Tensor spherical harmonics and tensor multipoles. I. Euclidean space

1976 ◽  
Vol 17 (10) ◽  
pp. 1903-1909 ◽  
Author(s):  
M. Daumens ◽  
P. Minnaert
Keyword(s):  
Euclidean Space ◽  
Spherical Harmonics ◽  
Tensor Spherical Harmonics

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.

Eigenvalues and degeneracies for n‐dimensional tensor spherical harmonics

1984 ◽  
Vol 25 (10) ◽  
pp. 2888-2894 ◽  
Author(s):  
Mark A. Rubin ◽  
Carlos R. Ordóñez
Keyword(s):  
Spherical Harmonics ◽  
Tensor Spherical Harmonics

Scalar, vector and tensor spherical harmonics

2013 ◽  
pp. 612-615
Author(s):  
Thomas W. Baumgarte ◽  
Stuart L. Shapiro
Keyword(s):  
Spherical Harmonics ◽  
Tensor Spherical Harmonics

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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