# §14.30 Spherical and Spheroidal Harmonics

## §14.30(i) Definitions

With $l$ and $m$ integers such that $0\leq m\leq l$, and $\theta$ and $\phi$ angles such that $0\leq\theta\leq\pi$, $0\leq\phi\leq 2\pi$,

 14.30.1 $\mathop{Y_{{l},{m}}\/}\nolimits\!\left(\theta,\phi\right)=\left(\frac{(l-m)!(2% l+1)}{4\pi(l+m)!}\right)^{1/2}e^{im\phi}\mathop{\mathsf{P}^{m}_{l}\/}\nolimits% \!\left(\mathop{\cos\/}\nolimits\theta\right),$ Defines: $\mathop{Y_{{\NVar{l}},{\NVar{m}}}\/}\nolimits\!\left(\NVar{\theta},\NVar{\phi}\right)$: spherical harmonic Symbols: $\mathop{\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathrm{e}$: base of exponential function, $!$: factorial (as in $n!$), $m$: nonnegative integer and $l$: nonnegative integer Referenced by: §14.30(i), §14.30(ii), §14.30(ii), §14.30(iv) Permalink: http://dlmf.nist.gov/14.30.E1 Encodings: TeX, pMML, png See also: Annotations for 14.30(i)
 14.30.2 $\mathop{Y_{l}^{m}\/}\nolimits\!\left(\theta,\phi\right)=\mathop{\cos\/}% \nolimits\!\left(m\phi\right)\mathop{\mathsf{P}^{m}_{l}\/}\nolimits\!\left(% \mathop{\cos\/}\nolimits\theta\right)\text{ or }\mathop{\sin\/}\nolimits\!% \left(m\phi\right)\mathop{\mathsf{P}^{m}_{l}\/}\nolimits\!\left(\mathop{\cos\/% }\nolimits\theta\right).$ Defines: $\mathop{Y_{\NVar{l}}^{\NVar{m}}\/}\nolimits\!\left(\NVar{\theta},\NVar{\phi}\right)$: surface harmonic of the first kind Symbols: $\mathop{\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$: Ferrers function of the first kind, $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sin\/}\nolimits\NVar{z}$: sine function, $m$: nonnegative integer and $l$: nonnegative integer Permalink: http://dlmf.nist.gov/14.30.E2 Encodings: TeX, pMML, png See also: Annotations for 14.30(i)

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

$\mathop{P^{m}_{n}\/}\nolimits\!\left(x\right)$ and $\mathop{Q^{m}_{n}\/}\nolimits\!\left(x\right)$ ($x>1$) are often referred to as the prolate spheroidal harmonics of the first and second kinds, respectively. $\mathop{P^{m}_{n}\/}\nolimits\!\left(ix\right)$ and $\mathop{Q^{m}_{n}\/}\nolimits\!\left(ix\right)$ ($x>0$) are known as oblate spheroidal harmonics of the first and second kinds, respectively. Segura and Gil (1999) introduced the scaled oblate spheroidal harmonics $R_{n}^{m}(x)=e^{-i\pi n/2}\mathop{P^{m}_{n}\/}\nolimits\!\left(ix\right)$ and $T_{n}^{m}(x)=ie^{i\pi n/2}\mathop{Q^{m}_{n}\/}\nolimits\!\left(ix\right)$ which are real when $x>0$ and $n=0,1,2,\dots$.

## §14.30(ii) Basic Properties

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

### Explicit Representation

 14.30.3 $\mathop{Y_{{l},{m}}\/}\nolimits\!\left(\theta,\phi\right)=\frac{(-1)^{l+m}}{2^% {l}l!}\left(\frac{(l-m)!(2l+1)}{4\pi(l+m)!}\right)^{1/2}e^{im\phi}\left(% \mathop{\sin\/}\nolimits\theta\right)^{m}\*\left(\frac{\mathrm{d}}{\mathrm{d}(% \mathop{\cos\/}\nolimits\theta)}\right)^{l+m}\left(\mathop{\sin\/}\nolimits% \theta\right)^{2l}.$

### Special Values

 14.30.4 $\mathop{Y_{{l},{m}}\/}\nolimits\!\left(0,\phi\right)=\begin{cases}\left(\dfrac% {2l+1}{4\pi}\right)^{1/2},&m=0,\\ 0,&m=1,2,3,\dots,\end{cases}$
 14.30.5 $\mathop{Y_{{l},{m}}\/}\nolimits\!\left(\tfrac{1}{2}\pi,\phi\right)=\begin{% cases}\dfrac{(-1)^{(l+m)/2}e^{im\phi}}{2^{l}\left(\frac{1}{2}l-\frac{1}{2}m% \right)!\left(\frac{1}{2}l+\frac{1}{2}m\right)!}\left(\dfrac{(l-m)!(l+m)!(2l+1% )}{4\pi}\right)^{1/2},&\frac{1}{2}l+\frac{1}{2}m\in\mathbb{Z},\\ 0,&\frac{1}{2}l+\frac{1}{2}m\notin\mathbb{Z}.\end{cases}$

