Showing papers on "Bessel function published in 1983"
TL;DR: In this paper, a unified treatment of elastic and electromagnetic (EM) wave propagation in horizontally layered media for which the parameters in the partial differential equations are piecewise continuous functions of only one spatial variable is provided.
Abstract: The objective of this paper is to provide a unified treatment of elastic and electromagnetic (EM) wave propagation in horizontally layered media for which the parameters in the partial differential equations are piece‐wise continuous functions of only one spatial variable. By applying a combination of Fourier, Laplace, and Bessel transforms to the partial differential equations describing the elastic or EM wave propagation I obtain a system of 2n linear ordinary differential equations. The 2n×2n coefficient matrix is partitioned into 4n×n submatrices. By a proper choice of variables, the diagonal submatrices are zero and the off‐diagonal submatrices are symmetric. All the results in the paper are derived from the symmetry properties of this general equation. In the appendices it is shown that three‐dimensional elastic waves, cylindrical P‐SV waves, acoustic waves, and electromagnetic waves in isotropic layered media can all be represented by an equation with the same properties. The symmetry properties of...
273 citations
Book•
18 Jan 1983
TL;DR: In this paper, the authors present a model for Second Order Systems with periodically time-varying parameters based on a matrix formulation of the Hill Equation and a triangulation of the waveform.
Abstract: Part-I Theory and Techniques.- 1 Historical Perspective.- 1.1 The Nature of Systems with Periodically Time-Varying Parameters.- 1.2 1831-1887 Faraday to Rayleigh-Early Experimentalists and Theorists.- 1.3 1918-1940 The First Applications.- 1.4 Second Generation Applications.- 1.5 Recent Theoretical Developments.- 1.6 Commonplace Illustrations of Parametric Behaviour.- References for Chapter 1.- Problems.- 2 The Equations and Their Properties.- 2.1 Hill Equations.- 2.2 Matrix Formulation of Hill Equations.- 2.3 The State Transition Matrix.- 2.4 Floquet Theory.- 2.5 Second Order Systems.- 2.6 Natural Modes of Solution.- 2.7 Concluding Comments.- References for Chapter 2.- Problems.- 3 Solutions to Periodic Differential Equations.- 3.1 Solutions Over One Period of the Coefficient.- 3.2 The Meissner Equation.- 3.3 Solution at Any Time for a Second Order Periodic Equation.- 3.4 Evaluation of ?(?, 0)m, m Integral.- 3.5 The Hill Equation with a Staircase Coefficient.- 3.6 The Hill Equation with a Sawtooth Waveform Coefficient.- 3.6.1 The Wronskian Matrix with z Negative.- 3.6.2 The Wronskian Matrix with z Zero.- 3.6.3 The Case of ? Negative.- 3.7 The Hill Equation with a Positive Slope, Sawtooth Waveform Coefficient.- 3.8 The Hill Equation with a Triangular Coefficient.- 3.9 The Hill Equation with a Trapezoidal Coefficient.- 3.10 Bessel Function Generation.- 3.11 The Hill Equation with a Repetitive Exponential Coefficient.- 3.12 The Hill Equation with a Coefficient in the Form of a Repetitive Sequence of Impulses.- 3.13 Equations of Higher Order.- 3.14 Response to a Sinusoidal Forcing Function.- 3.15 Phase Space Analysis.- 3.16 Concluding Comments.- References for Chapter 3.- Problems.- 4 Stability.- 4.1 Types of Stability.- 4.2 Stability Theorems for Periodic Systems.- 4.3 Second Order Systems.- 4.3.1 Stability and the Characteristic Exponent.- 4.3.2 The Meissner Equation.- 4.3.3 The Hill Equation with an Impulsive Coefficient.