球谐函数表

这里列出球谐函数 $Y_{l}^{m}$，以方程式表示为

$Y_\ell^m( \theta , \varphi ) = \sqrt{{(2\ell+1)\over 4\pi}{(\ell-m)!\over (\ell+m)!}} \, P_\ell^m ( \cos{\theta} ) \, e^{i m \varphi }$。

其中，$l$ 为正值整数，$m$ 为小于或等于 $l$ 的正值整数，$P_\ell ^{m}$ 是伴随勒让德多项式，以方程式表示为

$P_\ell ^{m} (x)= (-1)^m (1-x^2)^{m/2} \frac{\mathrm{d}^m P_\ell (x)}{\mathrm{d}x^m}$。

表内有些方程式也给出直角坐标版本。球坐标与直角坐标之间的变换关系是

$x = r \sin\theta\cos\varphi\,$。

$y = r \sin\theta\sin\varphi\,$。

$z = r \cos\theta\,$。

## $l=0$

$$Y_{0}^{0}(\theta,\varphi)={1\over 2}\sqrt{1\over \pi}$$

## $l=1$

$$Y_{1}^{-1}(\theta,\varphi)={1\over 2}\sqrt{3\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\quad={1\over 2}\sqrt{3\over 2\pi}\cdot{(x-iy)\over r}$$

$$Y_{1}^{0}(\theta,\varphi)={1\over 2}\sqrt{3\over \pi}\cdot\cos\theta\quad={1\over 2}\sqrt{3\over \pi}\cdot{z\over r}$$

$$Y_{1}^{1}(\theta,\varphi)={-1\over 2}\sqrt{3\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\quad={-1\over 2}\sqrt{3\over 2\pi}\cdot{(x+iy)\over r}$$

## $l=2$

$$Y_{2}^{-2}(\theta,\varphi) ={1\over 4}\sqrt{15\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\quad ={1\over 4}\sqrt{15\over 2\pi}\cdot{(x - iy)^2 \over r^{2}}$$

$$Y_{2}^{-1}(\theta,\varphi) ={1\over 2}\sqrt{15\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot\cos\theta\quad ={1\over 2}\sqrt{15\over 2\pi}\cdot{(x - iy)z \over r^{2}}$$

$$Y_{2}^{0}(\theta,\varphi) ={1\over 4}\sqrt{5\over \pi}\cdot(3\cos^{2}\theta-1)\quad ={1\over 4}\sqrt{5\over \pi}\cdot{(-x^{2}-y^{2}+2z^{2})\over r^{2}}$$

$$Y_{2}^{1}(\theta,\varphi) ={-1\over 2}\sqrt{15\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot\cos\theta\quad ={-1\over 2}\sqrt{15\over 2\pi}\cdot{(x + iy)z \over r^{2}}$$

$$Y_{2}^{2}(\theta,\varphi) ={1\over 4}\sqrt{15\over 2\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\quad ={1\over 4}\sqrt{15\over 2\pi}\cdot{(x + iy)^2 \over r^{2}}$$

## $l=3$

$$Y_{3}^{-3}(\theta,\varphi) = {1\over 8}\sqrt{35\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\quad = {1\over 8}\sqrt{35\over \pi}\cdot{(x - iy)^{3}\over r^{3}}$$

$$Y_{3}^{-2}(\theta,\varphi) = {1\over 4}\sqrt{105\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot\cos\theta\quad = {1\over 4}\sqrt{105\over 2\pi}\cdot{(x- iy)^2 z \over r^{3}}$$

$$Y_{3}^{-1}(\theta,\varphi) ={1\over 8}\sqrt{21\over \pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(5\cos^{2}\theta-1)\quad ={1\over 8}\sqrt{21\over \pi}\cdot{(x - iy)(4z^2- x^2 - y^2)\over r^{3}}$$

$$Y_{3}^{0}(\theta,\varphi) ={1\over 4}\sqrt{7\over \pi}\cdot(5\cos^{3}\theta-3\cos\theta)\quad ={1\over 4}\sqrt{7\over \pi}\cdot{z(2z^2 - 3x^2 - 3y^2)\over r^{3}}$$

