§14.30 Spherical and Spheroidal Harmonics

With $l$ and $m$ integers such that $|m|\le l$, and $\theta$ and $\varphi$ angles such that $0\le \theta \le \pi$, $0\le \varphi \le 2\pi$,

$Y_{l,m}(\theta ,\varphi )$ are known as spherical harmonics. $Y_l^m(\theta ,\varphi )$ are known as surface harmonics of the first kind: tesseral for and sectorial for $|m|=l$. Sometimes $Y_{l,m}(\theta ,\varphi )$ is denoted by ${\mathrm{i}}^{-l}{𝔇}_{lm}(\theta ,\varphi )$; also the definition of $Y_{l,m}(\theta ,\varphi )$ can differ from (14.30.1), for example, by inclusion of a factor ${(-1)}^m$.

$P_n^m\left(x\right)$ and $Q_n^m\left(x\right)$ ($x>1$) are often referred to as the prolate spheroidal harmonics of the first and second kinds, respectively. $P_n^m\left(\mathrm{i}x\right)$ and $Q_n^m\left(\mathrm{i}x\right)$ ($x>0$) are known as oblate spheroidal harmonics of the first and second kinds, respectively. Segura and Gil (1999) introduced the scaled oblate spheroidal harmonics $R_n^m(x)={\mathrm{e}}^{-\mathrm{i}\pi n/2}{P}_n^m\left(\mathrm{i}x\right)$ and $T_n^m(x)=\mathrm{i}{\mathrm{e}}^{\mathrm{i}\pi n/2}{Q}_n^m\left(\mathrm{i}x\right)$ which are real when $x>0$ and $n=0,1,2,\mathrm{\dots }$.

Most mathematical properties of $Y_{l,m}(\theta ,\varphi )$ can be derived directly from (14.30.1) and the properties of the Ferrers function of the first kind given earlier in this chapter.

In general, spherical harmonics are defined as the class of homogeneous harmonic polynomials. See Andrews et al. (1999, Chapter 9). The special class of spherical harmonics $Y_{l,m}(\theta ,\varphi )$, defined by (14.30.1), appear in many physical applications. As an example, Laplace’s equation ${\nabla }^{2}W=0$ in spherical coordinates (§1.5(ii)):

has solutions $W(\rho ,\theta ,\varphi )={\rho }^l{Y}_{l,m}(\theta ,\varphi )$, which are everywhere one-valued and continuous.

In the quantization of angular momentum the spherical harmonics $Y_{l,m}(\theta ,\varphi )$ are normalized solutions of the eigenvalue equations

and

where $\mathrm{\hslash }$ is the reduced Planck’s constant. Here, in spherical coordinates, $L^2$ is the squared angular momentum operator:

and ${\mathrm{L}}_z$ is the $z$ component of the angular momentum operator

see Edmonds (1974, §2.5).

For applications in geophysics see Stacey (1977, §§4.2, 6.3, and 8.1).