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 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 A Proof of the Addition Theorem for the Legendre Functions

1923 ◽  
Vol 42 ◽  
pp. 93-94
Author(s):  
T. M. MacRobert
Keyword(s):  
Positive Integer ◽  
Addition Theorem ◽  
Legendre Functions ◽  
Image Position

A Theorem of Dougall's. In Vol. XVIII. (p. 78) of these Proceedings Dr Dougall has established the theorem that, if m is a positive integer,

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.

scholarly journals On Neumann's Formula for the Legendre Functions

1952 ◽  
Vol 1 (1) ◽  
pp. 10-12 ◽  
Author(s):  
T. M. Macrobert
Keyword(s):  
Positive Integer ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Image Position

§1. Introductory. The formulawhere w is zero or a positive integer and | ζ | > 1, was given by F. E. Neumann “Crelle's Journal, XXXVII (1848), p. 24”. In § 2 of this paper some related formulae are given; the extension to the case when n is not integral is dealt with in § 3; while in § 4 the corresponding formulae for the Associated Legendre Functions when the sum of the degree and the order is a positive integer are established.

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.

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