# §14.30 Spherical and Spheroidal Harmonics

## §14.30(i) Definitions

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

 14.30.1 $Y_{{l},{m}}\left(\theta,\phi\right)=\left(\frac{(l-m)!(2l+1)}{4\pi(l+m)!}% \right)^{1/2}e^{im\phi}\mathsf{P}^{m}_{l}\left(\cos\theta\right),$ ⓘ Defines: $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$: spherical harmonic Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$: Ferrers function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $m$: 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 and Ch.14
 14.30.2 $Y_{l}^{m}\left(\theta,\phi\right)=\cos\left(m\phi\right)\mathsf{P}^{m}_{l}% \left(\cos\theta\right)\text{ or }\sin\left(m\phi\right)\mathsf{P}^{m}_{l}% \left(\cos\theta\right).$ ⓘ Defines: $Y_{\NVar{l}}^{\NVar{m}}\left(\NVar{\theta},\NVar{\phi}\right)$: surface harmonic of the first kind Symbols: $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$: Ferrers function of the first kind, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function, $m$: integer and $l$: nonnegative integer Referenced by: §14.30(i), Erratum Section 14.30 Permalink: http://dlmf.nist.gov/14.30.E2 Encodings: TeX, pMML, png See also: Annotations for §14.30(i), §14.30 and Ch.14

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

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

## §14.30(ii) Basic Properties

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

### Special Values

 14.30.4 $Y_{{l},{m}}\left(0,\phi\right)=\begin{cases}\left(\dfrac{2l+1}{4\pi}\right)^{1% /2},&m=0,\\ 0,&|m|=1,2,3,\dots,\end{cases}$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$: spherical harmonic, $m$: integer and $l$: nonnegative integer Referenced by: §14.30(ii), Erratum Section 14.30 Permalink: http://dlmf.nist.gov/14.30.E4 Encodings: TeX, pMML, png Correction (effective with 1.0.25): Previously this equation was only given for $m\geq 0$. In this update we have extended the definition of spherical harmonics to include negative integer values such that $|m|\leq l$. Hence in the second part of this equation, we have replaced $m$ with $|m|$. Suggested 2019-10-15 by Ching-Li Chai See also: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14
 14.30.5 $Y_{{l},{m}}\left(\tfrac{1}{2}\pi,\phi\right)=\begin{cases}\dfrac{(-1)^{(l+m)/2% }e^{im\phi}}{2^{l}\left(\frac{1}{2}l-\frac{1}{2}m\right)!\left(\frac{1}{2}l+% \frac{1}{2}m\right)!}\left(\dfrac{(l-m)!(l+m)!(2l+1)}{4\pi}\right)^{1/2},&% \frac{1}{2}l+\frac{1}{2}m\in\mathbb{Z},\\ 0,&\frac{1}{2}l+\frac{1}{2}m\notin\mathbb{Z}.\end{cases}$

### Symmetry

 14.30.6 $Y_{{l},{-m}}\left(\theta,\phi\right)=(-1)^{m}\overline{Y_{{l},{m}}\left(\theta% ,\phi\right)}.$ ⓘ Symbols: $\overline{\NVar{z}}$: complex conjugate, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$: spherical harmonic, $m$: integer and $l$: nonnegative integer Referenced by: §14.30(ii) Permalink: http://dlmf.nist.gov/14.30.E6 Encodings: TeX, pMML, png Notational Change (effective with 1.0.19): As a notational clarification, the spherical harmonic on the right-hand side has been explicity marked up as a complex conjugate. See also: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

### Parity Operation

 14.30.7 $Y_{{l},{m}}\left(\pi-\theta,\phi+\pi\right)=(-1)^{l}Y_{{l},{m}}\left(\theta,% \phi\right).$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$: spherical harmonic, $m$: integer and $l$: nonnegative integer Referenced by: §14.30(ii) Permalink: http://dlmf.nist.gov/14.30.E7 Encodings: TeX, pMML, png See also: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

### Orthogonality

 14.30.8 $\int_{0}^{2\pi}\!\!\int_{0}^{\pi}\overline{Y_{{l_{1}},{m_{1}}}\left(\theta,% \phi\right)}Y_{{l_{2}},{m_{2}}}\left(\theta,\phi\right)\sin\theta\mathrm{d}% \theta\mathrm{d}\phi=\delta_{l_{1},l_{2}}\delta_{m_{1},m_{2}}.$ ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$: Kronecker delta, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$: complex conjugate, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\sin\NVar{z}$: sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$: spherical harmonic, $m$: integer and $l$: nonnegative integer Referenced by: §14.30(ii) Permalink: http://dlmf.nist.gov/14.30.E8 Encodings: TeX, pMML, png Notational Change (effective with 1.0.19): As a notational clarification, the first spherical harmonic in the integral over $\theta$ been explicity marked up as a complex conjugate. See also: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

## §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 $\mathsf{P}_{l}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}% \cos\left(\phi_{1}-\phi_{2}\right)\right)=\frac{4\pi}{2l+1}\sum_{m=-l}^{l}% \overline{Y_{{l},{m}}\left(\theta_{1},\phi_{1}\right)}Y_{{l},{m}}\left(\theta_% {2},\phi_{2}\right).$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$: complex conjugate, $\cos\NVar{z}$: cosine function, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$: Ferrers function of the first kind, $\sin\NVar{z}$: sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$: spherical harmonic, $m$: integer and $l$: nonnegative integer Referenced by: §18.18(ii) Permalink: http://dlmf.nist.gov/14.30.E9 Encodings: TeX, pMML, png Notational Change (effective with 1.0.19): As a notational clarification, the first spherical harmonic in the sum over $m$ has been explicity marked up as a complex conjugate. See also: Annotations for §14.30(iii), §14.30(iii), §14.30 and Ch.14

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

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

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

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

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

 14.30.12 $\mathrm{L}^{2}=-\hbar^{2}\left(\frac{1}{\sin\theta}\frac{\partial}{\partial% \theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{{\sin^{% 2}}\theta}\frac{{\partial}^{2}}{{\partial\phi}^{2}}\right);$ ⓘ 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 $\sin\NVar{z}$: sine function Permalink: http://dlmf.nist.gov/14.30.E12 Encodings: TeX, pMML, png See also: Annotations for §14.30(iv), §14.30 and Ch.14

see Edmonds (1974, §2.5).

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