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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 complex spherical harmonics are represented on 2D plots with the azimuthal angle, , on the horizontal axis and the polar angle, , on the vertical axis.The saturation of the color at any point represents the magnitude of the spherical harmonic and the hue represents the phase.
If instead of the Helmholtz equation, the Laplace equation is solved in spheroidal coordinates using the method of separation of variables, the spheroidal wave functions reduce to the spheroidal harmonics. With oblate spheroidal coordinates, the solutions are called oblate harmonics and with prolate spheroidal coordinates, prolate harmonics.
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 k {\displaystyle \mathbf {k} } .
In mathematics and mathematical physics, Slater integrals are certain integrals of products of three spherical harmonics. They occur naturally when applying an orthonormal basis of functions on the unit sphere that transform in a particular way under rotations in three dimensions. Such integrals are particularly useful when computing properties ...
Here, properties of tensors, including high-rank moments as well, are considered to repeat basically features of solid spherical functions but having their own specifics. Using of invariant polynomial tensors in Cartesian coordinates , as shown in a number of recent studies, is preferable and simplifies the fundamental scheme of calculations ...
This gravitational model is complete to spherical harmonic degree and order 2159 (block diagonal), and contains additional coefficients extending to degree 2190 and order 2159. It provides a raster of 2.5′×2.5′ and an accuracy approaching 10 cm. 1'×1' is also available [ 7 ] in non-float but lossless PGM , [ 5 ] [ 8 ] but original .gsb ...
With the spherical-harmonic addition theorem the equation can be rewritten as = = = (^) (^), where . Y ℓ m are the spherical harmonics and; the superscript * denotes complex conjugation.