Solid harmonics

In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions $\mathbb {R} ^{3}\to \mathbb {C}$ . There are two kinds: the regular solid harmonics $R_{\ell }^{m}(\mathbf {r} )$ , which are well-defined at the origin and the irregular solid harmonics $I_{\ell }^{m}(\mathbf {r} )$ , 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:

R_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1}}}\;r^{\ell }Y_{\ell }^{m}(\theta ,\varphi )

I_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1}}}\;{\frac {Y_{\ell }^{m}(\theta ,\varphi )}{r^{\ell +1}}}

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