Showing papers on "Legendre polynomials published in 1971"
TL;DR: In this paper, the shape of the transuranium nucleus in the zeroth-order approximation is taken to be the Cassinian ovaloid, the deviation being expanded in a series of Legendre polynomials.
255 citations
TL;DR: In this paper, the steady motion of a viscous fluid contained between two concentric spheres which rotate about a common axis with different angular velocities is considered, and a high-order analytic perturbation solution, through terms of order Re7, is obtained for low Reynolds numbers.
Abstract: The steady motion of a viscous fluid contained between two concentric spheres which rotate about a common axis with different angular velocities is considered. A high-order analytic perturbation solution, through terms of order Re7, is obtained for low Reynolds numbers. For larger Reynolds numbers an approximate Legendre polynomial series representation is used to reduce the governing system of equations to a non-linear ordinary differential equation boundary-value problem which is solved numerically. The resulting flow pattern and the torque required to rotate the spheres are presented for various cases considered.
85 citations
TL;DR: The results were fitted with Legendre polynomials as discussed by the authors and the energy dependence of the resulting Legendre coefficients indicate resonance-like behaviour at energies corresponding to excitations of 16.6 MeV, 17.5 MeV and 11.4 MeV at lab angles between 0° and 160°.
40 citations
TL;DR: Differential cross sections for the 2 H(γ, p)n reaction were measured at seven laboratory angles from 30° to 150° for photon energies between 25 and 43 MeV, with additional results up to 55 MeV at the larger angles as mentioned in this paper.
38 citations
TL;DR: In this paper, the angular distribution coefficients for photonuclear and radiative capture reactions in which target and residual spins are ≤ 5/2 are presented in tabular form for angularmomentum theory in nuclear reactions.
24 citations
TL;DR: In this article, the authors measured the anomalous levels of neutron polarization at proton energies 3.0, 3.5, 4.0 and 5.5 MeV.
18 citations
TL;DR: In this article, the Boltzmann equation for electrons moving in a neutral gas under the influence of an externally applied field is solved by expanding the electron distribution function in terms of Legendre and Sonine polynomials.
Abstract: The Boltzmann equation for electrons moving in a neutral gas under the influence of an externally applied field is solved by expanding the electron distribution function in terms of Legendre and Sonine polynomials. The solution is given in terms of infinite matrices which have elements ordered by the Sonine polynomial index, and which are dependent upon the field strength. From the structure of the formulae, it is possible to infer that truncation of the Legendre polynomial expansion after two terms is a good approximation at all field strengths. This is supported by calculations of the electron drift velocity at low field strengths, which show that the error introduced by making the two-term approximation is small, even when the deviation from equilibrium is significant. The convergence of the Sonine polynomial expansion is shown to be strongly depende:r;J.t upon field strength, and large matrices are required in the drift velocity formula at even small field strengths.
13 citations
TL;DR: In this article, a superposition of single center static and nonspherical polarization potentials is used to approximate the elastic positron-hydrogen collision in Legendre polynomials.
Abstract: Elastic positron–hydrogen molecule collisions are considered below the first electronic excitation threshold. The interaction of the incoming positron with the hydrogen molecule is represented by a superposition of single center static and nonspherical polarization potentials. The static potential is expanded to second degree in Legendre polynomials. The Chebyshev polynomials are used to improve the truncated Legendre series approximation and the effect of including the fourth degree contribution is shown to be slight. Positronium formation effects are neglected. Two semiphenomenological polarization potentials are used, and it is shown that the parameters can be suitably adjusted to reproduce the results obtained if the Henry and Lane polarization potential is used. It is shown that the partial Ramsauer–Townsend effect is useful in determining the values of polarization potential parameters. Total elastic scattering cross sections are given for incoming positron energies between about 0.4 and 14 eV while...
10 citations
TL;DR: In this paper, the angular distribution of gamma rays from the reaction 12 C(α,γ) 16 O in the energy region E α = 7.6-8.1 MeV were measured with good accuracy.
9 citations
TL;DR: In this paper, the authors considered the inverse problem of scattering for a spherical vector scattering geometry, where the transverse scattered field components are related to the expansion coefficients of Hansen's vector wave expansion in terms of the scattered field matrix.
Abstract: The inverse problem of scattering for a spherical vector scattering geometry is considered. The transverse scattered field components are related to the expansion coefficients of Hansen's vector wave expansion in terms of the scattered field matrix. This matrix needs to be inverted to obtain the unknown expansion coefficients which are required to recover the shape of the target in question. Since the particular properties of the spherical vector wave expansion may cause highly instable matrix inversion, an analytical, closed form solution of the determinant associated with the scattered field matrix was sought. For vector scattering geometries representing the mth degree multipole cases such closed form solutions for the associated determinant of truncated order 2N are derived, using a novel complementary series expansion for the employed forms of the associated Legendre's functions of the first kind. A novel determinate optimization procedure is presented which enables the specification of the optimal d...