$$Y_{3}^{1}(\theta,\varphi) ={-1\over 8}\sqrt{21\over \pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(5\cos^{2}\theta-1)\quad ={-1\over 8}\sqrt{21\over \pi}\cdot{(x + iy) (4z^2 - x^2 - y^2) \over r^{3}}$$

$$Y_{3}^{2}(\theta,\varphi) ={1\over 4}\sqrt{105\over 2\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot\cos\theta\quad ={1\over 4}\sqrt{105\over 2\pi}\cdot{(x + iy)^2 z \over r^{3}}$$

$$Y_{3}^{3}(\theta,\varphi) ={-1\over 8}\sqrt{35\over \pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\quad ={-1\over 8}\sqrt{35\over \pi}\cdot{(x + iy)^3\over r^{3}}$$

## $l=4$

$$Y_{4}^{-4}(\theta,\varphi)={3\over 16}\sqrt{35\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta = \frac{3}{16} \sqrt{\frac{35}{2 \pi}} \cdot \frac{(x - i y)^4}{r^4}$$

$$Y_{4}^{-3}(\theta,\varphi)={3\over 8}\sqrt{35\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot\cos\theta = \frac{3}{8} \sqrt{\frac{35}{\pi}} \cdot \frac{(x - i y)^3 z}{r^4}$$

$$Y_{4}^{-2}(\theta,\varphi)={3\over 8}\sqrt{5\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(7\cos^{2}\theta-1) = \frac{3}{8} \sqrt{\frac{5}{2 \pi}} \cdot \frac{(x - i y)^2 \cdot (7 z^2 - r^2)}{r^4}$$

$$Y_{4}^{-1}(\theta,\varphi)={3\over 8}\sqrt{5\over \pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(7\cos^{3}\theta-3\cos\theta) = \frac{3}{8} \sqrt{\frac{5}{\pi}} \cdot \frac{(x - i y) \cdot z \cdot (7 z^2 - 3 r^2)}{r^4}$$

$$Y_{4}^{0}(\theta,\varphi)={3\over 16}\sqrt{1\over \pi}\cdot(35\cos^{4}\theta-30\cos^{2}\theta+3) = \frac{3}{16} \sqrt{\frac{1}{\pi}} \cdot \frac{(35 z^4 - 30 z^2 r^2 + 3 r^4)}{r^4}$$

$$Y_{4}^{1}(\theta,\varphi)={-3\over 8}\sqrt{5\over \pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(7\cos^{3}\theta-3\cos\theta) = \frac{- 3}{8} \sqrt{\frac{5}{\pi}} \cdot \frac{(x + i y) \cdot z \cdot (7 z^2 - 3 r^2)}{r^4}$$

$$Y_{4}^{2}(\theta,\varphi)={3\over 8}\sqrt{5\over 2\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot(7\cos^{2}\theta-1) = \frac{3}{8} \sqrt{\frac{5}{2 \pi}} \cdot \frac{(x + i y)^2 \cdot (7 z^2 - r^2)}{r^4}$$

$$Y_{4}^{3}(\theta,\varphi)={-3\over 8}\sqrt{35\over \pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\cdot\cos\theta = \frac{- 3}{8} \sqrt{\frac{35}{\pi}} \cdot \frac{(x + i y)^3 z}{r^4}$$

$$Y_{4}^{4}(\theta,\varphi)={3\over 16}\sqrt{35\over 2\pi}\cdot e^{4i\varphi}\cdot\sin^{4}\theta = \frac{3}{16} \sqrt{\frac{35}{2 \pi}} \cdot \frac{(x + i y)^4}{r^4}$$

