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