Showing papers on "Legendre polynomials published in 1976"
TL;DR: In this paper, a new formula for π is derived, which is a direct consequence of Gauss' arithmetic-geometric mean, the traditional method for calculating elliptic integrals.
Abstract: A new formula for π is derived. It is a direct consequence of Gauss’ arithmetic-geometric mean, the traditional method for calculating elliptic integrals, and of Legendre’s relation for elliptic integrals. The error analysis shows that its rapid convergence doubles the number of significant digits after each step. The new formula is proposed for use in a numerical computation of π, but no actual computational results are reported here.
134 citations
TL;DR: In this article, the angular dependence of the ab initio rigid-rotor surface of Yarkony et al. is fitted with a form appropriate for future quantum and semiclassical scattering calculations, namely a triple series in Legendre polynomials in the molecule-molecule body-fixed angles.
Abstract: The angular dependence of the ab initio rigid‐rotor HF–HF surface of Yarkony et al [J. Chem. Phys. 60, 855 (1974)] is fitted with a form appropriate for future quantum and semiclassical scattering calculations, namely a triple series in Legendre polynomials in the molecule–molecule body–fixed angles. The grid of ab initio points is not dense enough to permit a precise fit to the surface, particularly at small center‐of‐mass separations. Nevertheless, a reasonably accurate fit, which also gives a good description of the HF dimer, can be attained with ∼30 angular terms. The final body‐fixed expansion coefficients were transformed to give 20 angular terms in a space‐fixed (SF) frame. For the six largest SF terms, exponentials of various arguments as well as inverse powers were used to fit the dependence on the center‐of‐mass scattering coordinate. A total of 18 parameters were involved. Asymptotically, the interaction potential goes to the expected dipole–dipole plus dipole–quadrupole form. A simple, empiric...
55 citations
TL;DR: In this article, exact formulas for all the terms in van der Waals C7 and C8 coeffcients for the interaction of an Sstate atom with a Σ-state linear molecule are derived The angle dependence is presented in convenient Legendre polynominal expansion form.
Abstract: Exact formulas for all the terms in the van der Waals C7 and C8 coeffcients for the interaction of an S‐state atom with a Σ‐state linear molecule are derived The angle dependence is presented in convenient Legendre polynominal expansion form Procedures for the calculation or estimation of the coefficients are proposed, discussed, and applied to CO2–noble gas atom interactions Simple formulas are found to give good bounds for the angle‐dependent part of C6 Although the uncertainties in the estimates of the higher coefficients are large, using them instead of C6 alone improves the long range potential considerably
54 citations
TL;DR: In this article, a new formula for evaluating the truncation coefficient was derived from recurrence relations of Legendre polynomials, which was conveniently processed by an electronic computer, providing the value ofQn up to a degreen=49 which are exactly equal to those of Paul (1973).
Abstract: In this paper, a new formula for evaluating the truncation coefficientQn is derived from recurrence relations of Legendre polynomials. The present formula has been conveniently processed by an electronic computer, providing the value ofQn up to a degreen=49 which are exactly equal to those of Paul (1973).
35 citations
TL;DR: In this paper, a technique to obtain the energy group-to-group transfer cross sections for neutron elastic scattering is presented, which avoids the traditional Legendre polynomial expansions and can be used directly in a discrete ordinates formulation of the transport equation to give accurate numerical results for those situations in which the group to group transfer is highly anisotropic.
Abstract: A technique to obtain the energy group-to-group transfer cross sections for neutron elastic scattering is presented. This method, which avoids the traditional Legendre polynomial expansions, can be used directly in a discrete ordinates formulation of the transport equation to give accurate numerical results for those situations in which the group-to-group transfer is highly anisotropic. 1 table, 2 figures. (auth)
22 citations
TL;DR: In this article, the Boltzmann equation for Landau quasiparticles is solved for T ∼ 0 by a specialization of a method discussed by Sykes and Brooker, and the authors show that the first-sound solution is consistent with hydrodynamics, exhibiting viscosity η=μ s τ, while the zero-sound velocityc1=[ϱ−1(B1+4/3μs)]1/2, so that phenomenologically zero sound propagates like a longitudinal elastic wave in a glass.