8 citations
TL;DR: In this paper, cubature error bounds for the product rule generated by two Gauss-Legendre quadrature rules are derived for monospline kernels in the two-dimensional form of Peano's theorem.
Abstract: In this paper we derive cubature error bounds for the product rule generated by two Gauss-Legendre quadrature rules. An analogue to the fundamental theorem of algebra, for monosplines, is used to characterize the kernels in the two-dimensional form of Peano’s theorem.Tables are given which can be used to compute the error coefficients we obtain.
01 Jan 1971
TL;DR: In this article, the authors considered the inverse problem of scattering for a spherical vector scattering geometry, where the transverse scattered field components are related to the expansion coefficients of Hansen's vector wave expansion in terms of the scattered field matrix.
Abstract: The inverse problem of scattering for a spherical vector scattering geometry is considered. The transverse scattered field components are related to the expansion coefficients of Hansen's vector wave expansion in terms of the scattered field matrix. This matrix needs to be inverted to obtain the unknown expansion coefficients which are required to recover the shape of the target in question. Since the particular properties of the spherical vector wave expansion may cause highly instable matrix inversion, an analytical. closed form solution of the determinant associated with the scattered field matrix was sought. For vector scattering geometries representing the mth degree multipole cases such closed form solutions for the associated determinant of truncated order 2N are derived, using a novel complementary series expansion for the employed forms of the associated Legendre's functions of the firstkind. A novel determinate optimization procedure is presented which enables the specification of the optimal distribution of measurement aspect angles within any given finite measurement cone of the unit sphere of directions. The closed form solution for nonsymmetrical vector scattering geometries is presented in Appendix 111 only for the value N = 3 (m = 0 and 1) employing properties of quadratic forms as derived in Appendix 11. It is then shown that the electrical radius ka of a perfectly conducting spherical scatterer can be directly recovered from a finite number of contiguous expansion coefficients similar to the cylindrical case presented in Boerner, Vandenberghe, and Hamid. Furthermore, relationships between contiguous expansion coefficients of both electric and magnetic type result, which are relevant to the general inverse problem since the scattered field can be uniquely expressed in terms of only one set of expansion coefficients associated with either the electric or magnetic vector wave functions.
TL;DR: The coefficients of expansion of an arbitrary analytic function of the distance r between two points (r1, θ1, φ1) in terms of the Legendre polynomials Pl(cos θ12) are double Bessel transformed as discussed by the authors.
Abstract: The coefficients of expansion of an arbitrary analytic function of the distance r between two points (r1, θ1, φ1) and (r2, θ2, φ2) in terms of the Legendre polynomials Pl(cos θ12) are double Bessel transformed. Assuming that the transformed coefficients are diagonal, consistent with the differential equations satisfied by the original coefficients, we derive the explicit expressions for the latter coefficients. These formulas are identical to those derived by Sack from the solutions of the differential equations in terms of the hypergeometric functions.
TL;DR: In this paper, a solution for the potential due to an inner circular disk at a constant potential surrounded by an annualar ring at another constant potential is given, expressed both as an elliptic integral and as an expansion in Legendre polonomials.
Abstract: : A solution for the potential due to an inner circular disk at a constant potential surrounded by an annualar ring at another constant potential is given. The general solution is expressed both as an elliptic integral and as an expansion in Legendre polonomials. The case in which the two conductors are separated by a thin gap is also considered. An application of the solution to the collecting properties of a plasma probe is discussed. (Author)
TL;DR: In this article, the angular distribution of σ0 and of the product σ 0Ay were fitted with Legendre polynomials, and it was shown that more than one energy level of the 5Li compound nucleus is responsible for the broad anomaly in the cross section of 3He(d, p)4He reaction, observed at an excitation energy of ∼ 20 MeV in 5Li.
TL;DR: In this article, the angular distributions of the α0 and α1 groups from the 27Al(d, α)25Mg nuclear reaction have been measured at deuteron energiesE =650, 585 and 540 keV using plastics track detector techniques.
Abstract: The angular distributions of the α0 and α1 groups from the27Al(d, α)25Mg nuclear reaction have been measured at deuteron energiesE
d=650, 585 and 540 keV using plastics track detector techniques. The angular distributions which are nearly isotropic have been analysed in terms of the Legendre polynomials. Assuming statistical compound reaction mechanism the relative intensity ratio of the two measured alpha groups could be reproduced by a simple calculation giving the statistical weight factors for the alpha transitions concerned.
TL;DR: In this paper, a variational determination of the angular form of the pair density rho (2,r) at the surface of liquid argon at the triple point is given, which allows several of the thermodynamic functions to be expressed directly in terms of the hybridization of the harmonics.
