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The table of spherical harmonics contains a list of common spherical harmonics. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis , every function defined on the surface of a sphere can be written as a sum of these spherical harmonics.
This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree =. Some of these formulas are expressed in terms of the Cartesian expansion of the spherical harmonics into polynomials in x , y , z , and r .
The descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as harmonics .
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]
The potential V(R) at a point R outside the charge distribution, i.e. | R | > r max, can be expanded by the Laplace expansion: = | | = = = () = (), where () is an irregular solid harmonic (defined below as a spherical harmonic function divided by +) and () is a regular solid harmonic (a spherical harmonic times r ℓ). We define the spherical ...
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} } .
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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.