- 4.3.4 The Hill Equation with a Sawtooth Waveform Coefficient.- 4.3.5 The Hill Equation with a Triangular Waveform Coefficient.- 4.3.6 Hill Determinant Analysis.- 4.3.7 Parametric Frequencies for Second Order Systems.- 4.4 General Order Systems.- 4.4.1 Hill Determinant Analysis for General Order Systems.- 4.4.2 Residues of the Hill Determinant for q ? 0.- 4.4.3 Instability and Parametric Frequencies for General Systems.- 4.4.4 Stability Diagrams for General Order Systems.- 4.5 Natural Modes and Mode Diagrams.- 4.5.1 Nature of the Basis Solutions.- 4.5.2 P Type Solutions.- 4.5.3 C Type Solutions.- 4.5.4 N Type Solutions.- 4.5.5 Modes of Solution.- 4.5.6 The Modes of a Second Order Periodic System.- 4.5.7 Boundary Modes.- 4.5.8 Second Order System with Losses.- 4.5.9 Modes for Systems of General Order.- 4.5.10 Coexistence.- 4.6 Short Time Stability.- References for Chapter 4.- Problems.- 5 A Modelling Technique for Hill Equations.- 5.1 Convergence of the Hill Determinant and Significance of the Harmonics of the Periodic Coefficients.- 5.1.1 Second Order Systems.- 5.1.2 General Order Systems.- 5.2 A Modelling Philosophy for Intractable Hill Equations.- 5.3 The Frequency Spectrum of a Periodic Staircase Coefficient.- 5.4 Piecewise Linear Models.- 5.4.1 General Comments.- 5.4.2 Trapezoidal Models.- 5.5 Forced Response Modelling.- 5.6 Stability Diagram and Characteristic Exponent Modelling.- 5.7 Models for Nonlinear Hill Equations.- 5.8 A Note on Discrete Spectral Analysis.- 5.9 Concluding Remarks.- References for Chapter 5.- Problems.- 6 The Mathieu Equation.- 6.1 Classical Methods for Analysis and Their Limitations.- 6.1.1 Periodic Solutions.- 6.1.2 Mathieu Functions of Fractional Order.- 6.1.3 Fractional Order Unstable Solutions.- 6.1.4 Limitations of the Classical Method of Treatment.- 6.2 Numerical Solution of the Mathieu Equation.- 6.3 Modelling Techniques for Analysis.- 6.3.1 Rectangular Waveform Models.- 6.3.2 Trapezoidal Waveform Models.- 6.3.3 Staircase Waveform Models.- 6.3.4 Performance Comparison of the Models.- 6.4 Stability Diagrams for the Mathieu Equation.- 6.4.1 The Lossless Mathieu Equation.- 6.4.2 The Damped (Lossy) Mathieu Equation.- 6.4.3 Sufficient Conditions for the Stability of the Damped Mathieu Equation.- References for Chapter 6.- Problems.- II Applications.- 7 Practical Periodically Variable Systems.- 7.1 The Quadrupole Mass Spectrometer.- 7.1.1 Spatially Linear Electric Fields.- 7.1.2 The Quadrupole Mass Filter.- 7.1.3 The Monopole Mass Spectrometer.- 7.1.4 The Quadrupole Ion Trap.- 7.1.5 Simulation of Quadrupole Devices.- 7.1.6 Non idealities in Quadrupole Devices.- 7.2 Dynamic Buckling of Structures.- 7.3 Elliptical Waveguides.- 7.3.1 The Helmholtz Equation.- 7.3.2 Rectangular Waveguides.- 7.3.3 Circular Waveguides.- 7.3.4 Elliptical Waveguides.- 7.3.5 Computation of the Cut-off Frequencies for an Elliptical Waveguide..- 7.4 Wave Propagation in Periodic Media.- 7.4.1 Pass and Stop Bands.- 7.4.2 The ? - ?r (Brillouin) Diagram.- 7.4.3 Electromagnetic Wave Propagation in Periodic Media.- 7.4.4 Guided Electromagnetic Wave Propagation in Periodic Media.- 7.4.5 Electrons in Crystal Lattices.- 7.4.6 Other Examples of Waves in Periodic Media:.- 7.5 Electric Circuit Applications.- 7.5.1 Degenerate Parametric Amplification.- 7.5.2 Degenerate Parametric Amplification in High Order Periodic Networks.- 7.5.3 Nondegenerate Parametric Amplification.- 7.5.4 Parametric Up Converters.- 7.5.5 N-path Networks.- References for Chapter 7.- Problems.- Appendix Bessel Function Generation by Chebyshev Polynomial Methods.- References for Appendix.
