Solid harmonics

In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates. There are two kinds: the regular solid harmonics ${\displaystyle R_{\ell }^{m}(\mathbf {r} )}$, which vanish at the origin and the irregular solid harmonics ${\displaystyle I_{\ell }^{m}(\mathbf {r} )}$, which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately:

Introducing r, θ, and φ for the spherical polar coordinates of the 3-vector r, we can write the Laplace equation in the following form

where l² is the square of the nondimensional angular momentum operator,

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