Search results
Results From The WOW.Com Content Network
Using the orthonormality properties of the real unit-power spherical harmonic functions, it is straightforward to verify that the total power of a function defined on the unit sphere is related to its spectral coefficients by a generalization of Parseval's theorem (here, the theorem is stated for Schmidt semi-normalized harmonics, the ...
Below the real spherical harmonics are represented on polar plots. The magnitude of the spherical harmonic at particular polar and azimuthal angles is represented by the saturation of the color at that point and the phase is represented by the hue at that point. Visual Array of Real Spherical Harmonics Represented with Polar Plot
In many applications, vector spherical harmonics are defined as fundamental set of the solutions of vector Helmholtz equation in spherical coordinates. [6] [7] In this case, vector spherical harmonics are generated by scalar functions, which are solutions of scalar Helmholtz equation with the wavevector .
Unlike ordinary spherical harmonics, the spin-weighted harmonics are U(1) gauge fields rather than scalar fields: mathematically, they take values in a complex line bundle. The spin-weighted harmonics are organized by degree l , just like ordinary spherical harmonics, but have an additional spin weight s that reflects the additional U(1) symmetry.
A weakly harmonic function coincides almost everywhere with a strongly harmonic function, and is in particular smooth. A weakly harmonic distribution is precisely the distribution associated to a strongly harmonic function, and so also is smooth. This is Weyl's lemma. There are other weak formulations of Laplace's equation that are often useful.
The zonal spherical harmonics are rotationally invariant, meaning that () = () for every orthogonal transformation R.Conversely, any function f(x,y) on S n−1 ×S n−1 that is a spherical harmonic in y for each fixed x, and that satisfies this invariance property, is a constant multiple of the degree ℓ zonal harmonic.
The spherical harmonic functions form a complete orthonormal set of functions in the sense of Fourier series. Workers in the fields of geodesy, geomagnetism and spectral analysis use a different phase and normalization factor than given here (see spherical harmonics ).
Toggle Relation to spherical harmonics; Gaunt coefficients subsection ... which is a one-to-one correspondence between it and a 3-j symbol and assumes the properties ...