217 citations
TL;DR: In this paper, the authors present an algorithm for the automatic integration of the product of the kernel and Bessel functions between the asymptotic zero crossings of the latter and sums the series of partial integrations using a continued fraction expansion, equivalent to an analytic continuation of the series.
Abstract: An algorithm is presented for the accurate evaluation of Hankel (or Bessel) transforms of algebraically related kernel functions, defined here as the non-Bessel function portion of the integrand, that is more widely applicable than the standard digital filter methods without enormous increases in computational burden. The algorithm performs the automatic integration of the product of the kernel and Bessel functions between the asymptotic zero crossings of the latter and sums the series of partial integrations using a continued fraction expansion, equivalent to an analytic continuation of the series. The integrands may be saved to allow the rapid computation of related transforms without recalculating the kernel or Bessel functions, and the integration steps use interlacing quadrature formulas so that no function evaluations are wasted when it is necessary to increase the order of the quadrature rule. The continued fraction algorithm allows very slowly divergent or even formally divergent integrals to be computed quite easily. The local error is controlled at each step in the algorithm, and accuracy is limited largely by machine resolution. The algorithm is written in Fortran and is listed in an Appendix along with a driver program that illustrates its features. The driver program and subroutine are available from the SEG Business Offtce.
195 citations
TL;DR: In this article, the Fourier transform (FT) of a two-center RBF (reduced Bessel function) charge distribution permitting partial-wave analysis is derived with the use of Feynman's identity.
Abstract: A new formula for the Fourier transform (FT) of a two-center RBF (reduced Bessel function) charge distribution permitting partial-wave analysis is derived with the use of Feynman's identity. This formula is valid for all quantum numbers. It is also independent of the orientation of the coordinate axes. A new representation for the two-center overlap integral (to which the kinetic energy, two-center attraction and Coulomb repulsion integrals can be readily reduced) and for the three-center attraction integral is obtained with the help of the FT. It is stable for all values of the orbital exponents. The method developed by Graovac et al. for computing repulsion integrals for $s$ states is generalized to include all states. Numerical test values of several one- and two-electron integrals are also reported. A strategy which should enhance the efficiency of computation by making maximal use of the FT's already computed is suggested.
119 citations
TL;DR: In this paper, the smooth factor of the Bessel function is replaced by a truncated Chebyshev series approximation and the resulting integral is computed exactly, and the numerical aspects of this exact integration are discussed.
Abstract: The numerical evaluation of Bessel function integrals may be difficult when the Bessel function is rapidly oscillating in the interval of integration. In the method presented here, the smooth factor of the integrand is replaced by a truncated Chebyshev series approximation and the resulting integral is computed exactly. The numerical aspects of this exact integration are discussed.
72 citations
TL;DR: In this article, the Bouchon method with complex frequencies was extended to include simultaneous evaluation of multiple sources at different depths, which is the same feature as in Olson's finite element discrete Fourier Bessel code (DWFE) (Olson, 1982).
Abstract: Expressions for displacements on the surface of a layered half-space due to point force are given in terms of generalized reflection and transmission coefficient matrices (Kennett, 1980) and the discrete wavenumber summation method (Bouchon, 1981). The Bouchon method with complex frequencies yields accurate near-field dynamic and static solutions.
