases solvability conditions are determined and the unique solutions found. There are all together 12 Dirichlet kind boundary value problems for the inhomogeneous bi-harmonic equation treated. The investigation is restricted to the two dimensional case and complex notation is used.