scholarly journals Remarks on the Zeros of the Associated Legendre Functions with Integral Degree

2007 ◽  
Vol 99 (3) ◽  
pp. 293-299
Author(s):  
J. F. van Diejen
Keyword(s):  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Integral Degree

scholarly journals On the Reduction of Ferrers' Associated Legendre Function

1929 ◽  
Vol 1 (4) ◽  
pp. 241-243
Author(s):  
Hrishikesh Sircar
Keyword(s):  
Finite Number ◽  
Infinite Series ◽  
Infinite Number ◽  
Legendre Function ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Reduced Order ◽  
Zonal Harmonics ◽  
Integral Degree ◽  
Associated Legendre Function

Introduction. In the present paper a formula will be obtained to express a Ferrers' Associated Legendre Function of any integral degree and order as a sum of a finite number of Associated Legendre Functions of an order reduced by an even number. When the order is reduced by unity, an infinite series of the functions of reduced order is required. Thus a Ferrers' function can be expressed as the sum of a finite or infinite number of zonal harmonics according as the order of the function is even or odd.

The Analysis of the Associated Legendre Functions with Non-Integral Degree

2011 ◽  
Vol 130-134 ◽  
pp. 3001-3005
Author(s):  
Jian Qiang Wang ◽  
Hao Yuan Chen ◽  
Yin Fu Chen
Keyword(s):  
Spherical Harmonic ◽  
Legendre Function ◽  
Scalar Potential ◽  
Regional Model ◽  
Legendre Functions ◽  
Spherical Cap ◽  
Associated Legendre Functions ◽  
Core Content ◽  
Local Gravity ◽  
Integral Degree

Spherical cap harmonic (SCH) theory has been widely used to format regional model of fields that can be expressed as the gradient of a scalar potential. The functions of this method consist of trigonometric functions and associated Legendre functions with integral-order but non-integral degree. Evidently, the constructing and computing of Legendre functions are the core content of the spherical cap functions. In this paper,the approximated calculation method of the normalized association Legendre functions with non-integral degree is introduced and an analysis of the entire order of associated non-Legendre function calculation is presented. Besides, we use the Muller method to search out for all intrinsic values. The results showed that the highest order of spherical harmonic function for constructing regional model of fields is limited, thus high-resolution spherical harmonic structure of local gravity field need to be improved.

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 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

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