scholarly journals Some applications of Legendre numbers

1988 ◽  
Vol 11 (2) ◽  
pp. 405-412 ◽  
Author(s):  
Paul W. Haggard
Keyword(s):  
Legendre Polynomials ◽  
Legendre Functions ◽  
Linear Combinations ◽  
Associated Legendre Functions ◽  
In Series ◽  
Derivatives Of

The associated Legendre functions are defined using the Legendre numbers. From these the associated Legendre polynomials are obtained and the derivatives of these polynomials atx=0are derived by using properties of the Legendre numbers. These derivatives are then used to expand the associated Legendre polynomials andxnin series of Legendre polynomials. Other applications include evaluating certain integrals, expressing polynomials as linear combinations of Legendre polynomials, and expressing linear combinations of Legendre polynomials as polynomials. A connection between Legendre and Pascal numbers is also given.

scholarly journals Derivatives of addition theorems for Legendre functions

1995 ◽  
Vol 37 (2) ◽  
pp. 212-234 ◽  
Author(s):  
D.E. Winch ◽  
P.H. Roberts
Keyword(s):  
Spherical Harmonics ◽  
Chebyshev Polynomials ◽  
Legendre Polynomials ◽  
Addition Theorem ◽  
Legendre Functions ◽  
Addition Theorems ◽  
Associated Legendre Functions ◽  
Tensor Spherical Harmonics ◽  
Derivatives Of

AbstractDifferentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.

scholarly journals On Legendre numbers

1985 ◽  
Vol 8 (2) ◽  
pp. 407-411 ◽  
Author(s):  
Paul W. Haggard
Keyword(s):  
General Formula ◽  
Legendre Polynomials ◽  
Rational Numbers ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Infinite Set ◽  
Derivatives Of

The Legendre numbers, an infinite set of rational numbers are defined from the associated Legendre functions and several elementary properties are presented. A general formula for the Legendre numbers is given. Applications include summing certain series of Legendre numbers and evaluating certain integrals. Legendre numbers are used to obtain the derivatives of all orders of the Legendre polynomials atx=1.

XIII.—Some Integrals, with respect to their Degrees, of Associated Legendre Functions

1935 ◽  
Vol 54 ◽  
pp. 135-144 ◽  
Author(s):  
T. M. MacRobert
Keyword(s):  
Type A ◽  
Fourier Integral ◽  
Contour Integration ◽  
Legendre Functions ◽  
Integral Type ◽  
Integral Theorem ◽  
Associated Legendre Functions ◽  
In Series ◽  
Special Case

Little is known regarding the integration of Legendre Functions with respect to their degrees. In this paper several such integrals are evaluated, three different methods being employed. In § 2 proofs are given of a number of formulae which are required later. In § 3 an example is given of the evaluation of an integral by contour integration. The following section contains the proof of a formula of the Fourier Integral type, a special case of which was given in a previous paper (Proc. Roy. Soc. Edin., vol. li, 1931, p. 123). In § 5 an integral is evaluated by employing Fourier's Integral Theorem; while in § 6 other integrals are evaluated by. means of expansions in series.

scholarly journals Simplification of Geopotential Perturbing Force Acting on A Satellite

2008 ◽  
Vol 43 (2) ◽  
pp. 45-64 ◽  
Author(s):  
M. Eshagh ◽  
M. Abdollahzadeh ◽  
M. Najafi-Alamdari
Keyword(s):  
Maximum Degree ◽  
Legendre Functions ◽  
Orbital Elements ◽  
Associated Legendre Functions ◽  
Geopotential Models ◽  
Orbit Integration ◽  
Vector Differential Equation ◽  
Derivatives Of ◽  
Geopotential Coefficients ◽  
Short Arc