## $l=5$

$$Y_{5}^{-5}(\theta,\varphi)={3\over 32}\sqrt{77\over \pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta$$

$$Y_{5}^{-4}(\theta,\varphi)={3\over 16}\sqrt{385\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot\cos\theta$$

$$Y_{5}^{-3}(\theta,\varphi)={1\over 32}\sqrt{385\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(9\cos^{2}\theta-1)$$

$$Y_{5}^{-2}(\theta,\varphi)={1\over 8}\sqrt{1155\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(3\cos^{3}\theta-1\cos\theta)$$

$$Y_{5}^{-1}(\theta,\varphi)={1\over 16}\sqrt{165\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(21\cos^{4}\theta-14\cos^{2}\theta+1)$$

$$Y_{5}^{0}(\theta,\varphi)={1\over 16}\sqrt{11\over \pi}\cdot(63\cos^{5}\theta-70\cos^{3}\theta+15\cos\theta)$$

$$Y_{5}^{1}(\theta,\varphi)={-1\over 16}\sqrt{165\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(21\cos^{4}\theta-14\cos^{2}\theta+1)$$

$$Y_{5}^{2}(\theta,\varphi)={1\over 8}\sqrt{1155\over 2\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot(3\cos^{3}\theta-1\cos\theta)$$

$$Y_{5}^{3}(\theta,\varphi)={-1\over 32}\sqrt{385\over \pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\cdot(9\cos^{2}\theta-1)$$

$$Y_{5}^{4}(\theta,\varphi)={3\over 16}\sqrt{385\over 2\pi}\cdot e^{4i\varphi}\cdot\sin^{4}\theta\cdot\cos\theta$$

$$Y_{5}^{5}(\theta,\varphi)={-3\over 32}\sqrt{77\over \pi}\cdot e^{5i\varphi}\cdot\sin^{5}\theta$$

## $l=6$

$$Y_{6}^{-6}(\theta,\varphi)={1\over 64}\sqrt{3003\over \pi}\cdot e^{-6i\varphi}\cdot\sin^{6}\theta$$

$$Y_{6}^{-5}(\theta,\varphi)={3\over 32}\sqrt{1001\over \pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\cdot\cos\theta$$

$$Y_{6}^{-4}(\theta,\varphi)={3\over 32}\sqrt{91\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot(11\cos^{2}\theta-1)$$

$$Y_{6}^{-3}(\theta,\varphi)={1\over 32}\sqrt{1365\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(11\cos^{3}\theta-3\cos\theta)$$

$$Y_{6}^{-2}(\theta,\varphi)={1\over 64}\sqrt{1365\over \pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(33\cos^{4}\theta-18\cos^{2}\theta+1)$$

$$Y_{6}^{-1}(\theta,\varphi)={1\over 16}\sqrt{273\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(33\cos^{5}\theta-30\cos^{3}\theta+5\cos\theta)$$

$$Y_{6}^{0}(\theta,\varphi)={1\over 32}\sqrt{13\over \pi}\cdot(231\cos^{6}\theta-315\cos^{4}\theta+105\cos^{2}\theta-5)$$

$$Y_{6}^{1}(\theta,\varphi)={-1\over 16}\sqrt{273\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(33\cos^{5}\theta-30\cos^{3}\theta+5\cos\theta)$$

$$Y_{6}^{2}(\theta,\varphi)={1\over 64}\sqrt{1365\over \pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot(33\cos^{4}\theta-18\cos^{2}\theta+1)$$

$$Y_{6}^{3}(\theta,\varphi)={-1\over 32}\sqrt{1365\over \pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\cdot(11\cos^{3}\theta-3\cos\theta)$$

$$Y_{6}^{4}(\theta,\varphi)={3\over 32}\sqrt{91\over 2\pi}\cdot e^{4i\varphi}\cdot\sin^{4}\theta\cdot(11\cos^{2}\theta-1)$$

$$Y_{6}^{5}(\theta,\varphi)={-3\over 32}\sqrt{1001\over \pi}\cdot e^{5i\varphi}\cdot\sin^{5}\theta\cdot\cos\theta$$

$$Y_{6}^{6}(\theta,\varphi)={1\over 64}\sqrt{3003\over \pi}\cdot e^{6i\varphi}\cdot\sin^{6}\theta$$