The algorithm is extended to include simultaneous evaluation of multiple sources at different depths. This feature is the same as in Olson's finite element discrete Fourier Bessel code (DWFE) (Olson, 1982).
As numerical examples, we calculate some layered half-space problems. The results agree with synthetics generated with the Cagniard-de Hoop technique, P-SV modes, and DWFE codes. For a 10-layered crust upper mantle model with a bandwidth of 0 to 10 Hz, this technique requires one-tenth the time of the DWFE calculation. In the presence of velocity gradients, where finer layering is required, the DWFE code is more efficient.
66 citations
Book•
01 Jan 1983
TL;DR: In this paper, the representations of Bessel functions and BGL(2,K)$$γ$-functions were studied in the context of linear group representation theory.
Abstract: Preliminaries: Representation theory the general linear group The representations of $GL(2,K)$ $\Gamma$-functions and Bessel functions References Index.
59 citations
TL;DR: In this paper, the authors have shown that the general superposition of ordinary solitons and the various ripplons is possible for 2-dimensional Toda lattice equations.
Abstract: We have found new type exact solutions (ripplon solutions) of the 2-dimensional Toda lattice equation. The new solution reflects the effect of the essential multi-dimensionality of the system. We have shown that the general superposition of ordinary solitons and the various ripplons is possible.
52 citations
TL;DR: In this article, a second-order differential operator commuting with the reproducing kernel ∑ n − 0 T φ n (λ) φn (μ) h n each time that φ(n) is one of the classical orthogonal polynomials: Jacobi, Laguerre, Hermite and Bessel.
43 citations
01 Sep 1983
TL;DR: In this paper, the eigenfunctions are given in terms of Bessel functions and the coefficients of integration as well as eigenvalues are determined accurately such that the boundary conditions are satisfied.
Abstract: The solution of the three-dimensional linear hydrodynamic equations which describe wind-driven flow in a homogeneous sea are solved using the eigenfunction method. The eddy viscosity is taken to vary piecewise linearly in the vertical over an arbitrary number of layers. Using this formulation the eigenfunctions are given in terms of Bessel functions. The coefficients of integration as well as the eigenvalues are determined accurately such that the boundary conditions are satisfied. Values of the eigenfunctions at any depth can then be determined very fast and to a high degree of accuracy. Current profiles at any position can hence be computed accurately. The expansion of the horizontal component of current converges very fast at all depths.
42 citations
TL;DR: In this article, vertical hydromagnetic waves are considered in isothermal and non-isothermal atmospheres, with a constant external magnetic field, in the cases of a transversal Alfven-gravity wave and a longitudinal magnetosonic gravity wave.
Abstract: Vertical hydromagnetic waves are considered in isothermal and non-isothermal atmospheres, with a constant external magnetic field, in the cases of a transversal Alfven-gravity wave and a longitudinal magnetosonic-gravity wave It is shown that in any atmosphere, for propagating waves the velocity perturbation grows linearly and the magnetic field perturbation is asymptotically constant, and for standing modes the velocity perturbation is finite and the magnetic field perturbation decays exponentially to zero These properties are confirmed for isothermal atmospheres, in which case all asymptotic parameters can be calculated in terms of Bessel or hypergeometric functions The magnetosonic-gravity wave evolves from a hydrodynamic regime similar to acoustic-gravity waves, through a transition layer, to a hydromagnetic regime similar to Alfven-gravity waves The latter exhibits hydromagnetic behaviour at all altitudes, which is illustrate by plotting the waveforms for the first four standing modes, and the amplitudes and phases of propagating waves of four different wavelengths
TL;DR: In this paper, the concavity of zeros of bessel functions is investigated and it is shown that zeros are concave in the sense that the concaveness of a zeros is a function of the number of elements in the function.
Abstract: (1983). On the concavity of zeros of bessel functions. Applicable Analysis: Vol. 16, No. 4, pp. 261-278.
