Showing papers on "Bessel function published in 1991"
TL;DR: In this paper, the Coulomb forces over a charged particle by other charged particles are derived for the sums over Coulomb force exerted on a charge by other charge particles, the central cell system being repeated to infinity by periodic boundary conditions.
Abstract: Formulae are derived for the sums over Coulomb forces exerted on a charged particle by other charged particles, the central cell system being repeated to infinity by periodic boundary conditions. Such sums are needed in molecular dynamics simulations involving either ions or neutral molecules represented as bound conglomerates of charges, and in astrophysical simulations of gravitating masses. The derived sums are rapidly convergent, being expressed in terms of Bessel functions Kr(z), which decrease exponentially with z. The force expressions are integrated analytically to give the potential function, which may be used in Monte Carlo simulations. The geometries considered are: (i) systems confined between two parallel walls, and (ii) unconfined three-dimensional systems.
253 citations
TL;DR: In this paper, a rotational diffusion equation for the distribution function describing the onset and the decay of the induced optical and electro-optic properties is solved, with the help of the recurrence relation for spherical modified Bessel functions.
Abstract: Expansion of the orientational distribution function f(θ, t) of molecular dipoles in terms of Legendre polynomials with spherical modified Bessel functions in(μE/kT) as coefficients yields an analytic relation between the steady-state birefringence Δnz(ω) and the electro-optic coefficient χxxz(2)(-ω;ω,0) for a poled nonlinear optical system. A rotational diffusion equation, with the diffusion constant D, for the distribution function describing the onset and the decay of the induced optical and electro-optic properties is solved, with the help of the recurrence relation for spherical modified Bessel functions. It is found that the onset of birefringence involves at least two time constants, with rise times of 1/2D and 1/6D, while the onset of the electro-optic effect is dominated by the rise time of 1/2D. After removal of the dc poling field, the birefringence and the electro-optic effect are found to relax in time with different decay time constants, 1/6D and 1/2D, respectively. This is due to the difference in the tensor rank describing the birefringence and the electro-optic effect.
158 citations
TL;DR: Two-sided inequalities for the ratio of modified Bessel functions of first kind are given in this paper, which provide sharper upper and lower bounds than had been known earlier. But these inequalities are not applicable to the case of non-Bessel functions.
Abstract: Two-sided inequalties for the ratio of modified Bessel functions of first kind are given, which provide sharper upper and lower bounds than had been known earlier. Wronskian type inequalities for Bessel functions are proved, and in the sequel alternative proofs of Turan-type inequalities for Bessel and modified Bessel functions are also discussed. These then lead to a two-sided inequality for Bessel functions. Also incorporated in the discussion is an inequality for the ratio of two Bessel functions for 0 < x < 1. Verifications of these inequalities are pointed out numerically.
72 citations
TL;DR: In this paper, the short distance assymptotics of the τ-function associated to the 2-point function of the two-dimensional Ising model are computed as a function of integration constant defined from the long distance behavior of the ε-function.
Abstract: The short-distance assymptotics of the τ-function associated to the 2-point function of the two-dimensional Ising model is computed as a function of the integration constant defined from the long-distance behavior of the τ-function. The result is expressible in terms of the Barnes double gamma function (equivalently, the BarnesG-function).
63 citations
TL;DR: In this article, exact dual-series eigenfunction solutions and simple closed-form low-frequency asymptotic approximations are determined for the problems of TM and TE scattering from a semicircular channel in a perfectly conducting ground plane.
Abstract: Exact dual-series eigenfunction solutions, and simple closed-form low-frequency asymptotic approximations are determined for the problems of TM and TE scattering from a semicircular channel in a perfectly conducting ground plane. The eigenfunction solutions provide benchmarks for channel scattering, and the low-frequency solutions can be used to determine directly incremental length diffraction coefficients for narrow channels. >
62 citations
TL;DR: In this article, the modified Bessel functions I v (t ) and K v ( t ) were studied and inequalities were derived for the modified function w v = tI v(t )/ I v + 1 (t ), which is useful in some problems of finite elasticity.
58 citations
ENEA1
TL;DR: In this article, the generalized Bessel functions (GBF) were applied to the problem of scattering for which the dipole approximation is inadequate, and the results for the first-kind cylinder GBF in the preasymptotic region were presented.