## $l=7$

$$Y_{7}^{-7}(\theta,\varphi)={3\over 64}\sqrt{715\over 2\pi}\cdot e^{-7i\varphi}\cdot\sin^{7}\theta$$

$$Y_{7}^{-6}(\theta,\varphi)={3\over 64}\sqrt{5005\over \pi}\cdot e^{-6i\varphi}\cdot\sin^{6}\theta\cdot\cos\theta$$

$$Y_{7}^{-5}(\theta,\varphi)={3\over 64}\sqrt{385\over 2\pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\cdot(13\cos^{2}\theta-1)$$

$$Y_{7}^{-4}(\theta,\varphi)={3\over 32}\sqrt{385\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot(13\cos^{3}\theta-3\cos\theta)$$

$$Y_{7}^{-3}(\theta,\varphi)={3\over 64}\sqrt{35\over 2\pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(143\cos^{4}\theta-66\cos^{2}\theta+3)$$

$$Y_{7}^{-2}(\theta,\varphi)={3\over 64}\sqrt{35\over \pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(143\cos^{5}\theta-110\cos^{3}\theta+15\cos\theta)$$

$$Y_{7}^{-1}(\theta,\varphi)={1\over 64}\sqrt{105\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(429\cos^{6}\theta-495\cos^{4}\theta+135\cos^{2}\theta-5)$$

$$Y_{7}^{0}(\theta,\varphi)={1\over 32}\sqrt{15\over \pi}\cdot(429\cos^{7}\theta-693\cos^{5}\theta+315\cos^{3}\theta-35\cos\theta)$$

$$Y_{7}^{1}(\theta,\varphi)={-1\over 64}\sqrt{105\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(429\cos^{6}\theta-495\cos^{4}\theta+135\cos^{2}\theta-5)$$

$$Y_{7}^{2}(\theta,\varphi)={3\over 64}\sqrt{35\over \pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot(143\cos^{5}\theta-110\cos^{3}\theta+15\cos\theta)$$

$$Y_{7}^{3}(\theta,\varphi)={-3\over 64}\sqrt{35\over 2\pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\cdot(143\cos^{4}\theta-66\cos^{2}\theta+3)$$

$$Y_{7}^{4}(\theta,\varphi)={3\over 32}\sqrt{385\over 2\pi}\cdot e^{4i\varphi}\cdot\sin^{4}\theta\cdot(13\cos^{3}\theta-3\cos\theta)$$

$$Y_{7}^{5}(\theta,\varphi)={-3\over 64}\sqrt{385\over 2\pi}\cdot e^{5i\varphi}\cdot\sin^{5}\theta\cdot(13\cos^{2}\theta-1)$$

$$Y_{7}^{6}(\theta,\varphi)={3\over 64}\sqrt{5005\over \pi}\cdot e^{6i\varphi}\cdot\sin^{6}\theta\cdot\cos\theta$$

$$Y_{7}^{7}(\theta,\varphi)={-3\over 64}\sqrt{715\over 2\pi}\cdot e^{7i\varphi}\cdot\sin^{7}\theta$$

## $l=8$

$$Y_{8}^{-8}(\theta,\varphi)={3\over 256}\sqrt{12155\over 2\pi}\cdot e^{-8i\varphi}\cdot\sin^{8}\theta$$

$$Y_{8}^{-7}(\theta,\varphi)={3\over 64}\sqrt{12155\over 2\pi}\cdot e^{-7i\varphi}\cdot\sin^{7}\theta\cdot\cos\theta$$

$$Y_{8}^{-6}(\theta,\varphi)={1\over 128}\sqrt{7293\over \pi}\cdot e^{-6i\varphi}\cdot\sin^{6}\theta\cdot(15\cos^{2}\theta-1)$$

$$Y_{8}^{-5}(\theta,\varphi)={3\over 64}\sqrt{17017\over 2\pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\cdot(5\cos^{3}\theta-1\cos\theta)$$

$$Y_{8}^{-4}(\theta,\varphi)={3\over 128}\sqrt{1309\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot(65\cos^{4}\theta-26\cos^{2}\theta+1)$$

