1 基本的な性質

以下に球面調和関数 ${Y_l}^m(\theta,\phi)$の性質を挙げておく。

$${Y_l}^m(\theta,\phi) =(-1)^m \sqrt{ \frac{2l+1}{4\pi} \frac{(l-m)!}{(l+m)!}}\, {P_l}^m(\cos\theta)e^{im\phi}, \qquad \left(m\geq 0\right)$$ (46)

$${Y_l}^m(\theta,\phi) =(-1)^{\vert m\vert} {{Y_l}^{\vert m\vert}}^*(\theta,\phi), \qquad \left( m <0\right)$$ (47)

$${P_l}^m(\cos\theta) =\left(1-\cos^2\theta\right)^{m/2}\frac{d^m}{d(\cos\theta)^m} P_l(\cos\theta), \qquad \left(m\geq 0\right)$$ (48)

$$P_l(\cos\theta) = \frac{(-1)^l}{2^l l!} \frac{ d^l}{d(\cos\theta)^l}\left(1-\cos^2\theta\right)^l,$$ (49)

$$\int {{Y_{l'}}^{m'}}^*(\theta,\phi) {Y_l}^m(\theta,\phi) d\Omega = \delta_{ll'}\delta_{mm'}$$ (50)

$${Y_0}^0(\theta,\phi) = \frac{1}{4\pi},\quad {Y_1}^0(\theta,\phi) ... ...}^{\pm 1}(\theta,\phi) = \mp \sqrt{\frac{3}{8\pi}}\, \sin\theta e^{\pm im\phi}$$

$${Y_2}^0(\theta,\phi) = \sqrt{\frac{5}{16\pi}}\, \left(3\cos^2\the... ...heta,\phi) = \sqrt{\frac{15}{32\pi}}\, \left(\sin^2\theta\right)e^{\pm 2i\phi}$$