Simplification of Geopotential Perturbing Force Acting on A SatelliteOne of the aspects of geopotential models is orbit integration of satellites. The geopotential acceleration has the largest influence on a satellite with respect to the other perturbing forces. The equation of motion of satellites is a second-order vector differential equation. These equations are further simplified and developed in this study based on the geopotential force. This new expression is much simpler than the traditional one as it does not derivatives of the associated Legendre functions and the transformations are included in the equations. The maximum degree and order of the geopotential harmonic expansion must be selected prior to the orbit integration purposes. The values of the maximum degree and order of these coefficients depend directly on the satellite's altitude. In this article, behaviour of orbital elements of recent geopotential satellites, such as CHAMP, GRACE and GOCE is considered with respect to the different degree and order of geopotential coefficients. In this case, the maximum degree 116, 109 and 175 were derived for the Earth gravitational field in short arc orbit integration of the CHAMP, GRACE and GOCE, respectively considering millimeter level in perturbations.

V.—The Addition Theorem for the Legendre Functions of the Second Kind

1927 ◽  
Vol 46 ◽  
pp. 30-35
Author(s):  
T. M. MacRobert
Keyword(s):  
Positive Integer ◽  
Legendre Polynomials ◽  
Addition Theorem ◽  
Legendre Functions ◽  
Associated Legendre Functions

In a previous paper the author has employed certain formulæ of Dr Dougall's connecting the Associated Legendre Functions Pnm, where m is a positive integer and n is not integral, with the polynomials Ppm in which p is a positive integer, to deduce the Addition Theorem for the Legendre Functions of the first kind from the corresponding theorem for the Legendre Polynomials.

The complex wavenumber eigenvalues of Laplace’s tidal equations for oceans bounded by meridians

1995 ◽  
Vol 449 (1935) ◽  
pp. 51-64 ◽  
Keyword(s):  
Gravity Waves ◽  
Homogeneous System ◽  
Legendre Functions ◽  
Constant Depth ◽  
Eigenvalue Spectrum ◽  
Associated Legendre Functions ◽  
Complex Eigenvalues ◽  
Basin Scale ◽  
Tide Height ◽  
In Series

The complex wavenumber eigenvalues of Laplace’s tidal equations are determined for an ocean of constant depth bounded by meridians. A Galerkin method is used to expand the tide height and velocities in series of associated Legendre functions. A homogeneous system of equations results from the continuity and momentum equations. The frequency and depth are fixed, so that the meridional wavenumbers are the eigenvalues. This gives rise to a generalized eigenvalue problem that must be solved numerically by iteration. The eigenvalues are not integers and represent inertia-gravity wave solutions at the specified tidal forcing frequency that can be excited by the presence of meridional boundaries. Those complex eigenvalues represent solutions that decay away from meridional boundaries. The eigenvalue spectrum is investigated for the semi-diurnal, fortnightly, and monthly tides. One complex wavenumber for the semi-diurnal tide explains the amphidromic systems within 20° of a north-south coastline. The fortnightly and monthly tides have only real wavenumber eigenvalues. The basin scale deviation of these tides from equilibrium is attributed to low wavenumber divergent inertia-gravity waves.

Deformation of Open Spherical Shells Under Arbitrarily Located Concentrated Loads

1966 ◽  
Vol 33 (2) ◽  
pp. 305-312 ◽  
Author(s):  
J. P. Wilkinson ◽  
A. Kalnins
Keyword(s):  
Spherical Shell ◽  
Green’S Function ◽  
Green's Function ◽  
Driving Frequency ◽  
Legendre Functions ◽  
Spherical Shells ◽  
Concentrated Load ◽  
Associated Legendre Functions ◽  
Rotationally Symmetric ◽  
In Series

An exact solution is derived for the Green’s function of an open rotationally symmetric spherical shell subjected to any consistent boundary conditions. The fundamental singularity of the Green’s function is expanded in series according to the addition theorems of Legendre functions, and the solution for a spherical shell subjected to an arbitrarily situated, harmonically oscillating, normal, concentrated load is obtained explicitly in terms of associated Legendre functions. The corresponding static Green’s function is obtained simply by setting the driving frequency equal to zero. Numerical results for the displacements and stress resultants of an example are presented in detail.

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