Table of spherical harmonics

This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree $$\ell = 10$$. Some of these formulas are expressed in terms of the Cartesian expansion of the spherical harmonics into polynomials in x, y, z, and r. For purposes of this table, it is useful to express the usual spherical to Cartesian transformations that relate these Cartesian components to $$\theta$$ and $$\varphi$$ as

$$ \begin{cases} \cos(\theta) & = z/r\\ e^{\pm i\varphi} \cdot \sin(\theta) & = (x \pm iy)/r \end{cases} $$

Complex spherical harmonics
For ℓ = 0, …, 5, see.

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

ℓ = 1
$$ \begin{align} Y_{1}^{-1}(\theta,\varphi) &= & & {1\over 2}\sqrt{3\over 2\pi}\cdot e^{-i\varphi}\cdot\sin\theta & &= & &{1\over 2}\sqrt{3\over 2\pi} \cdot{(x-iy)\over r} \\ Y_{1}^{ 0}(\theta,\varphi) &= & & {1\over 2}\sqrt{3\over \pi}\cdot                   \cos\theta & &= & &{1\over 2}\sqrt{3\over  \pi} \cdot{z\over r} \\ Y_{1}^{ 1}(\theta,\varphi) &= &-& {1\over 2}\sqrt{3\over 2\pi}\cdot e^{i\varphi}\cdot \sin\theta & &= &-&{1\over 2}\sqrt{3\over 2\pi} \cdot{(x+iy)\over r} \end{align} $$

ℓ = 2
$$ \begin{align} Y_{2}^{-2}(\theta,\varphi)&=& &{1\over 4}\sqrt{15\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\quad                       &&=& &{1\over 4}\sqrt{15\over 2\pi}\cdot{(x - iy)^2 \over r^{2}}&\\ Y_{2}^{-1}(\theta,\varphi)&=& &{1\over 2}\sqrt{15\over 2\pi}\cdot e^{-i\varphi}\cdot\sin   \theta\cdot  \cos\theta\quad       &&=& &{1\over 2}\sqrt{15\over 2\pi}\cdot{(x - iy) \cdot z \over r^{2}}&\\ Y_{2}^{ 0}(\theta,\varphi)&=& &{1\over 4}\sqrt{ 5\over \pi}\cdot                                       (3\cos^{2}\theta-1)\quad&&=& &{1\over 4}\sqrt{ 5\over  \pi}\cdot{(3z^{2}-r^{2})\over r^{2}}&\\ Y_{2}^{ 1}(\theta,\varphi)&=&-&{1\over 2}\sqrt{15\over 2\pi}\cdot e^{ i\varphi}\cdot\sin    \theta\cdot  \cos\theta\quad       &&=&-&{1\over 2}\sqrt{15\over 2\pi}\cdot{(x + iy) \cdot z \over r^{2}}&\\ Y_{2}^{ 2}(\theta,\varphi)&=& &{1\over 4}\sqrt{15\over 2\pi}\cdot e^{ 2i\varphi}\cdot\sin^{2}\theta\quad                       &&=& &{1\over 4}\sqrt{15\over 2\pi}\cdot{(x + iy)^2 \over r^{2}}& \end{align}$$

