In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions R 3 → C {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } . There are two kinds: the regular solid harmonics R ℓ m ( r ) {\displaystyle R_{\ell }^{m}(\mathbf {r} )} , which are well-defined at the origin and the irregular solid harmonics I ℓ m ( r ) {\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: R ℓ m ( r ) ≡ 4 π 2 ℓ + 1 r ℓ Y ℓ m ( θ , φ ) {\displaystyle R_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1}}}\;r^{\ell }Y_{\ell }^{m}(\theta ,\varphi )}
I ℓ m ( r ) ≡ 4 π 2 ℓ + 1 Y ℓ m ( θ , φ ) r ℓ + 1 {\displaystyle I_{\ell }^{m}(\mathbf {r} )\equiv {\sqrt {\frac {4\pi }{2\ell +1}}}\;{\frac {Y_{\ell }^{m}(\theta ,\varphi )}{r^{\ell +1}}}}
Derivation, relation to spherical harmonics Introducing r, θ, and φ for the spherical polar coordinates of the 3-vector r, and assuming that Φ {\displaystyle \Phi } is a (smooth) function R 3 → C {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } , we can write the Laplace equation in the following form
∇ 2 Φ ( r ) = ( 1 r ∂ 2 ∂ r 2 r − L ^ 2 r 2 ) Φ ( r ) = 0 , r ≠ 0 , {\displaystyle \nabla ^{2}\Phi (\mathbf {r} )=\left({\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}r-{\frac {{\hat {L}}^{2}}{r^{2}}}\right)\Phi (\mathbf {r} )=0,\qquad \mathbf {r} \neq \mathbf {0} ,}
where L2 is the square of the nondimensional angular momentum operator,
L ^ = − i ( r × ∇ ) . {\displaystyle \mathbf {\hat {L}} =-i\,(\mathbf {r} \times \mathbf {\nabla } ).}
It is known that spherical harmonics Ymℓ are eigenfunctions of L2:
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