Abstract: In this paper we continue the systematic study of the generalized Bessel functions (GBF) recently introduced and often encountered in problems of scattering for which the dipole approximation is inadequate. We analyse the relations among different GBF and discuss their importance for the solution of differential finite-difference equation of the Raman-Nath type. We present numerical results for the first-kind cylinder GBF in the preasymptotic region and also a preliminary analysis of the asymptotic properties of the modified GBF.
51 citations
TL;DR: In this article, the spectrum of zeros of the polynomial solutions of the biconfluent Heun differential equation is investigated by two different methods: spectral Newton sums (i.e., the sums of the r th powers of the zeros) are given in a rigorous and recurrent way.
47 citations
TL;DR: The algorithm developed based on these improvements proves to be reliable and efficient, without size nor refractive index limitations, and the user has a choice to fix in advance the desired precision in the results.
Abstract: New improvements to compute Mie scattering quantities are presented. They are based on a detailed analysis of the various sources of error in Mie computations and on mathematical justifications. The algorithm developed based on these improvements proves to be reliable and efficient, without size (x = 2 nor refractive index (m = mR - imI) limitations, and the user has a choice to fix in advance the desired precision in the results. It also includes a new and efficient method to initiate the downward recurrences of Bessel functions.
46 citations
TL;DR: In this paper, the error of the physical optics solution for the E-polarized plane wave incidence in connection with diffraction by an arbitrary-angled dielectric wedge is corrected by calculating the nonuniform current distributed along the dielectrics interfaces.
Abstract: For pt.I see ibid., vol.39, no.9, p.1272-81 (1991). The error of the physical optics solution for the E-polarized plane wave incidence in connection with diffraction by an arbitrary-angled dielectric wedge is corrected by calculating the nonuniform current distributed along the dielectric interfaces. Two kinds of series expansions to the nonuniform current are employed. One is an asymptotic expansion as the multipole line source located at the edge of the dielectric wedge, since the correction field seems to be a cylindrical wave emanating from the edge in the far-field region. The other is arbitrary electric and magnetic surface currents expanded by infinite series of the Bessel functions, i.e. the Neumann expansion, of which fractional order is chosen to satisfy the edge condition near the edge of the dielectric wedge in the static limit. Both of the two different expansion coefficients for a wedge angle of 45 degrees , relative dielectric constants 2, 10, and 100, and the E-polarized incident angle of 150 degrees are evaluated by solving the dual series equation numerically after finite truncation. >
TL;DR: In this article, the authors derived an analytic expression for the infinite integral of three spherical Bessel functions, and used this result, together with the closure relation, to show how in principle one can derive an analytic function integral for any number of spherical functions.
Abstract: Integrals of several spherical Bessel functions occur frequently in nuclear physics. They are difficult to evaluate using standard numerical techniques, because of their slowly decreasing oscillatory form. The authors derive an analytic expression for the infinite integral of three spherical Bessel functions. They then use this result, together with the closure relation for spherical Bessel functions, to show how in principle one can derive an analytic expression for the integral of any number of spherical Bessel functions. They demonstrate this by deriving an analytic expression for the integral of four spherical Bessel functions. As with all of these analytic formulae, the results require that all angular momenta corresponding to the spherical Bessel functions can be coupled together to give an overall scalar quantity and conserve parity. The authors discuss the numerical accuracy and stability of this procedure.
TL;DR: In this article, a q -difference version of the Toda lattice equation is proposed, which admits solutions expressed by the q -Bessel function, and it is shown that the reduced equation admits solutions expressing the q − Bessel function.
Abstract: A q -difference version of the two-dimensional Toda lattice equation is proposed. Through a suitable reduction, it reduces to the q -difference version of the cylindrical Toda lattice equation. It is shown that the reduced equation admits solutions expressed by the q -Bessel function.
TL;DR: In this article, the explicit equations for the field components of the TE and TM waveguides are given, and the modal patterns for a number of modes are plotted, showing that the electric and magnetic fields of the modes TE/sub 1n/ and TM/sub n/ display edge singularity when the sectoral angle is greater than pi.
Abstract: Hollow cylindrical waveguides that have pie-shaped cross sections are described. The explicit equations for the field components of the TE (transverse electric) and TM (transverse magnetic) waves are given, and the modal patterns for a number of modes are plotted. The relation between the field distributions inside the circular and sectoral waveguides is discussed. It is shown that the electric and magnetic fields of the modes TE/sub 1n/ and TM/sub 1n/ display edge singularity when the sectoral angle is greater than pi . Formulas convenient for the evaluation of Bessel functions of fractional order on a personal computer are given. >
TL;DR: The use of local Taylor series expansions for determining the accuracy of computer programs for special functions, including testing of programs for exponential integrals, is discussed.