ℓ = 3
$$ \begin{align} Y_{3}^{-3}(\theta,\varphi) &=& &{1\over 8}\sqrt{ 35\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\quad& &=& & {1\over 8}\sqrt{35\over \pi}\cdot{(x - iy)^{3}\over r^{3}}&\\ Y_{3}^{-2}(\theta,\varphi) &=& &{1\over 4}\sqrt{105\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot\cos\theta\quad& &=& & {1\over 4}\sqrt{105\over 2\pi}\cdot{(x- iy)^2 \cdot z \over r^{3}}&\\ Y_{3}^{-1}(\theta,\varphi) &=& &{1\over 8}\sqrt{ 21\over \pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(5\cos^{2}\theta-1)\quad& &=& &{1\over 8}\sqrt{21\over \pi}\cdot{(x - iy) \cdot (5z^2- r^2)\over r^{3}}&\\ Y_{3}^{ 0}(\theta,\varphi) &=& &{1\over 4}\sqrt{ 7\over  \pi}\cdot(5\cos^{3}\theta-3\cos\theta)\quad& &=& &{1\over 4}\sqrt{7\over \pi}\cdot{(5z^3 - 3zr^2)\over r^{3}}&\\ Y_{3}^{ 1}(\theta,\varphi) &=&-&{1\over 8}\sqrt{ 21\over \pi}\cdot e^ { i\varphi}\cdot\sin\theta\cdot(5\cos^{2}\theta-1)\quad& &=& &{-1\over 8}\sqrt{21\over \pi}\cdot{(x + iy) \cdot (5z^2 - r^2) \over r^{3}}&\\ Y_{3}^{ 2}(\theta,\varphi) &=& &{1\over 4}\sqrt{105\over 2\pi}\cdot e^ {2i\varphi}\cdot\sin^{2}\theta\cdot\cos\theta\quad& &=& &{1\over 4}\sqrt{105\over 2\pi}\cdot{(x + iy)^2 \cdot z \over r^{3}}&\\ Y_{3}^{ 3}(\theta,\varphi) &=&-&{1\over 8}\sqrt{ 35\over \pi}\cdot e^ {3i\varphi}\cdot\sin^{3}\theta\quad& &=& &{-1\over 8}\sqrt{35\over \pi}\cdot{(x + iy)^3\over r^{3}}& \end{align}$$

ℓ = 4
$$ \begin{align} Y_{4}^{-4}(\theta,\varphi)&={ 3\over 16}\sqrt{35\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta= \frac{3}{16} \sqrt{\frac{35}{2 \pi}} \cdot \frac{(x - i y)^4}{r^4}\\ Y_{4}^{-3}(\theta,\varphi)&={ 3\over 8}\sqrt{35\over  \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot\cos\theta= \frac{3}{8} \sqrt{\frac{35}{\pi}} \cdot \frac{(x - i y)^3 z}{r^4}\\ Y_{4}^{-2}(\theta,\varphi)&={ 3\over 8}\sqrt{ 5\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(7\cos^{2}\theta-1)= \frac{3}{8} \sqrt{\frac{5}{2 \pi}} \cdot \frac{(x - i y)^2 \cdot (7 z^2 - r^2)}{r^4}\\ Y_{4}^{-1}(\theta,\varphi)&={ 3\over 8}\sqrt{ 5\over  \pi}\cdot e^{- i\varphi}\cdot\sin\theta\cdot(7\cos^{3}\theta-3\cos\theta)= \frac{3}{8} \sqrt{\frac{5}{\pi}} \cdot \frac{(x - i y) \cdot (7 z^3 - 3 z r^2)}{r^4}\\ Y_{4}^{ 0}(\theta,\varphi)&={ 3\over 16}\sqrt{ 1\over \pi}\cdot(35\cos^{4}\theta-30\cos^{2}\theta+3)= \frac{3}{16} \sqrt{\frac{1}{\pi}} \cdot \frac{(35 z^4 - 30 z^2 r^2 + 3 r^4)}{r^4}\\ Y_{4}^{ 1}(\theta,\varphi)&={-3\over 8}\sqrt{ 5\over  \pi}\cdot e^{  i\varphi}\cdot\sin\theta\cdot(7\cos^{3}\theta-3\cos\theta)= \frac{- 3}{8} \sqrt{\frac{5}{\pi}} \cdot \frac{(x + i y) \cdot (7 z^3 - 3 z r^2)}{r^4}\\ Y_{4}^{ 2}(\theta,\varphi)&={ 3\over 8}\sqrt{ 5\over 2\pi}\cdot e^{ 2i\varphi}\cdot\sin^{2}\theta\cdot(7\cos^{2}\theta-1)= \frac{3}{8} \sqrt{\frac{5}{2 \pi}} \cdot \frac{(x + i y)^2 \cdot (7 z^2 - r^2)}{r^4}\\ Y_{4}^{ 3}(\theta,\varphi)&={-3\over 8}\sqrt{35\over  \pi}\cdot e^{ 3i\varphi}\cdot\sin^{3}\theta\cdot\cos\theta= \frac{- 3}{8} \sqrt{\frac{35}{\pi}} \cdot \frac{(x + i y)^3 z}{r^4}\\ Y_{4}^{ 4}(\theta,\varphi)&={ 3\over 16}\sqrt{35\over 2\pi}\cdot e^{ 4i\varphi}\cdot\sin^{4}\theta= \frac{3}{16} \sqrt{\frac{35}{2 \pi}} \cdot \frac{(x + i y)^4}{r^4} \end{align} $$