Abstract: The Boltzmann equation for Landau quasiparticles is solved for T ∼ 0 by a specialization of a method discussed by Sykes and Brooker. The quasiparticle distribution function is expanded in Legendre polynomials, assuming a boundary condition which imposes axial symmetry, and even-order terms are assumed to relax together with relaxation time τ e , odd-order terms with relaxation time τ o . By letting wavelength λ → ∞, with τ finite, one obtains a first-sound solution, and by lettingT → 0, and then λ → ∞, one obtains a zero-sound solution. When these solutions are used to calculate the pressure, it is found that the first-sound solution is consistent with hydrodynamics, exhibiting viscosity η=μ s τ, while the zero-sound velocityc1=[ϱ−1(B1+4/3μs)]1/2, so that phenomenologically zero-sound propagates like a longitudinal elastic wave in a glass. A higher zero-sound mode is also predicted, but is heavily damped. The heat flux is calculated and found to obey Vernotte's equation, which contains an intertial term, added to Fourier's law, that becomes significant asT → 0.
17 citations
TL;DR: In this article, it was shown that all known classes of low-pass filters with monotonic passband magnitude response (Butterworth, Legendre, LSM, and Halpern filters) are obtained as special cases of this type of approximation.
Abstract: New classes of low-pass filters with no finite zeros having a monotonic passband response are introduced by maximizing or minimizing the mean square error and changing the boundaries of the error integral. It is shown that all known classes of filters with monotonic passband magnitude response (Butterworth, Legendre, LSM, and Halpern filters) are obtained as special cases of this type of approximation.
13 citations
TL;DR: In this article, the authors deal with the approximate analysis of second order linear systems with variable coefficients through the application of orthogonal polynomials, such that the approximate differential equations thus obtained have known closed form solutions.
12 citations
TL;DR: In this paper, the authors proved the positivity of the coefficients of the expansion in Legendre polynomials of (σ A dt ) ( s, cos θ ), the absorptive contribution to the elastic unpolarized differential cross-section for particles with arbitrary spins.
11 citations
TL;DR: In this article, the authors presented the nomographs for additional classical filters, including ultraspherical, Legendre, modified associated Legendre and Papoulis, Halpern, Bessel, Gaussian and synchronously turned.
Abstract: This paper presents the nomographs for additional classical filters, including ultraspherical, Legendre, modified associated Legendre, Papoulis, Halpern, Bessel, Gaussian, and synchronously-turned. It also identifies inaccuracies in the earlier nomographs. The basic theory of nomographs and their utilization in developing filter nomographs is presented.
10 citations
TL;DR: In this paper, the critical rotating speed causing buckling of an axially prestressed rotating rod attached to the rim of a ring is investigated by using the extended Galerkin procedure with shape functions represented by a complete set of Legendre polynomials.
09 Feb 1976
TL;DR: An engineering interpretation of and some minor corrections to previously completed work on a uniform representation of the gravitational potential and its derivatives (with emphasis on avoiding the usual singularity at the pole) were presented in this paper.
Abstract: An engineering interpretation of and some minor corrections to previously completed work on a uniform representation of the gravitational potential and its derivatives (with emphasis on avoiding the usual singularity at the pole) were presented. The physical meaning of the variables was explained, the derivation of results was separated into smaller parts for easier reading, some additional recurrence relations for the derived Legendre polynomials were included and compared, and a computer program implementing this formulation was presented. Numerical experiments have shown that the use of this representation, besides removing the singularity, substantially increases the speed of the computation.
01 Jun 1976-Metallurgical and Materials Transactions B-process Metallurgy and Materials Processing Science
01 Oct 1976-Journal of Research of the National Bureau of Standards, Section B: Mathematical Sciences
TL;DR: In this article, the authors developed a uniform asymptotic theory of second-order linear differential equations with two coalescing simple turning points, and applied the results to the associated Legendre equation.
Abstract: Recently [2] 1, [3] , I developed a uniform asymptotic theory of second-order linear differential equations with two coalescing simple turning points, and applied the results to the associated Legendre equation_ Subsequently, in the course of writing a paper on connection formulas for multiple turning points [4] it became clear how to effect some improvements in two of the four cases treated in [2]. The purpose of the present note is to describe these improvements. It will be assumed that the reader is familiar with the results presented in [2], and the same notation will be used except where indicated otherwise. The next section introduces new auxiliary functions for the solutions of the modified Weber equation. The new form of the general approximation theorem is stated and discussed in the third (and concluding) section. An application of the results to the approximation of Whittaker functions with both parameters large will be published in due course.
TL;DR: In this paper, the time-dependent spatial-Legendre moments for the monoenergetic transport equation in spherical geometry are expressed in terms of the half and logarithmically weighted-legendre moments of the flux in plane geometry.
Abstract: The time-dependent spatial-Legendre moments for the monoenergetic transport equation in spherical geometry are expressed in terms of the half- and logarithmically weighted-Legendre moments of the flux in plane geometry. In cylindrical geometry, the moments are shown to be directly proportional to those in spherical geometry.
TL;DR: In this paper, a model-independent method of performing energy-dependent scattering amplitude analysis is presented, which makes use of previously developed two-variable expansions of scattering amplitudes but involves only summations and no integrals.
