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The statement of the parity of spherical harmonics is then (,) (, +) = (,) (This can be seen as follows: The associated Legendre polynomials gives (−1) ℓ+m and from the exponential function we have (−1) m, giving together for the spherical harmonics a parity of (−1) ℓ.)
The functions , (,) are the spherical harmonics, and the quantity in the square root is a normalizing factor. Recalling the relation between the associated Legendre functions of positive and negative m , it is easily shown that the spherical harmonics satisfy the identity [ 5 ]
Legendre wavelet. In functional analysis, compactly supported wavelets derived from Legendre polynomials are termed Legendre wavelets or spherical harmonic wavelets. [1] Legendre functions have widespread applications in which spherical coordinate system is appropriate. [2][3][4] As with many wavelets there is no nice analytical formula for ...
The Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre [3] as the coefficients in the expansion of the Newtonian potential where r and r′ are the lengths of the vectors x and x′ respectively and γ is the angle between those two vectors. The series converges when r > r′.
The amplitude of the spherical harmonic (magnitude and sign) at a particular polar and azimuthal angle is represented by the elevation of the plot at that point above or below the surface of a uniform sphere. The magnitude is also represented by the saturation of the color at a given point. The phase is represented by the hue at a given point.
The general Legendre equation reads ″ ′ + [(+)] =, where the numbers λ and μ may be complex, and are called the degree and order of the relevant function, respectively. . The polynomial solutions when λ is an integer (denoted n), and μ = 0 are the Legendre polynomials P n; and when λ is an integer (denoted n), and μ = m is also an integer with | m | < n are the associated Legendre ...
Geopotential spherical harmonic model. This article includes a , , or , but its sources remain unclear because it lacks . In geophysics and physical geodesy, a geopotential model is the theoretical analysis of measuring and calculating the effects of Earth 's gravitational field (the geopotential). The Earth is not exactly spherical, mainly ...
It was introduced in 1927 by Eugene Wigner, and plays a fundamental role in the quantum mechanical theory of angular momentum. The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The letter Dstands for Darstellung, which means "representation" in German.