ℓ = 5
$$ \begin{align} Y_{5}^{-5}(\theta,\varphi)&={ 3\over 32}\sqrt{ 77\over  \pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\\ Y_{5}^{-4}(\theta,\varphi)&={ 3\over 16}\sqrt{ 385\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot\cos\theta\\ Y_{5}^{-3}(\theta,\varphi)&={ 1\over 32}\sqrt{ 385\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(9\cos^{2}\theta-1)\\ Y_{5}^{-2}(\theta,\varphi)&={ 1\over 8}\sqrt{1155\over 2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(3\cos^{3}\theta-\cos\theta)\\ Y_{5}^{-1}(\theta,\varphi)&={ 1\over 16}\sqrt{ 165\over 2\pi}\cdot e^{- i\varphi}\cdot\sin   \theta\cdot(21\cos^{4}\theta-14\cos^{2}\theta+1)\\ Y_{5}^{ 0}(\theta,\varphi)&={ 1\over 16}\sqrt{ 11\over  \pi}\cdot                                       (63\cos^{5}\theta-70\cos^{3}\theta+15\cos\theta)\\ Y_{5}^{ 1}(\theta,\varphi)&={-1\over 16}\sqrt{ 165\over 2\pi}\cdot e^{ i\varphi}\cdot\sin    \theta\cdot(21\cos^{4}\theta-14\cos^{2}\theta+1)\\ Y_{5}^{ 2}(\theta,\varphi)&={ 1\over 8}\sqrt{1155\over 2\pi}\cdot e^{ 2i\varphi}\cdot\sin^{2}\theta\cdot(3\cos^{3}\theta-\cos\theta)\\ Y_{5}^{ 3}(\theta,\varphi)&={-1\over 32}\sqrt{ 385\over \pi}\cdot e^{ 3i\varphi}\cdot\sin^{3}\theta\cdot(9\cos^{2}\theta-1)\\ Y_{5}^{ 4}(\theta,\varphi)&={ 3\over 16}\sqrt{ 385\over 2\pi}\cdot e^{ 4i\varphi}\cdot\sin^{4}\theta\cdot\cos\theta\\ Y_{5}^{ 5}(\theta,\varphi)&={-3\over 32}\sqrt{ 77\over  \pi}\cdot e^{ 5i\varphi}\cdot\sin^{5}\theta \end{align}$$

