Solid Spherical Harmonic

A solid spherical harmonic of degree n is a homogeneous function u_n in the rectangular coordinates x, y, and z that satisfies Laplace’s equation:

There are 2n + 1 linearly independent homogeneous polynomials in x, y, and z of integral positive degree n that are solid spherical harmonics. Their linear combination represents the general form of such a polynomial of degree n. For example, the general forms of homogeneous polynomials of degree 0, 1, and 2 that are solid spherical harmonics are represented by

u₀ = a

u₁ = ax – by + cz

u₂ = a(x² – z²) + b(y² – z²) + cxy + dyz + ezx

where a, b, c, d, and e are arbitrary constants.

If the spherical coordinates r, θ, and φ are introduced instead of the rectangular coordinates x, y, and z, then solid spherical harmonics may be expressed in terms of the spherical surface harmonics Y_n(θ, φ) according to the formula

u_n = rⁿY_n(θ, φ)

To every solid spherical harmonic u_n of degree n there corresponds a solid spherical harmonic r^{– 2n–1}u_n of degree – n – 1.

Solid spherical harmonics are used in finding the general solution of Laplace’s equation and in solving problems of mathematical physics for regions bounded by spherical surfaces.