TL;DR: In this article, the problem of finding a representation for the product Ym1l1(∇)Fm2l2(r) with Ym∇ specifying a solid harmonic whose argument is the nabla operator ∂/∂r instead of the vector r is reduced to the determination of the radial functions generated by the product.
Abstract: In this article, we analyze representations for the product Ym1l1(∇)Fm2l2(r) with Yml(∇) specifying a solid harmonic whose argument is the nabla operator ∂/∂r instead of the vector r. Since both Ym1l1(∇) and Fm2l2(r) are irreducible spherical tensors, we can use angular momentum algebra for evaluating the product. Accordingly, the problem of finding a representation for the product is reduced to the determination of the radial functions generated by the product. Analytical expressions for these radial functions are derived by direct differentiation and with the help of Fourier transforms. Closely related to the spherical tensor gradient Yml(∇) is the spherical delta function δml(r). We derive new representations for δml by considering convolution integrals involving B functions. These functions are closely related to the modified Bessel functions and also to the Yukawa potential e−αr/r. We show that the definition of the B functions can be extended to include a large class of derivatives of the delta func...
01 Nov 1983
TL;DR: It is a classical result in the theory of special functions that Bessel functions are limits in an appropriate sense of Legendre polynomials as discussed by the authors, which is also known for certain other special functions.
Abstract: It is a classical result in the theory of special functions that Bessel functions are limits in an appropriate sense of Legendre polynomials. For example in (11), § 17.4, the following result is attributed to Heine:such limiting formulae are also known for certain other special functions (cf. (1), (5)). Apart from their intrinsic interest, these formulae have been used in the theory of special functions to obtain product formulae, etc. for the limit function from those of the approximating sequence.
TL;DR: An approximate technique for the inversion of Laplace transforms is represented and some simple applications are given in this article, where the limitations of the method were explored and it was clear that functions which have oscillatory inverses present difficulty for the method.
Abstract: An approximate technique for the inversion of Laplace transforms is represented and some simple applications are given. The limitations of the method were explored and it is clear that functions which have oscillatory inverses present difficulty for the method. Better approximations result from the use of computers with longer word lengths and there is considerable improvement when an averaging algorithm is employed. Eight-bit microcomputers are generally sufficiently accurate for nonoscillatory time functions but transforms for lightly damped sinusoids and Bessel functions require large main-frame computers with relatively long word length.
TL;DR: In this paper, the authors present a method for calculating complete theoretical seismograms in earth models whose velocity, density and attenuation profiles are arbitrary piecewise-continuous functions of depth only.
Abstract: Summary. We present a method for calculating complete theoretical seismograms in earth models whose velocity, density and attenuation profiles are arbitrary piecewise-continuous functions of depth only. A form of attenuation valid for low loss situations is included by allowing the seismic velocities to be complex, and frequency is also allowed to be complex to avoid wraparound problems in the time-domain seismograms. Solutions for the stressdisplacement vectors in the medium are expanded in terms of orthogonal cylindrical functions. A seismic source is applied at the Earth’s surface and a radiation condition is applied at depth. The resulting two-point boundary value problem for the expansion coefficients is solved by a collocation technique which works best for those cases that other methods, e.g. propagator matrices, work most poorly, i.e. highly evanescent solutions. Solutions for the expansion coefficients are obtained in the depth, frequency and horizontal wavenumber domain. Phase velocity filtering may be effected at this point by restricting the portion of the frequency-wavenumber plane in which solutions are sought. The transformed strain tensor at depth is formed by taking linear combinations of the solutions. This strain tensor is transformed back into the space and time domain by successive application of a Bessel transform, a fast Fourier transform, and by multiplication of the time signal by a growing exponential to remove the exponential decay introduced by the use of complex frequency. The strain tensor is contracted with a seismic moment tensor, and a reciprocity relation for Green’s functions is used to obtain displacements at the Earth’s surface caused by a buried moment tensor source.