$$Y_{8}^{-3}(\theta,\varphi)={1\over 64}\sqrt{19635\over 2\pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(39\cos^{5}\theta-26\cos^{3}\theta+3\cos\theta)$$

$$Y_{8}^{-2}(\theta,\varphi)={3\over 128}\sqrt{595\over \pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(143\cos^{6}\theta-143\cos^{4}\theta+33\cos^{2}\theta-1)$$

$$Y_{8}^{-1}(\theta,\varphi)={3\over 64}\sqrt{17\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(715\cos^{7}\theta-1001\cos^{5}\theta+385\cos^{3}\theta-35\cos\theta)$$

$$Y_{8}^{0}(\theta,\varphi)={1\over 256}\sqrt{17\over \pi}\cdot(6435\cos^{8}\theta-12012\cos^{6}\theta+6930\cos^{4}\theta-1260\cos^{2}\theta+35)$$

$$Y_{8}^{1}(\theta,\varphi)={-3\over 64}\sqrt{17\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(715\cos^{7}\theta-1001\cos^{5}\theta+385\cos^{3}\theta-35\cos\theta)$$

$$Y_{8}^{2}(\theta,\varphi)={3\over 128}\sqrt{595\over \pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot(143\cos^{6}\theta-143\cos^{4}\theta+33\cos^{2}\theta-1)$$

$$Y_{8}^{3}(\theta,\varphi)={-1\over 64}\sqrt{19635\over 2\pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\cdot(39\cos^{5}\theta-26\cos^{3}\theta+3\cos\theta)$$

$$Y_{8}^{4}(\theta,\varphi)={3\over 128}\sqrt{1309\over 2\pi}\cdot e^{4i\varphi}\cdot\sin^{4}\theta\cdot(65\cos^{4}\theta-26\cos^{2}\theta+1)$$

$$Y_{8}^{5}(\theta,\varphi)={-3\over 64}\sqrt{17017\over 2\pi}\cdot e^{5i\varphi}\cdot\sin^{5}\theta\cdot(5\cos^{3}\theta-1\cos\theta)$$

$$Y_{8}^{6}(\theta,\varphi)={1\over 128}\sqrt{7293\over \pi}\cdot e^{6i\varphi}\cdot\sin^{6}\theta\cdot(15\cos^{2}\theta-1)$$

$$Y_{8}^{7}(\theta,\varphi)={-3\over 64}\sqrt{12155\over 2\pi}\cdot e^{7i\varphi}\cdot\sin^{7}\theta\cdot\cos\theta$$

$$Y_{8}^{8}(\theta,\varphi)={3\over 256}\sqrt{12155\over 2\pi}\cdot e^{8i\varphi}\cdot\sin^{8}\theta$$

## $l=9$

$$Y_{9}^{-9}(\theta,\varphi)={1\over 512}\sqrt{230945\over \pi}\cdot e^{-9i\varphi}\cdot\sin^{9}\theta$$

$$Y_{9}^{-8}(\theta,\varphi)={3\over 256}\sqrt{230945\over 2\pi}\cdot e^{-8i\varphi}\cdot\sin^{8}\theta\cdot\cos\theta$$

$$Y_{9}^{-7}(\theta,\varphi)={3\over 512}\sqrt{13585\over \pi}\cdot e^{-7i\varphi}\cdot\sin^{7}\theta\cdot(17\cos^{2}\theta-1)$$

$$Y_{9}^{-6}(\theta,\varphi)={1\over 128}\sqrt{40755\over \pi}\cdot e^{-6i\varphi}\cdot\sin^{6}\theta\cdot(17\cos^{3}\theta-3\cos\theta)$$

$$Y_{9}^{-5}(\theta,\varphi)={3\over 256}\sqrt{2717\over \pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\cdot(85\cos^{4}\theta-30\cos^{2}\theta+1)$$

$$Y_{9}^{-4}(\theta,\varphi)={3\over 128}\sqrt{95095\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot(17\cos^{5}\theta-10\cos^{3}\theta+1\cos\theta)$$

$$Y_{9}^{-3}(\theta,\varphi)={1\over 256}\sqrt{21945\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(221\cos^{6}\theta-195\cos^{4}\theta+39\cos^{2}\theta-1)$$