### Symmetry

 14.30.6 $\mathop{Y_{{l},{-m}}\/}\nolimits\!\left(\theta,\phi\right)=(-1)^{m}{\mathop{Y_% {{l},{m}}\/}\nolimits^{*}}\!\left(\theta,\phi\right).$ Symbols: $\mathop{Y_{{\NVar{l}},{\NVar{m}}}\/}\nolimits\!\left(\NVar{\theta},\NVar{\phi}\right)$: spherical harmonic, $m$: nonnegative integer and $l$: nonnegative integer Referenced by: §14.30(ii) Permalink: http://dlmf.nist.gov/14.30.E6 Encodings: TeX, pMML, png See also: Annotations for 14.30(ii)

### Parity Operation

 14.30.7 $\mathop{Y_{{l},{m}}\/}\nolimits\!\left(\pi-\theta,\phi+\pi\right)=(-1)^{l}% \mathop{Y_{{l},{m}}\/}\nolimits\!\left(\theta,\phi\right).$

### Orthogonality

 14.30.8 $\int_{0}^{2\pi}\!\!\int_{0}^{\pi}{\mathop{Y_{{l_{1}},{m_{1}}}\/}\nolimits^{*}}% \!\left(\theta,\phi\right)\mathop{Y_{{l_{2}},{m_{2}}}\/}\nolimits\!\left(% \theta,\phi\right)\mathop{\sin\/}\nolimits\theta\mathrm{d}\theta\mathrm{d}\phi% =\delta_{l_{1},l_{2}}\delta_{m_{1},m_{2}};$

here and elsewhere in this section the asterisk (*) denotes complex conjugate.

## §14.30(iii) Sums

### Distributional Completeness

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 $\mathop{\mathsf{P}_{l}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta_{1}% \mathop{\cos\/}\nolimits\theta_{2}+\mathop{\sin\/}\nolimits\theta_{1}\mathop{% \sin\/}\nolimits\theta_{2}\mathop{\cos\/}\nolimits\!\left(\phi_{1}-\phi_{2}% \right)\right)=\frac{4\pi}{2l+1}\sum_{m=-l}^{l}{\mathop{Y_{{l},{m}}\/}% \nolimits^{*}}\!\left(\theta_{1},\phi_{1}\right)\mathop{Y_{{l},{m}}\/}% \nolimits\!\left(\theta_{2},\phi_{2}\right).$

## §14.30(iv) Applications

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 $\mathop{Y_{{l},{m}}\/}\nolimits\!\left(\theta,\phi\right)$, 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 ${\frac{1}{\rho^{2}}\frac{\partial}{\partial\rho}\left(\rho^{2}\frac{\partial W% }{\partial\rho}\right)+\frac{1}{\rho^{2}\mathop{\sin\/}\nolimits\theta}\frac{% \partial}{\partial\theta}\left(\mathop{\sin\/}\nolimits\theta\frac{\partial W}% {\partial\theta}\right)}+\frac{1}{\rho^{2}{\mathop{\sin\/}\nolimits^{2}}\theta% }\frac{{\partial}^{2}W}{{\partial\phi}^{2}}=0,$

has solutions $W(\rho,\theta,\phi)=\rho^{l}\mathop{Y_{{l},{m}}\/}\nolimits\!\left(\theta,\phi\right)$, which are everywhere one-valued and continuous.

In the quantization of angular momentum the spherical harmonics $\mathop{Y_{{l},{m}}\/}\nolimits\!\left(\theta,\phi\right)$ are normalized solutions of the eigenvalue equation

 14.30.11 $\mathrm{L}^{2}\mathop{Y_{{l},{m}}\/}\nolimits=\hbar^{2}l(l+1)\mathop{Y_{{l},{m% }}\/}\nolimits,$

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

 14.30.12 $\mathrm{L}^{2}=-\hbar^{2}\left(\frac{1}{\mathop{\sin\/}\nolimits\theta}\frac{% \partial}{\partial\theta}\left(\mathop{\sin\/}\nolimits\theta\frac{\partial}{% \partial\theta}\right)+\frac{1}{{\mathop{\sin\/}\nolimits^{2}}\theta}\frac{{% \partial}^{2}}{{\partial\phi}^{2}}\right);$ Defines: $\mathrm{L}$: angular momentum operator (locally) Symbols: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$: partial derivative of $f$ with respect to $x$, $\partial\NVar{x}$: partial differential of $x$ and $\mathop{\sin\/}\nolimits\NVar{z}$: sine function Permalink: http://dlmf.nist.gov/14.30.E12 Encodings: TeX, pMML, png See also: Annotations for 14.30(iv)

see Edmonds (1974, §2.5).

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