TL;DR: In this paper, the asymptotic expansion for a spectral function of the Laplacian operator involving geometrical properties of the domain is demonstrated by direct calculation for the case of a doubly-connected region in the form of a narrow annular membrane.
Abstract: The asymptotic expansion for a spectral function of the Laplacian operator, involving geometrical properties of the domain, is demonstrated by direct calculation for the case of a doubly-connected region in the form of a narrow annular membrane. By utilizing a known formula for the zeros of the eigenvalue equation containing Bessel functions, the area, total perimeter and connectivity are all extracted explicitly.
TL;DR: In this article, a modification of Ambarzumian's method was 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 with second order Legendre phase function scattering.
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 with second order Legendre phase function scattering. The incident radiation is collimated, normal to the top surface, and is dependent only on the radial coordinate. Boundary conditions which vary as a Bessel function and as a Gaussian distribution are investigated. The Gaussian distribution approximates a laser beam. Numerical results are presented in graphical and tabular forms for a Rayleigh scattering medium. The results are compared with those of isotropic scattering.
TL;DR: In this paper, a hydrogen-like wave function is expanded in terms of a series of hydrogenlike wave functions referred to an arbitrary center in space, and the expansion coefficient is given as a function of the distance between two centers.
Abstract: A hydrogenlike wave function is expanded in terms of a series of hydrogenlike wave functions referred to an arbitrary center in space. The expansion coefficient is given as a function of the distance between two centers. The coefficient for the radial part contains an integral of a spherical Bessel function with two Gegenbauer polynomials of different arguments. This integral is evaluated analytically to be a primitive polynomial multiplied by an exponential function of the distance. The coefficient for the angular part is represented in terms of the spherical harmonics.
TL;DR: In this article, the evaluation of Bessel functions of the first and second kinds, covering a wide range of complex arguments and integer orders, is required in the determination of the intensity of acoustic reflection from absorbing bodies.
TL;DR: In the small-angle x-ray diffraction pattern of the living relaxed anterior byssus retractor muscle of Mytilus edulis, the thin filaments showed the following features.
Journal Article•
TL;DR: This FORTRAN subroutine calculates sequences of modified Bessel functions of the first kind, I/sub n+a/(x), n = 0, 1, ..., NB - 1, where 0 less than or equal to a < 1, and 0 less more than orequal to x less than and equal to EXPARG or 0 less less thanor equal to xLess than or Equal to XLARGE, as appropriate.
Abstract: This FORTRAN subroutine calculates sequences of modified Bessel functions of the first kind, I/sub n+a/(x), n = 0, 1, ..., NB - 1 or e/sup -x/I/sub n+a/(x), n = 0, 1, ..., NB - 1, where 0 less than or equal to a < 1, and 0 less than or equal to x less than or equal to EXPARG or 0 less than or equal to x less than or equal to XLARGE, as appropriate. The method is based on the backward-recurrence algorithm of Miller as modified by Olver. Olver's contribution is an efficient method for determining the proper starting point for backward recurrence to meet a requested accuracy. The scheme uses a modified forward recurrence to perform Gaussian elimination on a set of simultaneous linear equations related to the usual backward-recurrence equations. The distance of the forward sweep, that is, the proper starting point for the backward recurrence, is determined dynamically from estimates of the corresponding error in the solution to the linear system. Backward substitution in these equations then returns the necessary function values with guaranteed error bounds. 7 references, 1 table.
TL;DR: In this article, a fully kinetic, non-local stability analysis of the Bennett equilibrium for a cylindrical plasma column is presented, where perturbations of the equilibrium distribution function constructed from the constants of motion may be divided into the adiabatic and non-adiabatic parts.