ℓ = 6
$$ \begin{align} Y_{6}^{-6}(\theta,\varphi)&= {1\over 64}\sqrt{3003\over \pi}\cdot e^{-6i\varphi}\cdot\sin^{6}\theta\\ Y_{6}^{-5}(\theta,\varphi)&= {3\over 32}\sqrt{1001\over \pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\cdot\cos\theta\\ Y_{6}^{-4}(\theta,\varphi)&= {3\over 32}\sqrt{ 91\over 2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot(11\cos^{2}\theta-1)\\ Y_{6}^{-3}(\theta,\varphi)&= {1\over 32}\sqrt{1365\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(11\cos^{3}\theta-3\cos\theta)\\ Y_{6}^{-2}(\theta,\varphi)&= {1\over 64}\sqrt{1365\over \pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(33\cos^{4}\theta-18\cos^{2}\theta+1)\\ Y_{6}^{-1}(\theta,\varphi)&= {1\over 16}\sqrt{ 273\over 2\pi}\cdot e^{- i\varphi}\cdot\sin   \theta\cdot(33\cos^{5}\theta-30\cos^{3}\theta+5\cos\theta)\\ Y_{6}^{ 0}(\theta,\varphi)&= {1\over 32}\sqrt{ 13\over  \pi}\cdot                                       (231\cos^{6}\theta-315\cos^{4}\theta+105\cos^{2}\theta-5)\\ Y_{6}^{ 1}(\theta,\varphi)&=-{1\over 16}\sqrt{ 273\over 2\pi}\cdot e^{ i\varphi}\cdot\sin    \theta\cdot(33\cos^{5}\theta-30\cos^{3}\theta+5\cos\theta)\\ Y_{6}^{ 2}(\theta,\varphi)&= {1\over 64}\sqrt{1365\over \pi}\cdot e^{ 2i\varphi}\cdot\sin^{2}\theta\cdot(33\cos^{4}\theta-18\cos^{2}\theta+1)\\ Y_{6}^{ 3}(\theta,\varphi)&=-{1\over 32}\sqrt{1365\over \pi}\cdot e^{ 3i\varphi}\cdot\sin^{3}\theta\cdot(11\cos^{3}\theta-3\cos\theta)\\ Y_{6}^{ 4}(\theta,\varphi)&= {3\over 32}\sqrt{ 91\over 2\pi}\cdot e^{ 4i\varphi}\cdot\sin^{4}\theta\cdot(11\cos^{2}\theta-1)\\ Y_{6}^{ 5}(\theta,\varphi)&=-{3\over 32}\sqrt{1001\over \pi}\cdot e^{ 5i\varphi}\cdot\sin^{5}\theta\cdot\cos\theta\\ Y_{6}^{ 6}(\theta,\varphi)&= {1\over 64}\sqrt{3003\over \pi}\cdot e^{ 6i\varphi}\cdot\sin^{6}\theta \end{align}$$

ℓ = 7
$$ \begin{align} 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 \end{align}$$

ℓ = 8
$$ \begin{align} 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-\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-\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 \end{align}$$

ℓ = 9
$$ \begin{align} 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+\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+\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 \end{align}$$

ℓ = 10
$$ \begin{align} 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 \end{align}$$

2D polar/azimuthal angle maps
Below the complex spherical harmonics are represented on 2D plots with the azimuthal angle, $$\phi$$, on the horizontal axis and the polar angle, $$\theta$$, on the vertical axis. The saturation of the color at any point represents the magnitude of the spherical harmonic and the hue represents the phase.



Polar plots
Below the complex spherical harmonics are represented on polar plots. The magnitude of the spherical harmonic at particular polar and azimuthal angles is represented by the saturation of the color at that point and the phase is represented by the hue at that point.



Polar plots with magnitude as radius
Below the complex spherical harmonics are represented on polar plots. The magnitude of the spherical harmonic at particular polar and azimuthal angles is represented by the radius of the plot at that point and the phase is represented by the hue at that point.



Real spherical harmonics
For each real spherical harmonic, the corresponding atomic orbital symbol (s, p, d, f) is reported as well.

For ℓ = 0, …, 3, see.

ℓ = 0
$$Y_{00} = s = Y_0^0 = \frac{1}{2} \sqrt{\frac{1}{\pi}}$$

ℓ = 1
$$\begin{align} Y_{1,-1} & = p_y = i \sqrt{\frac{1}{2}} \left( Y_1^{- 1} + Y_1^1 \right) = \sqrt{\frac{3}{4 \pi}} \cdot \frac{y}{r} = \sqrt{\frac{3}{4 \pi}} \sin( \theta) \sin \varphi  \\ Y_{1,0} & = p_z = Y_1^0 = \sqrt{\frac{3}{4 \pi}} \cdot \frac{z}{r} = \sqrt{\frac{3}{4 \pi}} \cos( \theta) \\ Y_{1,1} & = p_x = \sqrt{\frac{1}{2}} \left( Y_1^{- 1} - Y_1^1 \right) = \sqrt{\frac{3}{4 \pi}} \cdot \frac{x}{r} = \sqrt{\frac{3}{4 \pi}} \sin( \theta) \cos \varphi \end{align}$$