Abstract: This paper discusses the use of local Taylor series expansions for determining the accuracy of computer programs for special functions. The main example is testing of programs for exponential integrals. Additional applicaitons include testing of programs for certain Bessel functions, Dawson's integral, and error functions.
TL;DR: In this paper, the rho -algorithm, a nonlinear transformation, is shown to be applicable to monotonic series and the results of applying the algorithm to a series involving the zeroth-order Hankel function of the second kind and its associated Fourier transform are given.
Abstract: The rho -algorithm, a nonlinear transformation, is shown to be applicable to monotonic series. The results of applying the algorithm to a series involving the zeroth-order Hankel function of the second kind and its associated Fourier transform are given. It is shown that the algorithm performs better than the epsilon -algorithm derived from Shanks' transform (1955). Numerical results include a relative error measure versus number of terms taken in the series. >
TL;DR: In this article, a boundary integral formulation for the 2D Helmholtz equation involving kernels in the form of modified Bessel functions is presented, and accurate schemes for evaluating integrals of the kernels and their derivatives are presented.
Abstract: Boundary integral formulations for the 2D Helmholtz equation involve kernels in the form of modified Bessel functions. Accurate schemes for evaluating integrals of the kernels and their derivatives are presented. Special attention is paid to integrals involving singular and near singular kernels. Both boundary and domain integrals are considered. It is shown that, with the use of series expansion functions for the modified Bessel functions, the boundary integrals can be evaluated analytically in the neighbourhood of the singularity. For domain integrals, the behaviour of the kernels in the vicinity of the singularity is used to construct accurate numerical quadrature schemes. A transient heat conduction problem is formulated as a Helmholtz equation, solved, and compared against analytic solution to demonstrate the effectiveness of these schemes in relation to traditional methods. References are made to previous work to advocate the utility of the boundary integral method for non-linear and time-transient problems.
TL;DR: In this article, asymptotic expansions for conical Legendre functions of order µ and degree −½ + iτ are derived, where µ and τ are non-negative real parameters, and these expansions are uniformly valid for 0 ≦ µ ≦ Aτ (A an arbitrary positive constant).
Abstract: Uniform asymptotic expansions are derived for conical functions, Legendre functions of order µ and degree −½ + iτ, where µ and τ are non-negative real parameters. As τ → ∞, expansions are furnished for the conical functions which involve Bessel functions of order µ. These expansions are uniformly valid for 0 ≦ µ ≦ Aτ (A an arbitrary positive constant), and are also uniformly valid for Re (z) ≧ 0 in the complex argument case, and 0 ≦ z < ∞ in the real argument case. The case µ → ∞ is also considered, and expansions are furnished which are uniformly valid in the same z regions for 0 ≦ τ ≧ Bµ (B an arbitrary positive constant); in the cases where Re(z) ≧ 0 and 1 ≦ z < ∞, the expansions involve Bessel functions of purely imaginary order iτ, and in the case where 0 ≦ z < 1 the expansions involve elementary functions.
TL;DR: In this article, a simplified Radon transformation of Bessel basis functions in the method of series expansion in 2D tomography has been derived, where the integrand is written in terms of simple sinusoidal functions with no singularities.
Abstract: A simplified formula for the Radon transformation of Bessel basis functions in the method of series expansion in 2‐D tomography has been derived. The result is p 1 Jm (xmlr) Tm(p/r)r dr √r2−p2 0 cos−1 p dθ cos(mθ) sin[xml( cos θ−p)]. A numerical comparison demonstrates at least an order of magnitude CPU time reduction using the simplified version because the integrand is written in terms of simple sinusoidal functions with no singularities.
TL;DR: In this paper, the authors considered magneto-acoustic-gravity waves in an isothermal atmosphere, in a uniform magnetic field tilted at an arbitrary angle to the vertical, and with wave vector in the plane of gravity and the magnetic field.
