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 $,

14.30.1 | $${Y}_{l,m}(\theta ,\varphi )={\left(\frac{(l-m)!(2l+1)}{4\pi (l+m)!}\right)}^{1/2}{\mathrm{e}}^{\mathrm{i}m\varphi}{\U0001d5af}_{l}^{m}\left(\mathrm{cos}\theta \right),$$ | ||

14.30.2 | $${Y}_{l}^{m}(\theta ,\varphi )=\mathrm{cos}\left(m\varphi \right){\U0001d5af}_{l}^{m}\left(\mathrm{cos}\theta \right)\text{or}\mathrm{sin}\left(m\varphi \right){\U0001d5af}_{l}^{m}\left(\mathrm{cos}\theta \right).$$ | ||

${Y}_{l,m}(\theta ,\varphi )$ 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
${\mathrm{i}}^{-l}{\U0001d507}_{lm}(\theta ,\varphi )$; 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.

14.30.3 | $${Y}_{l,m}(\theta ,\varphi )=\begin{array}{l}\frac{{(-1)}^{l+m}}{{2}^{l}l!}{\left(\frac{(l-m)!(2l+1)}{4\pi (l+m)!}\right)}^{1/2}{\mathrm{e}}^{\mathrm{i}m\varphi}\\ \phantom{\rule{2em}{0ex}}\times {\left(\mathrm{sin}\theta \right)}^{m}{\left(\frac{d}{d(\mathrm{cos}\theta )}\right)}^{l+m}{\left(\mathrm{sin}\theta \right)}^{2l}.\end{array}$$ | ||

14.30.4 | $${Y}_{l,m}(0,\varphi )=\{\begin{array}{cc}{\left(\frac{2l+1}{4\pi}\right)}^{1/2},\hfill & m=0,\hfill \\ 0,\hfill & |m|=1,2,3,\mathrm{\dots},\hfill \end{array}$$ | ||

14.30.5 | $${Y}_{l,m}(\frac{1}{2}\pi ,\varphi )=\{\begin{array}{cc}\frac{{(-1)}^{(l+m)/2}{\mathrm{e}}^{\mathrm{i}m\varphi}}{{2}^{l}\left(\frac{1}{2}l-\frac{1}{2}m\right)!\left(\frac{1}{2}l+\frac{1}{2}m\right)!}{\left(\frac{(l-m)!(l+m)!(2l+1)}{4\pi}\right)}^{1/2},\hfill & \frac{1}{2}l+\frac{1}{2}m\in \mathbb{Z},\hfill \\ 0,\hfill & \frac{1}{2}l+\frac{1}{2}m\notin \mathbb{Z}.\hfill \end{array}$$ | ||

14.30.6 | $${Y}_{l,-m}(\theta ,\varphi )={(-1)}^{m}\overline{{Y}_{l,m}(\theta ,\varphi )}.$$ | ||

14.30.7 | $${Y}_{l,m}(\pi -\theta ,\varphi +\pi )={(-1)}^{l}{Y}_{l,m}(\theta ,\varphi ).$$ | ||

14.30.8 | $${\int}_{0}^{2\pi}{\int}_{0}^{\pi}\overline{{Y}_{{l}_{1},{m}_{1}}(\theta ,\varphi )}{Y}_{{l}_{2},{m}_{2}}(\theta ,\varphi )\mathrm{sin}\theta d\theta d\varphi ={\delta}_{{l}_{1},{l}_{2}}{\delta}_{{m}_{1},{m}_{2}}.$$ | ||

The following is the Herglotz generating function

14.30.8_5 | $${\mathrm{e}}^{t\mathbf{a}\cdot \mathbf{x}}=\sqrt{4\pi}\sum _{n=0}^{\mathrm{\infty}}\sum _{m=-n}^{n}\frac{{t}^{n}{r}^{n}{\lambda}^{m}{Y}_{n,m}(\theta ,\varphi )}{\sqrt{(2n+1)(n+m)!(n-m)!}},$$ | ||

where $\mathbf{a}=(\frac{1}{2\lambda}-\frac{\lambda}{2},-\frac{\mathrm{i}}{2\lambda}-\frac{\mathrm{i}\lambda}{2},1)$ and $\mathbf{x}=(r\mathrm{sin}\theta \mathrm{cos}\varphi ,r\mathrm{sin}\theta \mathrm{sin}\varphi ,r\mathrm{cos}\theta )$.

For a series representation of the product of two Dirac deltas in terms of products of spherical harmonics see §1.17(iii).

14.30.9 | $$\begin{array}{l}{\U0001d5af}_{l}\left(\mathrm{cos}{\theta}_{1}\mathrm{cos}{\theta}_{2}+\mathrm{sin}{\theta}_{1}\mathrm{sin}{\theta}_{2}\mathrm{cos}\left({\varphi}_{1}-{\varphi}_{2}\right)\right)\\ \phantom{\rule{2em}{0ex}}=\frac{4\pi}{2l+1}\sum _{m=-l}^{l}\overline{{Y}_{l,m}({\theta}_{1},{\varphi}_{1})}{Y}_{l,m}({\theta}_{2},{\varphi}_{2}).\end{array}$$ | ||

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)):

14.30.10 | $$\frac{1}{{\rho}^{2}}\frac{\partial}{\partial \rho}\left({\rho}^{2}\frac{\partial W}{\partial \rho}\right)+\frac{1}{{\rho}^{2}\mathrm{sin}\theta}\frac{\partial}{\partial \theta}\left(\mathrm{sin}\theta \frac{\partial W}{\partial \theta}\right)+\frac{1}{{\rho}^{2}{\mathrm{sin}}^{2}\theta}\frac{{\partial}^{2}W}{{\partial \varphi}^{2}}=0,$$ | ||

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

14.30.11 | $${L}^{2}{Y}_{l,m}={\mathrm{\hslash}}^{2}l(l+1){Y}_{l,m},$$ | ||

$l=0,1,2,\mathrm{\dots}$, | |||

and

14.30.11_5 | $${\mathrm{L}}_{z}{Y}_{l,m}=\mathrm{\hslash}m{Y}_{l,m},$$ | ||

$m=-l,-1+1,\mathrm{\dots},0,\mathrm{\dots},l-1,l$, | |||

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

14.30.12 | $${L}^{2}=-{\mathrm{\hslash}}^{2}\left(\frac{1}{\mathrm{sin}\theta}\frac{\partial}{\partial \theta}\left(\mathrm{sin}\theta \frac{\partial}{\partial \theta}\right)+\frac{1}{{\mathrm{sin}}^{2}\theta}\frac{{\partial}^{2}}{{\partial \varphi}^{2}}\right),$$ | ||

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

14.30.13 | $${\mathrm{L}}_{z}=-\mathrm{i}\mathrm{\hslash}\frac{\partial}{\partial \varphi};$$ | ||

see Edmonds (1974, §2.5).

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