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 R ℓ m ( r ) {displaystyle R_{ell }^{m}(mathbf {r} )} , which vanish 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: 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 R ℓ m ( r ) {displaystyle R_{ell }^{m}(mathbf {r} )} , which vanish 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: 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, It is known that spherical harmonics Yml are eigenfunctions of l2: Substitution of Φ(r) = F(r) Yml into the Laplace equation gives, after dividing out the spherical harmonic function, the following radial equation and its general solution, The particular solutions of the total Laplace equation are regular solid harmonics: and irregular solid harmonics: Racah's normalization (also known as Schmidt's semi-normalization) is applied to both functions