Abstract: We consider magneto-acoustic-gravity waves in an isothermal atmosphere, in a uniform magnetic field tilted at an arbitrary angle to the vertical, and with wavevector in the plane of gravity and the magnetic field. It is shown that the velocity perturbation transverse to the magnetic field and gravity satisfies a decoupled second-order Alfven wave equation, and the fourth-order wave equation for the coupled slow-fast mode is solved exactly, using as variable the velocity perturbation transverse to the magnetic field, in the plane of the latter and gravity. The solution uses the method of two characteristic polynomials (Campos, 1987; Campos and Leitao 1988), and is checked by comparison with particular cases in the literature expressible in terms of elementary, Bessel or hypergeometric functions. It is shown that there is no critical layer, both for vertical and oblique waves, except in the case of a horizontal magnetic field; it is shown that for oblique waves in an oblique magnetic field there ar...
TL;DR: In this paper, the authors considered the problem of computing the smallest positive zero of a Bessel function and its derivatives of a prescribed order within a prescribed relative error, and they also considered the inverse problem of finding a zero of Bessel functions with a given positive value at a prescribed positive value.
TL;DR: In this paper, the Radon transformation of Bessel basis functions in 2D tomography is further reduced to an analytical series expansion in terms of sinusoidal functions with known coefficients, which is simpler and faster to compute than the previous expression.
Abstract: Based on a previously simplified formula [Wang and Granetz, Rev. Sci. Instrum. 62, 842 (1991)], the Radon transformation of Bessel basis functions in 2‐D tomography is further reduced to an analytical series expansion in terms of sinusoidal functions with known coefficients. Because there is no integration it is simpler and faster to compute than the previous expression.
TL;DR: In this article, the full relativistic dielectric tensor for a Maxwellian plasma in the electron-cyclotron range of frequencies was investigated and a new representation for arbitrary values of the wave frequency and direction of propagation of the anti-Hermitian part was presented that avoids the standard expansions of the Bessel functions.
Abstract: The full relativistic dielectric tensor for a Maxwellian plasma in the electron-cyclotron range of frequencies is investigated. A new representation for arbitrary values of the wave frequency and direction of propagation of the anti-Hermitian part is presented that avoids the standard expansions of the Bessel functions. A compact form of the wave damping for ω≥2ω c is obtained
TL;DR: In this article, an improved canonical plane-wave and line-source solution with modified diffraction coefficients is derived via a straightforward extension of the canonical line source solution given by Jones (1963), which results in convergence at the shadow boundary provided that the source and observation point are not both located at asymptotic distances from the scatterer.
Abstract: A number of canonical plane-wave and line-source solutions involving circular cylinders have led to a widely accepted uniform geometrical theory of diffraction (GTD) formulation for the scattered electromagnetic field in the shadow region of a smooth convex surface. The current solution does not constitute a properly formulated asymptotic high-frequency theory in the sense that it becomes increasingly inaccurate with increasing frequency. This inaccuracy results from transition-function dominance in the deep shadow region over the Pekeris caret function that is used. An improved formulation that circumvents this difficulty by avoiding use of a transition function is derived via a straightforward extension of the canonical line-source solution given by Jones (1963). This new solution takes the form of Keller-type modes with modified diffraction coefficients that result in convergence at the shadow boundary provided that the source and observation point are not both located at asymptotic distances from the scatterer. >
ENEA1
TL;DR: In this paper, a generalization of Bessel-type functions for multivariables and one-index functions is discussed. But the authors focus on the case where the dipole approximation does not hold and many higher harmonics are simultaneously operating.
Abstract: In this note we introduce a further generalization of Bessel-type functions, discussing the case of a multivariables and one-index function. This kind of function can be usefully exploited in problems in which the dipole approximation does not hold and many higher harmonics are simultaneously operating. We analyse the relevant recurrence properties, the modified forms and the generating functions.
TL;DR: In this article, a new method for numerical evaluation of ∫ 1 0 (1-x) α x β J v (ax)dx is proposed, where, α, β, ν, a are given constants, and Jν is the Bessel function of the first kind and of order ν.
TL;DR: In this paper, the Kernel function of the integral equation relating the pressure to the normal-wash distribution in unsteady potential subsonic flow was studied and exact solutions of the involved integrals of the kernel function were given in terms of new functions.
Abstract: We deal with the Kernel function of the integral equation relating the pressure to the normal-wash distribution in unsteady potential subsonic flow. Exact solutions of the involved integrals of the Kernel function are given in terms of new functions. Efficient and accurate numerical evaluation of these functions are described.