$$Y_{9}^{-2}(\theta,\varphi)={3\over 128}\sqrt{1045\over \pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(221\cos^{7}\theta-273\cos^{5}\theta+91\cos^{3}\theta-7\cos\theta)$$

$$Y_{9}^{-1}(\theta,\varphi)={3\over 256}\sqrt{95\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(2431\cos^{8}\theta-4004\cos^{6}\theta+2002\cos^{4}\theta-308\cos^{2}\theta+7)$$

$$Y_{9}^{0}(\theta,\varphi)={1\over 256}\sqrt{19\over \pi}\cdot(12155\cos^{9}\theta-25740\cos^{7}\theta+18018\cos^{5}\theta-4620\cos^{3}\theta+315\cos\theta)$$

$$Y_{9}^{1}(\theta,\varphi)={-3\over 256}\sqrt{95\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(2431\cos^{8}\theta-4004\cos^{6}\theta+2002\cos^{4}\theta-308\cos^{2}\theta+7)$$

$$Y_{9}^{2}(\theta,\varphi)={3\over 128}\sqrt{1045\over \pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot(221\cos^{7}\theta-273\cos^{5}\theta+91\cos^{3}\theta-7\cos\theta)$$

$$Y_{9}^{3}(\theta,\varphi)={-1\over 256}\sqrt{21945\over \pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\cdot(221\cos^{6}\theta-195\cos^{4}\theta+39\cos^{2}\theta-1)$$

$$Y_{9}^{4}(\theta,\varphi)={3\over 128}\sqrt{95095\over 2\pi}\cdot e^{4i\varphi}\cdot\sin^{4}\theta\cdot(17\cos^{5}\theta-10\cos^{3}\theta+1\cos\theta)$$

$$Y_{9}^{5}(\theta,\varphi)={-3\over 256}\sqrt{2717\over \pi}\cdot e^{5i\varphi}\cdot\sin^{5}\theta\cdot(85\cos^{4}\theta-30\cos^{2}\theta+1)$$

$$Y_{9}^{6}(\theta,\varphi)={1\over 128}\sqrt{40755\over \pi}\cdot e^{6i\varphi}\cdot\sin^{6}\theta\cdot(17\cos^{3}\theta-3\cos\theta)$$

$$Y_{9}^{7}(\theta,\varphi)={-3\over 512}\sqrt{13585\over \pi}\cdot e^{7i\varphi}\cdot\sin^{7}\theta\cdot(17\cos^{2}\theta-1)$$

$$Y_{9}^{8}(\theta,\varphi)={3\over 256}\sqrt{230945\over 2\pi}\cdot e^{8i\varphi}\cdot\sin^{8}\theta\cdot\cos\theta$$

$$Y_{9}^{9}(\theta,\varphi)={-1\over 512}\sqrt{230945\over \pi}\cdot e^{9i\varphi}\cdot\sin^{9}\theta$$

## $l=10$

$$Y_{10}^{-10}(\theta,\varphi)={1\over 1024}\sqrt{969969\over \pi}\cdot e^{-10i\varphi}\cdot\sin^{10}\theta$$

$$Y_{10}^{-9}(\theta,\varphi)={1\over 512}\sqrt{4849845\over \pi}\cdot e^{-9i\varphi}\cdot\sin^{9}\theta\cdot\cos\theta$$

$$Y_{10}^{-8}(\theta,\varphi)={1\over 512}\sqrt{255255\over 2\pi}\cdot e^{-8i\varphi}\cdot\sin^{8}\theta\cdot(19\cos^{2}\theta-1)$$

$$Y_{10}^{-7}(\theta,\varphi)={3\over 512}\sqrt{85085\over \pi}\cdot e^{-7i\varphi}\cdot\sin^{7}\theta\cdot(19\cos^{3}\theta-3\cos\theta)$$

$$Y_{10}^{-6}(\theta,\varphi)={3\over 1024}\sqrt{5005\over \pi}\cdot e^{-6i\varphi}\cdot\sin^{6}\theta\cdot(323\cos^{4}\theta-102\cos^{2}\theta+3)$$

