Table of spherical harmonics

From Exampleproblems

This is a table of spherical harmonics up to l=10.

1 Spherical harmonics with l = 0
2 Spherical harmonics with l = 1
3 Spherical harmonics with l = 2
4 Spherical harmonics with l = 3
5 Spherical harmonics with l = 4
6 Spherical harmonics with l = 5
7 Spherical harmonics with l = 6
8 Spherical harmonics with l = 7
9 Spherical harmonics with l = 8
10 Spherical harmonics with l = 9
11 Spherical harmonics with l = 10
12 See also

Spherical harmonics with l = 0

$Y_{0}^{0}(x)={1\over 2}\sqrt{1\over \pi}$

Spherical harmonics with l = 1

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

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

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

Spherical harmonics with l = 2

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

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

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

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

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

Spherical harmonics with l = 3

$Y_{3}^{-3}(x)={1\over 8}\sqrt{35\over \pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\quad={1\over 8}\sqrt{35\over \pi}\cdot{(x^{3}-3ix^{2}y-3xy^{2}+iy^{3})\over r^{3}}$

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

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

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

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

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

$Y_{3}^{3}(x)={-1\over 8}\sqrt{35\over \pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\quad={-1\over 8}\sqrt{35\over \pi}\cdot{(x^{3}+3ix^{2}y-3xy^{2}-iy^{3})\over r^{3}}$

Spherical harmonics with l = 4

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

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

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

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

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

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

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

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

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

Spherical harmonics with l = 5

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

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

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

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

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

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

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

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

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

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

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

Spherical harmonics with l = 6

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

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

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

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

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

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

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

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

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

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

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

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

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

Spherical harmonics with l = 7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Spherical harmonics with l = 8

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Spherical harmonics with l = 9

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Spherical harmonics with l = 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Table of spherical harmonics

From Exampleproblems

Contents

Spherical harmonics with l = 0

Spherical harmonics with l = 1

Spherical harmonics with l = 2

Spherical harmonics with l = 3

Spherical harmonics with l = 4

Spherical harmonics with l = 5

Spherical harmonics with l = 6

Spherical harmonics with l = 7

Spherical harmonics with l = 8

Spherical harmonics with l = 9

Spherical harmonics with l = 10

See also

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