Abstract: A fully kinetic, non-local stability analysis of the Bennett equilibrium for a cylindrical plasma column is presented. The perturbations of the equilibrium distribution function constructed from the constants of motion may be divided into the adiabatic and non-adiabatic parts. The dominant trajectories of the particles are betatron orbits, and from an integration over these orbits the radial eigenmode equation is obtained. Expressing the electrostatic potential in terms of the vacuum eigenfunctions, i.e. the Bessel functions, yields a matrix dispersion relation. The wave-particle resonance in the analysis is spatially localized, owing to the betatron nature of the particle orbits. The eigenvalues of the problem are computed numerically.
TL;DR: In this paper, an oscillatory flow of a viscous incompressible fluid in an elastic tube of variable cross section has been investigated at low Reynolds number under the assumption that the variation of the cross-section is slow in the axial direction for a tethered tube.
Abstract: An oscillatory flow of a viscous incompressible fluid in an elastic tube of variable cross section has been investigated at low Reynolds number. The equations governing, the flow are derived under the assumption that the variation of the cross-section is slow in the axial direction for a tethered tube. The problem is then reduced to that of solving for the excess pressure from a second order ordinary differential equation with complex valued Bessel functions as the coefficients. This equation has been solved numerically for geometries of physiological interest and a comparison is made with some of the known theoretical and experimental results.
TL;DR: In this article, the problem of deconvolving the spectrum of ultrarelativistic source electrons from the observed photon spectrum is treated from the viewpoint of deconvolution of synchrotron radiation spectra.
Abstract: The problem of synchrotron radiation spectra is treated from the viewpoint of deconvolving the spectrum of ultrarelativistic source electrons from the observed photon spectrum. It is shown that for homogeneous sources the problem amounts to inversion of a Meijer transform with a modified Bessel finction kernel. A precise analytic inversion is only possible in the complex plane but Meijer transform tables are available for a wide range of functions. More convenient inversion formulae prove possible by use of a Laplace transform approximation or by analysing the spectra in terms of their integral moments. The filtering property of the transform is also established showing that the contribution to the synchrotron spectrum of high frequency components in the electron spectrum declines exponentially with their frequency. Thus, as with other Laplace-like transforms, only a few terms in an electron spectrum expansion can be deconvolved for any plausible noise level in the synchrotron spectrum.
TL;DR: In this article, the Bessel functions Jvk, yvk and cvk were extended to k = 2, 3, and 4, respectively, for the case of the first positive zero jvk.
TL;DR: In this paper, a recurrence relation is described for a certain ratio of Riccati-Bessel functions required for Mie scattering calculations, which improves the speed of calculation and yields similar accuracy to previous calculations.
Abstract: A recurrence relation is described for a certain ratio of Riccati-Bessel functions required for Mie scattering calculations. This improves the speed of calculation and yields similar accuracy to previous calculations.
TL;DR: In this article, the Hermite polynomials, the Bessel function of order ± 1/4 and the Airy function, were described by three exact solutions of the coupled nonlinear Schroedinger equation of Benney-Roskes.
Abstract: We have found new exact solutions of the coupled nonlinear Schroedinger equation of Benney-Roskes which is written as i u t -β u x x +γ u y y +δ u * u u -2 w u =0, β w x x +γ w y y -βδ( u * u ) x x =0. Three types of new solutions have been obtained. They are described by the Hermite polynomials, the Bessel function of order ±1/4 and ±1/3 (Airy function). Physically these solutions represent exploding and decaying solitons.
TL;DR: In this article, the first quadrant of the complex plane is divided into six sectors, and separate approximations are given for I z I 8 can also be used to evaluate the Bessel functions Yo(z), Y,(z) and the Hankel functions of the first and second kinds.
Abstract: Polynomial-based approximations for JO(z) and J1(z) are presented. The first quadrant of the complex plane is divided into six sectors, and separate approximations are given for I z I 8 can also be used to evaluate the Bessel functions Yo(z) and Y,(z) and the Hankel functions of the first and second kinds.