$$Y_{10}^{-5}(\theta,\varphi)={3\over 256}\sqrt{1001\over \pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\cdot(323\cos^{5}\theta-170\cos^{3}\theta+15\cos\theta)$$

$$Y_{10}^{-4}(\theta,\varphi)={3\over 256}\sqrt{5005\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot(323\cos^{6}\theta-255\cos^{4}\theta+45\cos^{2}\theta-1)$$

$$Y_{10}^{-3}(\theta,\varphi)={3\over 256}\sqrt{5005\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(323\cos^{7}\theta-357\cos^{5}\theta+105\cos^{3}\theta-7\cos\theta)$$

$$Y_{10}^{-2}(\theta,\varphi)={3\over 512}\sqrt{385\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(4199\cos^{8}\theta-6188\cos^{6}\theta+2730\cos^{4}\theta-364\cos^{2}\theta+7)$$

$$Y_{10}^{-1}(\theta,\varphi)={1\over 256}\sqrt{1155\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(4199\cos^{9}\theta-7956\cos^{7}\theta+4914\cos^{5}\theta-1092\cos^{3}\theta+63\cos\theta)$$

$$Y_{10}^{0}(\theta,\varphi)={1\over 512}\sqrt{21\over \pi}\cdot(46189\cos^{10}\theta-109395\cos^{8}\theta+90090\cos^{6}\theta-30030\cos^{4}\theta+3465\cos^{2}\theta-63)$$

$$Y_{10}^{1}(\theta,\varphi)={-1\over 256}\sqrt{1155\over 2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(4199\cos^{9}\theta-7956\cos^{7}\theta+4914\cos^{5}\theta-1092\cos^{3}\theta+63\cos\theta)$$

$$Y_{10}^{2}(\theta,\varphi)={3\over 512}\sqrt{385\over 2\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot(4199\cos^{8}\theta-6188\cos^{6}\theta+2730\cos^{4}\theta-364\cos^{2}\theta+7)$$

$$Y_{10}^{3}(\theta,\varphi)={-3\over 256}\sqrt{5005\over \pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\cdot(323\cos^{7}\theta-357\cos^{5}\theta+105\cos^{3}\theta-7\cos\theta)$$

$$Y_{10}^{4}(\theta,\varphi)={3\over 256}\sqrt{5005\over 2\pi}\cdot e^{4i\varphi}\cdot\sin^{4}\theta\cdot(323\cos^{6}\theta-255\cos^{4}\theta+45\cos^{2}\theta-1)$$

$$Y_{10}^{5}(\theta,\varphi)={-3\over 256}\sqrt{1001\over \pi}\cdot e^{5i\varphi}\cdot\sin^{5}\theta\cdot(323\cos^{5}\theta-170\cos^{3}\theta+15\cos\theta)$$

$$Y_{10}^{6}(\theta,\varphi)={3\over 1024}\sqrt{5005\over \pi}\cdot e^{6i\varphi}\cdot\sin^{6}\theta\cdot(323\cos^{4}\theta-102\cos^{2}\theta+3)$$

$$Y_{10}^{7}(\theta,\varphi)={-3\over 512}\sqrt{85085\over \pi}\cdot e^{7i\varphi}\cdot\sin^{7}\theta\cdot(19\cos^{3}\theta-3\cos\theta)$$

$$Y_{10}^{8}(\theta,\varphi)={1\over 512}\sqrt{255255\over 2\pi}\cdot e^{8i\varphi}\cdot\sin^{8}\theta\cdot(19\cos^{2}\theta-1)$$

$$Y_{10}^{9}(\theta,\varphi)={-1\over 512}\sqrt{4849845\over \pi}\cdot e^{9i\varphi}\cdot\sin^{9}\theta\cdot\cos\theta$$

$$Y_{10}^{10}(\theta,\varphi)={1\over 1024}\sqrt{969969\over \pi}\cdot e^{10i\varphi}\cdot\sin^{10}\theta$$