Surface harmonics and solid harmonics

A surface harmonic of degree $n$ (where $n$ is a non-negative integer), denoted ${\cal S}_n(\theta,\phi)$, is defined as a well-behaved solution to

$$r^{\,2}\,\nabla^{\,2} {\cal S}_n + n\,(n+1)\,{\cal S}_n = 0$$ (C.1)

on the surface of a sphere (i.e., $r=$ constant). Here, $r$, $\theta$, $\phi$ are standard spherical coordinates. It follows that

$${\cal S}_n(\theta,\phi) = \sum_{m=-n,+n}\,\alpha_{n,m}\,Y_n^{\,m}(\theta,\phi),$$ (C.2)

where the $\alpha_{n,m}$ are arbitrary coefficients, and the $Y_n^{\,m}(\theta,\phi)$ are spherical harmonics. (See Section A.12.)

A solid harmonic of degree $n$ (where $n$ is a non-negative integer), denoted ${\cal R}_n(r,\theta,\phi)$, is defined as a well-behaved solution to

$$\nabla^{\,2} {\cal R}_n = 0$$ (C.3)

in the interior of a sphere (i.e., the region $r<$ constant). It follows that (Riley 1974c)

$${\cal R}_n(r,\theta,\phi) \propto r^{\,n}\,{\cal S}_n(\theta,\phi).$$ (C.4)

Note that the Cartesian coordinates $x_i$ (where $i$ runs from $1$ to $3$) are solid harmonics of degree $1$. Moreover, $\partial {\cal R}_n/\partial x_i$ is a solid harmonic of degree $n-1$. Here, we have employed standard tensor notation (Riley 1974e).

The following results regarding solid harmonics are helpful:

$$x_i\,\frac{\partial {\cal R}_n}{\partial x_i} = r\,\frac{\partial {\cal R}_n}{\partial r} = n\,{\cal R}_n,$$ (C.5)

$$\nabla^{\,2}(x_i\,{\cal R}_n) = \nabla\cdot({\cal R}_n\,\nabla x_i + x_i\,\nabla{\cal R}_n) = 2\,\nabla x_i\cdot\nabla{\cal R}_n = 2\,\frac{\partial {\cal R}_n}{\partial x_i},$$ (C.6)

$$\nabla^{\,2}(r^{\,m}\,{\cal R}_n) = \nabla^{\,2}(r^{\,m+n}\,{\cal S}_n) = \frac{1}{r^{\,2}}\,\frac{... ...,\frac{d}{dr}(r^{\,m+n}\,{\cal S}_n)\right] - n\,(n+1)\,r^{\,m+n-2}\,{\cal S}_n$$

$$= m\,(m+2\,n+1)\,r^{\,m-2}\,{\cal R}_n.$$ (C.7)

In deriving these results, use has been made of standard vector field theory (Fitzpatrick 2008). In addition, we have adopted the well-known Einstein summation convention that repeated indices are implicitly summed from 1 to 3 (Riley 1974e).