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 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 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.

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.

scholarly journals Spatially Restricted Integrals in Gradiometric Boundary Value Problems

2009 ◽  
Vol 44 (4) ◽  
pp. 131-148 ◽  
Author(s):  
M. Eshagh
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Legendre Functions ◽  
Gravity Gradiometry ◽  
Satellite Gravity Gradiometry ◽  
Satellite Gravity ◽  
Earth’S Gravity Field ◽  
Associated Legendre Functions ◽  
Tensor Spherical Harmonics ◽  
Slepian Functions

Spatially Restricted Integrals in Gradiometric Boundary Value ProblemsThe spherical Slepian functions can be used to localize the solutions of the gradiometric boundary value problems on a sphere. These functions involve spatially restricted integral products of scalar, vector and tensor spherical harmonics. This paper formulates these integrals in terms of combinations of the Gaunt coefficients and integrals of associated Legendre functions. The presented formulas for these integrals are useful in recovering the Earth's gravity field locally from the satellite gravity gradiometry data.

scholarly journals The Asymptotic Expansions of the Spherical Harmonics

1922 ◽  
Vol 41 ◽  
pp. 82-93
Author(s):  
T. M. MacRobert
Keyword(s):  
Real Axis ◽  
Spherical Harmonics ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
The Real ◽  
Image Position

Associated Legendre Functions as Integrals involving Bessel Functions. Let,where C denotes a contour which begins at −∞ on the real axis, passes positively round the origin, and returns to −∞, amp λ=−π initially, and R(z)>0, z being finite and ≠1. [If R(z)>0 and z is finite, then R(z±)>0.] Then if I−m (λ) be expanded in ascending powers of λ, and if the resulting expression be integrated term by term, it is found that

On the expansion of a Coulomb potential in spherical harmonics

1950 ◽  
Vol 46 (4) ◽  
pp. 626-633 ◽  
Author(s):  
B. C. Carlson ◽  
G. S. Rushbrooke
Keyword(s):  
Spherical Harmonics ◽  
Coulomb Potential ◽  
Addition Theorem ◽  
Electrostatic Potentials ◽  
Legendre Functions ◽  
Expansion Formula ◽  
Image Position

The addition theorem for Legendre functions leads, as is well known, to a useful expansion formula of importance in the theory of electrostatic potentials,or, in an alternative notation,

Mathematical Aspects of Spectral Models

1998 ◽  
Author(s):  
T. N. Krishnamurti ◽  
H. S. Bedi ◽  
V. M. Hardiker
Keyword(s):  
Spherical Harmonics ◽  
Basis Functions ◽  
Spectral Model ◽  
Water Model ◽  
Data Sets ◽  
Legendre Functions ◽  
Global Spectral Model ◽  
Associated Legendre Functions ◽  
Spectral Models ◽  
Legendre Transforms

In this chapter we provide an introduction to the topic of spherical harmonics as basis functions for a global spectral model. The spherical harmonics are made up of trigonometric functions along the zonal direction and associated Legendre functions in the meridional direction. A number of properties of these functions need to be understood for the formulation of a spectral model. This chapter describes some useful properties that will be used to illustrate the procedure for the representation of data sets over a sphere with spherical harmonics as basis functions. The calculations of Fourier and Legendre transforms and their inverse transforms are an important part of global spectral modeling, and these are covered in some detail in this chapter. Finally, this chapter addresses the formulation of two simple spectral models. One of these is a single-level barotropic model, and the other is a shallow-water model.

THEORY OF MULTIPOLE RADIATIONS

1951 ◽  
Vol 29 (5) ◽  
pp. 393-402 ◽  
Author(s):  
P. R. Wallace
Keyword(s):  
Magnetic Field ◽  
Spherical Harmonics ◽  
Wave Length ◽  
Momentum Density ◽  
Systematic Analysis ◽  
Electric And Magnetic Field ◽  
Tensor Spherical Harmonics ◽  
Vector Identity ◽  
Derivatives Of ◽  
Flux Energy

We develop a systematic analysis of the radiation from a given oscillating system of charges and currents, without any approximations. Using a simple vector identity, the vector potential is separated into irrotational and solenoidal parts. The field may be expressed in terms of the latter alone. A similar vector identity involving the operator L = r × grad (the rotation operator) permits the separation of the field into parts in which the radial components of the electric and magnetic field, respectively, vanish. The energy flux, energy density, and angular momentum density may in each case be expressed in terms of the angular operators L, L2. Expansion in the eigenfunctions of these operators, the spherical harmonics, corresponds to the separation into electric and magnetic multipoles of all orders. Introduction of "tensor spherical harmonics" enables us to exhibit these radiations in terms of natural multipoles (derivatives of 1/r). All calculations are carried out without restriction as to size of radiating system relative to wave length, in the induction as well as the radiation region.

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