ℓ = 2
$$ \begin{align} Y_{2,-2} & = d_{xy} = i \sqrt{\frac{1}{2}} \left( Y_2^{- 2} - Y_2^2\right) = \frac{1}{2} \sqrt{\frac{15}{\pi}} \cdot \frac{x y}{r^2} = \frac{1}{4} \sqrt{\frac{15}{\pi}} \sin^{2}\theta \sin(2\varphi) \\ Y_{2,-1} & = d_{yz} = i \sqrt{\frac{1}{2}} \left( Y_2^{- 1} + Y_2^1 \right) = \frac{1}{2} \sqrt{\frac{15}{\pi}} \cdot \frac{y \cdot z}{r^2} = \frac{1}{4} \sqrt{\frac{15}{\pi}} \sin(2 \theta) \sin \varphi \\ Y_{2,0} & = d_{z^2} = Y_2^0 = \frac{1}{4} \sqrt{\frac{5}{\pi}} \cdot \frac{3z^2 - r^2}{r^2} = \frac{1}{4} \sqrt{\frac{5}{\pi}} (3\cos^{2}\theta -1)\\ Y_{2,1} & = d_{xz} = \sqrt{\frac{1}{2}} \left( Y_2^{- 1} - Y_2^1 \right) = \frac{1}{2} \sqrt{\frac{15}{\pi}} \cdot \frac{x \cdot z}{r^2} = \frac{1}{4} \sqrt{\frac{15}{\pi}} \sin(2 \theta) \cos \varphi\\ Y_{2,2} & = d_{x^2-y^2} = \sqrt{\frac{1}{2}} \left( Y_2^{- 2} + Y_2^2 \right) = \frac{1}{4} \sqrt{\frac{15}{\pi}} \cdot \frac{x^2 - y^2 }{r^2} = \frac{1}{4} \sqrt{\frac{15}{\pi}} \sin^{2}\theta \cos(2\varphi) \end{align}$$

ℓ = 3
$$ \begin{align} Y_{3,-3} & = f_{y(3x^2-y^2)} = i \sqrt{\frac{1}{2}} \left( Y_3^{- 3} + Y_3^3 \right) = \frac{1}{4} \sqrt{\frac{35}{2 \pi}} \cdot \frac{y \left( 3 x^2 - y^2 \right)}{r^3} \\ Y_{3,-2} & = f_{xyz} = i \sqrt{\frac{1}{2}} \left( Y_3^{- 2} - Y_3^2 \right) = \frac{1}{2} \sqrt{\frac{105}{\pi}} \cdot \frac{xy \cdot z}{r^3} \\ Y_{3,-1} & = f_{yz^2} = i \sqrt{\frac{1}{2}} \left( Y_3^{- 1} + Y_3^1 \right) = \frac{1}{4} \sqrt{\frac{21}{2 \pi}} \cdot \frac{y \cdot (5 z^2 - r^2)}{r^3} \\ Y_{3,0} & = f_{z^3} = Y_3^0 = \frac{1}{4} \sqrt{\frac{7}{\pi}} \cdot \frac{5 z^3 - 3 z r^2}{r^3} \\ Y_{3,1} & = f_{xz^2} = \sqrt{\frac{1}{2}} \left( Y_3^{- 1} - Y_3^1 \right) = \frac{1}{4} \sqrt{\frac{21}{2 \pi}} \cdot \frac{x \cdot (5 z^2 - r^2)}{r^3} \\ Y_{3,2} & = f_{z(x^2-y^2)} = \sqrt{\frac{1}{2}} \left( Y_3^{- 2} + Y_3^2 \right) = \frac{1}{4} \sqrt{\frac{105}{\pi}} \cdot \frac{\left( x^2 - y^2 \right) \cdot z}{r^3} \\ Y_{3,3} & = f_{x(x^2-3y^2)} = \sqrt{\frac{1}{2}} \left( Y_3^{- 3} - Y_3^3 \right) = \frac{1}{4} \sqrt{\frac{35}{2 \pi}} \cdot \frac{x \left( x^2 - 3 y^2 \right)}{r^3} \end{align}$$