Abstract: A variational determination of the angular form of the pair density rho (2)(z,r) at the surface of liquid argon at the triple point is given. It is found that a description in terms of the first and second unassociated Legendre harmonic functions allows several of the thermodynamic functions to be expressed directly in terms of the hybridization of the harmonics. The harmonic mode analysis of the pair density may be simply extended to more complex conditions of atomic anisotropy.
TL;DR: In this article, the best set of angles and corresponding weights of observations was determined by minimizing the variance for the case of W ( ϑ ) = A 0 + A 2 P 2 (cos ϑ)+ A 4 P 4 (cons ϑ ).
TL;DR: In this article, a density expansion technique for treating k-nad ω-dependent current-density correlation functions is discussed and carried out for dilute gases, and the results are compared to forms obtained from the Navier-Stokes and the Burnett equations.
TL;DR: In this article, the combined nonlinear and collisional effects on the Landau damping of longitudinal electron oscillations are studied using the Lorentz gas model to represent electron neutral and electron ion interactions.
Abstract: The combined nonlinear and collisional effects on the Landau damping of longitudinal electron oscillations are studied using the Lorentz gas model to represent electron‐neutral and electron‐ion interactions. Three types of interactions are considered: hard‐sphere collisions, inverse fifth‐power interactions, and inverse square interactions. A perturbation analysis of the nonlinear kinetic equation is carried out following Montgomery's expansion technique. For hard‐sphere collisions, the collisional damping and the nonlinear change in Landau damping are determined analytically. For inverse fifth‐power interactions and inverse square interactions, the same results are obtained numerically by expanding the distribution function in Legendre polynomials to sufficiently high order to adequately represent its variations in the resonance region.
TL;DR: In this article, the exposure angular distributions of scattered gamma rays at points along the axis of plane-disk isotropic 60Co sources, imbedded in an infinite air medium (air density = 1.293 g/liter), have been calculated using the moments method solution to the gamma-ray transport equation.
Abstract: Exposure angular distributions of scattered gamma rays at points along the axis of plane-disk isotropic 60Co sources, imbedded in an infinite air medium (air density = 1.293 g/liter), have been calculated using the moments method solution to the gamma-ray transport equation. The method is based on the Legendre-moments transformation of the transport equation for scattered energy flux density at a height z above an infinite-plane isotropic source. Coefficients of the Legendre expansions were reconstructed using standard biorthogonal polynomial techniques. An extrapolation technique is developed to extend the number of Legendre coefficients to smooth resulting distributions. Results are given for disks of radius 100 ft to infinity at detector heights of 3 to 1000 ft above the source plane.
TL;DR: In this article, the angular distribution and polarizations of the two-body final states of the K−p system were investigated at eight incident K− momenta from 450 to 670 MeV/c.
Abstract: Angular distributions and polarizations of the two-body final states of the K−p system have been investigated at eight incident K− momenta from 450 to 670 MeV/c. The measured Legendre coefficients appear to be smooth functions of the energy, an upper limit of 5% for the elasticity of the Σ(1616) has been estimated. The result of a multichannel partial-wave analysis is also discussed.
01 Jan 1971
TL;DR: In this paper, the authors studied analogous integrals in which the contours of integration are indeterminate, called incomplete cylindrical functions, by analogy to incomplete elliptic integrals of Legendre or to the incomplete gamma function.
Abstract: Basic information about the cylindrical functions of Bessel, Neumann, and Hankel was given in the preceeding chapter. These functions are all solutions of Bessel’s differential equation, and have integral representations in the Poisson and Bessel forms for which the contours of integration are completely determined. In practice, however, it frequently becomes necessary to study analogous integrals in which the contours are indeterminate. Such functions, by analogy to the incomplete elliptic integrals of Legendre or to the incomplete gamma function [12], may be called incomplete cylindrical functions.
TL;DR: For any continuous univariate population with finite variance there is a mathematical relation which expresses the variate-value z as a convergent series of Legendre polynomials in (2F-1), where F≡F(z) is the distribution function of the population, and the coefficients in this series are the expectations of homogeneous linear forms in the order statistics of random samples from the population as mentioned in this paper.
Abstract: For any continuous univariate population with finite variance there is a mathematical relation which expresses the variate-value z as a convergent series of Legendre polynomials in (2F—1), where F≡F(z) is the distribution function of the population, and the coefficients in this series are the expectations of homogeneous linear forms in the order statistics of random samples from the population. The relation is well adapted for estimating the median and other percentile points when nothing more is known about the population, but a random sample from it is available. The variances of these estimates can be estimated from the data. A somewhat similar relation which expresses z as a series of Chebyshev polynomials is also discussed briefly. Finally a modification of the Legendre polynomial relation enables prior knowledge of a finite extremity of the population range to be used, and a numerical illustration is given.
TL;DR: The sums of series of Legendre polynomials can be reduced to quadratures and on this basis the properties of these sums are investigated in this article, where they are shown to be quadratic.
Abstract: The sums of series of Legendre polynomials can be reduced to quadratures and on this basis the properties of these sums are investigated.