§14.30 Spherical and Spheroidal Harmonics
Contents
§14.30(i) Definitions
With and
integers such that
, and
and
angles such that
,
,
![\mathop{Y_{{{l},{m}}}\/}\nolimits\!\left(\theta,\phi\right)=\left(\frac{(l-m)!(2l+1)}{4\pi(l+m)!}\right)^{{1/2}}e^{{im\phi}}\mathop{\mathsf{P}^{{m}}_{{l}}\/}\nolimits\!\left(\mathop{\cos\/}\nolimits\theta\right),](https://webarchive.library.unt.edu/web/20130301103453im_/http://dlmf.nist.gov/14/30/E1.png)
![\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).](https://webarchive.library.unt.edu/web/20130301103453im_/http://dlmf.nist.gov/14/30/E2.png)
are known as spherical
harmonics.
are known as surface
harmonics of the first kind: tesseral for
and sectorial
for
. Sometimes
is denoted by
; also the definition of
can differ from
(14.30.1), for example, by inclusion of a factor
.
and
(
) are often referred
to as the prolate spheroidal harmonics of the first and second kinds,
respectively.
and
(
) are
known as oblate spheroidal harmonics of the first and second kinds,
respectively. Segura and Gil (1999) introduced the scaled oblate spheroidal
harmonics
and
which are real when
and
.
§14.30(ii) Basic Properties
Most mathematical properties of 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
![\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{d}{d(\mathop{\cos\/}\nolimits\theta)}\right)^{{l+m}}\left(\mathop{\sin\/}\nolimits\theta\right)^{{2l}}.](https://webarchive.library.unt.edu/web/20130301103453im_/http://dlmf.nist.gov/14/30/E3.png)
¶ Special Values
![\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}](https://webarchive.library.unt.edu/web/20130301103453im_/http://dlmf.nist.gov/14/30/E4.png)
![\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\Integer,\\
0,&\frac{1}{2}l+\frac{1}{2}m\notin\Integer.\end{cases}](https://webarchive.library.unt.edu/web/20130301103453im_/http://dlmf.nist.gov/14/30/E5.png)
¶ Symmetry
![\mathop{Y_{{{l},{-m}}}\/}\nolimits\!\left(\theta,\phi\right)=(-1)^{m}{\mathop{Y_{{{l},{m}}}\/}\nolimits^{{*}}}\!\left(\theta,\phi\right).](https://webarchive.library.unt.edu/web/20130301103453im_/http://dlmf.nist.gov/14/30/E6.png)
¶ Parity Operation
![\mathop{Y_{{{l},{m}}}\/}\nolimits\!\left(\pi-\theta,\phi+\pi\right)=(-1)^{l}\mathop{Y_{{{l},{m}}}\/}\nolimits\!\left(\theta,\phi\right).](https://webarchive.library.unt.edu/web/20130301103453im_/http://dlmf.nist.gov/14/30/E7.png)
§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(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 ,
defined by (14.30.1), appear in many physical
applications. As an example, Laplace’s equation
in spherical
coordinates (§1.5(ii)):
![{\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,](https://webarchive.library.unt.edu/web/20130301103453im_/http://dlmf.nist.gov/14/30/E10.png)
has solutions , which are everywhere one-valued
and continuous.
In the quantization of angular momentum the spherical harmonics
are normalized solutions of the
eigenvalue equation
![\mathrm{L}^{2}\mathop{Y_{{{l},{m}}}\/}\nolimits=\hbar^{2}l(l+1)\mathop{Y_{{{l},{m}}}\/}\nolimits,](https://webarchive.library.unt.edu/web/20130301103453im_/http://dlmf.nist.gov/14/30/E11.png)
where is the reduced Planck’s constant, and
is the angular
momentum operator in spherical coordinates:
![\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);](https://webarchive.library.unt.edu/web/20130301103453im_/http://dlmf.nist.gov/14/30/E12.png)
see Edmonds (1974, §2.5).
For applications in geophysics see Stacey (1977, §§4.2, 6.3, and 8.1).