Abstract: A model-independent method of performing energy-dependent scattering- amplitude analysis is presented. It makes use of previously developed two- variable expansions of scattering amplitudes but involves only summations and no integrals. The energy and angle dependence are displayed in known functions. The angle, as usual, figures in Legendre polynomials; the energy is contained in Gegenbauer polynomials, if the data analysis is performed over a finite energy region and in specifically constructed basis functions when the analysis concerns the entire energy region. The purpose of the expansions is to make it possible to analyze all data for a given two-body reaction simultaneously (for all energies and angles) and to store the obtained information in the expansion coefficients. These then characterize the dynamics of a specific reaction, rather than a certain kinematic situation. (AIP)
TL;DR: In this article, the authors adapted the cranking model to the four dimentional ( ϵ 2, ϵ 3, ϵ 4 and ϵ 5 ) nuclear shape parametrization of the Nilsson model.
TL;DR: In this paper, it was shown that up to six redundancies can occur if reactions initiated by tensor polarized particles are described in terms of associated Legendre polynomials.
TL;DR: In this paper, a method for calculating the asymptotic diffusion lengths (the so-called discrete eigenvalues) without use of the usual Legendre expansions is presented for the case of one speed particle transport in a highly anisotropic scattering medium.
TL;DR: In this article, the buckling loads of cantilever bars supported laterally by an elastic foundation were determined by using the Legendre polynomials in conjunction with the Galerkin procedure.
Abstract: Legendre polynomials in conjunction with the Galerkin procedure are used to determine the buckling loads of cantilever bars supported laterally by an elastic foundation. The numerical results of example problems converge rapidly, and the use of the Legendre polynomials to represent the buckled deformation is found to be very desirable. Although the example problem considers only the axial compression to vary along the length of the bar, the general procedure can be extended to account for cases having variable cross sections or foundation stiffness, or both.
TL;DR: In this paper, the impression of two identical axisymmetric stamps in an elastic sphere is considered, where the surface of the sphere outside the stamps is stress-free, while there are no shear stresses under the stamps.
01 Oct 1976
01 Mar 1976
TL;DR: In this article, the Dougall's identities for Legendre associated functions were extended to give several new identities for non-integral order Legendre functions of integral order and degree, and the uniform convergence of each new identity is considered in detail.
Abstract: The identities for Legendre associated functions $P_v^m (x)$ of nonintegral order v, known as Dougall’s identities, are extended to give several new identities for Legendre associated functions $P_n^m (x)$ of integral order and degree. Such identities are required in the simplification and evaluation of expansions arising from the use of Green’s functions. The uniform convergence of each new identity is considered in detail.
Journal Article•
TL;DR: In this article, a solution to the motion of a satellite in an eccentric orbit and in resonance with one or more of the longitude-dependent harmonics of the central planet is developed.
Abstract: The solution to the motion of a satellite in an eccentric orbit and in resonance with one or more of the longitude-dependent harmonics of the central planet is developed. The method of solution parallels the well known von Zeipel method of general perturbations. The solution consists of expressions for the variations of the Delaunay variables. These expressions are composed of the perturbations developed by Brouwer in 1959 for the motion of an artificial satellite plus first-order resonant perturbations due to longitude-dependent harmonics (in terms of Legendre normal elliptic integrals of the first and second kind).
01 May 1976
TL;DR: In this article, a Ge(Li) detector and a fission detector were used to measure elemental differential cross section excitation functions for fast-neutron gamma-ray production from iron relative to fast neutron fission of /sup 235/U.
Abstract: A Ge(Li) detector and a fission detector were used to measure elemental differential cross section excitation functions for fast-neutron gamma-ray production from iron relative to fast-neutron fission of /sup 235/U. Data were acquired at approximately 50 keV intervals with approximately 50 keV neutron-energy resolution from near threshold to approximately 2 MeV. Angular distributions for the 0.847-MeV gamma ray were measured at 0.93, 0.98, 1.08, 1.18, 1.28, 1.38, 1.59, 1.68, 1.79, 1.85 and 2.03 MeV. Significant fourth-order terms were required for the Legendre polynomial expansions used in fitting several of these angular distributions. This casts doubt on the accuracy of the commonly used approximation that the integrated gamma-ray production cross section is essentially equal to 4..pi.. times the 55-degree (or 125-degree) differential cross section. The method employed in processing these data is described. Comparison is made between results from the present work and some previously reported data sets. The uncertainties associated with energy scales, neutron-energy resolution and other experimental factors for these various measurements make it difficult to draw conclusions concerning the observed differences in the values reported for these fluctuating cross sections.