Spherical harmonics

The spherical harmonics, denoted $Y_l^{\,m}(\theta,\phi)$, where $l$ is a non-negative integer, and $m$ an integer lying in the range $-l\leq m\leq l$, are the well-behaved solutions to

$\displaystyle r^{\,2}\,\nabla^{\,2} Y_l^{\,m} + l\,(l+1)\,Y_l^{\,m} = 0$ (A.159)

on the surface of a sphere (i.e., $r=$ constant). Here, $\nabla^{\,2}$ denotes a Laplacian (Riley 1974a), and $r$, $\theta $, $\phi $ are standard spherical coordinates. The spherical harmonics take the form (Jackson 1975)

$\displaystyle Y_l^{\,m}(\theta,\phi) = \sqrt{\frac{(2\,l+1)}{4\pi}\,\frac{(l-m)!}{(l+m)!}}\,P_l^{\,m}(\cos\theta)\,\,{\rm e}^{\,{\rm i}\,m\,\phi},$ (A.160)

where the $P_l^{\,m}(x)$ are associated Legendre polynomials (Abramowitz and Stegun 1965a). In particular,

$\displaystyle Y_l^{\,0}(\theta,\phi) = \sqrt{\frac{2\,l+1}{4\pi}}\,P_l(\cos\theta),$ (A.161)

where the $P_l(x)$ are the Legendre polynomials introduced in Section 3.4. The spherical harmonics satisfy

$\displaystyle Y_l^{\,m\,\ast} = (-1)^m\,Y_l^{\,-m},$ (A.162)

and have the property that they are orthonormal when integrated over the surface of a sphere; that is,

$\displaystyle \int_0^\pi\oint Y_l^{\,m}\,Y_{l'}^{\,m'\ast}\,\sin\theta\,d\theta\,d\phi = \delta_{ll'}\,\delta_{mm'}.$ (A.163)

The first few spherical harmonics are:

$\displaystyle Y_0^{\,0}$ $\displaystyle = \sqrt{\frac{1}{4\pi}},$ (A.164)
$\displaystyle Y_1^{\,-1}$ $\displaystyle = \sqrt{\frac{3}{8\pi}}\,\sin\theta\,\,{\rm e}^{-{\rm i}\,\phi},$ (A.165)
$\displaystyle Y_1^{\,0}$ $\displaystyle = \sqrt{\frac{3}{4\pi}}\,\cos\theta,$ (A.166)
$\displaystyle Y_1^{\,1}$ $\displaystyle =- \sqrt{\frac{3}{8\pi}}\,\sin\theta\,\,{\rm e}^{\,{\rm i}\,\phi},$ (A.167)
$\displaystyle Y_2^{\,-2}$ $\displaystyle = \sqrt{\frac{15}{32\pi}}\,\sin^2\theta\,\,{\rm e}^{-{\rm i}\,2\,\phi},$ (A.168)
$\displaystyle Y_2^{\,-1}$ $\displaystyle = \sqrt{\frac{15}{8\pi}}\,\sin\theta\,\cos\theta\,\,{\rm e}^{-{\rm i}\,\phi},$ (A.169)
$\displaystyle Y_2^{\,0}$ $\displaystyle = \sqrt{\frac{5}{16\pi}}\,(3\,\cos^2\theta-1),$ (A.170)
$\displaystyle Y_2^{\,1}$ $\displaystyle = -\sqrt{\frac{15}{8\pi}}\,\sin\theta\,\cos\theta\,\,{\rm e}^{\,{\rm i}\,\phi},$ (A.171)
$\displaystyle Y_2^{\,2}$ $\displaystyle = \sqrt{\frac{15}{32\pi}}\,\sin^2\theta\,\,{\rm e}^{\,{\rm i}\,2\,\phi}.$ (A.172)