ℓ = 4
$$ \begin{align} Y_{4,-4} & = i \sqrt{\frac{1}{2}} \left( Y_4^{- 4} - Y_4^4 \right) = \frac{3}{4} \sqrt{\frac{35}{\pi}} \cdot \frac{xy \left( x^2 - y^2 \right)}{r^4} \\ Y_{4,-3} & = i \sqrt{\frac{1}{2}} \left( Y_4^{- 3} + Y_4^3 \right) = \frac{3}{4} \sqrt{\frac{35}{2 \pi}} \cdot \frac{y (3 x^2 - y^2) \cdot z}{r^4} \\ Y_{4,-2} & = i \sqrt{\frac{1}{2}} \left( Y_4^{- 2} - Y_4^2 \right) = \frac{3}{4} \sqrt{\frac{5}{\pi}} \cdot \frac{xy \cdot (7 z^2 - r^2)}{r^4} \\ Y_{4,-1} & = i \sqrt{\frac{1}{2}} \left( Y_4^{- 1} + Y_4^1\right) = \frac{3}{4} \sqrt{\frac{5}{2 \pi}} \cdot \frac{y \cdot (7 z^3 - 3 z r^2)}{r^4} \\ Y_{4,0} & = Y_4^0 = \frac{3}{16} \sqrt{\frac{1}{\pi}} \cdot \frac{35 z^4 - 30 z^2 r^2 + 3 r^4}{r^4} \\ Y_{4,1} & = \sqrt{\frac{1}{2}} \left( Y_4^{- 1} - Y_4^1 \right) = \frac{3}{4} \sqrt{\frac{5}{2 \pi}} \cdot \frac{x \cdot (7 z^3 - 3 z r^2)}{r^4} \\ Y_{4,2} & = \sqrt{\frac{1}{2}} \left( Y_4^{- 2} + Y_4^2 \right) = \frac{3}{8} \sqrt{\frac{5}{\pi}} \cdot \frac{(x^2 - y^2) \cdot (7 z^2 - r^2)}{r^4} \\ Y_{4,3} & = \sqrt{\frac{1}{2}} \left( Y_4^{- 3} - Y_4^3 \right) = \frac{3}{4} \sqrt{\frac{35}{2 \pi}} \cdot \frac{x(x^2 - 3 y^2) \cdot z}{r^4} \\ Y_{4,4} & = \sqrt{\frac{1}{2}} \left( Y_4^{- 4} + Y_4^4 \right) = \frac{3}{16} \sqrt{\frac{35}{\pi}} \cdot \frac{x^2 \left( x^2 - 3 y^2 \right) - y^2 \left( 3 x^2 - y^2 \right)}{r^4} \end{align}$$

2D polar/azimuthal angle maps
Below the real spherical harmonics are represented on 2D plots with the azimuthal angle, $$\phi$$, on the horizontal axis and the polar angle, $$\theta$$, on the vertical axis. The saturation of the color at any point represents the magnitude of the spherical harmonic and the hue represents the phase.



Polar plots
Below the real spherical harmonics are represented on polar plots. The magnitude of the spherical harmonic at particular polar and azimuthal angles is represented by the saturation of the color at that point and the phase is represented by the hue at that point.



Polar plots with magnitude as radius
Below the real spherical harmonics are represented on polar plots. The magnitude of the spherical harmonic at particular polar and azimuthal angles is represented by the radius of the plot at that point and the phase is represented by the hue at that point.



Polar plots with amplitude as elevation
Below the real spherical harmonics are represented on polar plots. The amplitude of the spherical harmonic (magnitude and sign) at a particular polar and azimuthal angle is represented by the elevation of the plot at that point above or below the surface of a uniform sphere. The magnitude is also represented by the saturation of the color at a given point. The phase is represented by the hue at a given point.



General references

 * See section 3 in (see section 3.3)
 * For complex spherical harmonics, see also SphericalHarmonicY[l,m,theta,phi at Wolfram Alpha], especially for specific values of l and m.