球諧函數表

這裏列出球諧函數 $Y_l^m$ ，以方程式表示為

$Y_{\ell }^m(\theta ,\phi )=\sqrt{\frac{(2\ell +1)}{4\pi }\frac{(\ell -m)!}{(\ell +m)!}}{P}_{\ell }^m(\cos \theta )e^{im\phi }$ ；

其中，$l$ 為正值整數，$m$ 為小於或等於 $l$ 的正值整數，$P_{\ell }^m$ 是伴隨勒讓德多項式，以方程式表示為

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

表內有些方程式也給出直角坐標版本。球坐標與直角坐標之間的變換關係是

$x=r\sin \theta \cos \phi$ 、

$y=r\sin \theta \sin \phi$ 、

$z=r\cos \theta$ 。

$l=0$

$Y_0^0(\theta ,\phi )=\frac{1}{2}\sqrt{\frac{1}{\pi }}$

$l=1$

$Y_1^{-1}(\theta ,\phi )=\frac{1}{2}\sqrt{\frac{3}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta =\frac{1}{2}\sqrt{\frac{3}{2\pi }}\cdot \frac{(x-iy)}{r}$

$Y_1^0(\theta ,\phi )=\frac{1}{2}\sqrt{\frac{3}{\pi }}\cdot \cos \theta =\frac{1}{2}\sqrt{\frac{3}{\pi }}\cdot \frac{z}{r}$

$Y_1^1(\theta ,\phi )=\frac{-1}{2}\sqrt{\frac{3}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta =\frac{-1}{2}\sqrt{\frac{3}{2\pi }}\cdot \frac{(x+iy)}{r}$

$l=2$

$Y_2^{-2}(\theta ,\phi )=\frac{1}{4}\sqrt{\frac{15}{2\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta =\frac{1}{4}\sqrt{\frac{15}{2\pi }}\cdot \frac{(x-iy{)}^2}{r^2}$

$Y_2^{-1}(\theta ,\phi )=\frac{1}{2}\sqrt{\frac{15}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot \cos \theta =\frac{1}{2}\sqrt{\frac{15}{2\pi }}\cdot \frac{(x-iy)z}{r^2}$

$Y_2^0(\theta ,\phi )=\frac{1}{4}\sqrt{\frac{5}{\pi }}\cdot (3{\cos }^2\theta -1)=\frac{1}{4}\sqrt{\frac{5}{\pi }}\cdot \frac{(-x^2-y^2+2z^2)}{r^2}$

$Y_2^1(\theta ,\phi )=\frac{-1}{2}\sqrt{\frac{15}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot \cos \theta =\frac{-1}{2}\sqrt{\frac{15}{2\pi }}\cdot \frac{(x+iy)z}{r^2}$

$Y_2^2(\theta ,\phi )=\frac{1}{4}\sqrt{\frac{15}{2\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta =\frac{1}{4}\sqrt{\frac{15}{2\pi }}\cdot \frac{(x+iy{)}^2}{r^2}$

$l=3$

$Y_3^{-3}(\theta ,\phi )=\frac{1}{8}\sqrt{\frac{35}{\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta =\frac{1}{8}\sqrt{\frac{35}{\pi }}\cdot \frac{(x-iy{)}^3}{r^3}$

$Y_3^{-2}(\theta ,\phi )=\frac{1}{4}\sqrt{\frac{105}{2\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot \cos \theta =\frac{1}{4}\sqrt{\frac{105}{2\pi }}\cdot \frac{(x-iy{)}^{2}z}{r^3}$

$Y_3^{-1}(\theta ,\phi )=\frac{1}{8}\sqrt{\frac{21}{\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (5{\cos }^2\theta -1)=\frac{1}{8}\sqrt{\frac{21}{\pi }}\cdot \frac{(x-iy)(4z^2-x^2-y^2)}{r^3}$

$Y_3^0(\theta ,\phi )=\frac{1}{4}\sqrt{\frac{7}{\pi }}\cdot (5{\cos }^3\theta -3\cos \theta )=\frac{1}{4}\sqrt{\frac{7}{\pi }}\cdot \frac{z(2z^2-3x^2-3y^2)}{r^3}$

$Y_3^1(\theta ,\phi )=\frac{-1}{8}\sqrt{\frac{21}{\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (5{\cos }^2\theta -1)=\frac{-1}{8}\sqrt{\frac{21}{\pi }}\cdot \frac{(x+iy)(4z^2-x^2-y^2)}{r^3}$

$Y_3^2(\theta ,\phi )=\frac{1}{4}\sqrt{\frac{105}{2\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot \cos \theta =\frac{1}{4}\sqrt{\frac{105}{2\pi }}\cdot \frac{(x+iy{)}^{2}z}{r^3}$

$Y_3^3(\theta ,\phi )=\frac{-1}{8}\sqrt{\frac{35}{\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta =\frac{-1}{8}\sqrt{\frac{35}{\pi }}\cdot \frac{(x+iy{)}^3}{r^3}$

$l=4$

$Y_4^{-4}(\theta ,\phi )=\frac{3}{16}\sqrt{\frac{35}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta =\frac{3}{16}\sqrt{\frac{35}{2\pi }}\cdot \frac{(x-iy{)}^4}{r^4}$

$Y_4^{-3}(\theta ,\phi )=\frac{3}{8}\sqrt{\frac{35}{\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot \cos \theta =\frac{3}{8}\sqrt{\frac{35}{\pi }}\cdot \frac{(x-iy{)}^{3}z}{r^4}$

$Y_4^{-2}(\theta ,\phi )=\frac{3}{8}\sqrt{\frac{5}{2\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (7{\cos }^2\theta -1)=\frac{3}{8}\sqrt{\frac{5}{2\pi }}\cdot \frac{(x-iy{)}^2\cdot (7z^2-r^2)}{r^4}$

$Y_4^{-1}(\theta ,\phi )=\frac{3}{8}\sqrt{\frac{5}{\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (7{\cos }^3\theta -3\cos \theta )=\frac{3}{8}\sqrt{\frac{5}{\pi }}\cdot \frac{(x-iy)\cdot z\cdot (7z^2-3r^2)}{r^4}$

$Y_4^0(\theta ,\phi )=\frac{3}{16}\sqrt{\frac{1}{\pi }}\cdot (35{\cos }^4\theta -30{\cos }^2\theta +3)=\frac{3}{16}\sqrt{\frac{1}{\pi }}\cdot \frac{(35z^4-30z^2{r}^2+3r^4)}{r^4}$

$Y_4^1(\theta ,\phi )=\frac{-3}{8}\sqrt{\frac{5}{\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (7{\cos }^3\theta -3\cos \theta )=\frac{-3}{8}\sqrt{\frac{5}{\pi }}\cdot \frac{(x+iy)\cdot z\cdot (7z^2-3r^2)}{r^4}$

$Y_4^2(\theta ,\phi )=\frac{3}{8}\sqrt{\frac{5}{2\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (7{\cos }^2\theta -1)=\frac{3}{8}\sqrt{\frac{5}{2\pi }}\cdot \frac{(x+iy{)}^2\cdot (7z^2-r^2)}{r^4}$

$Y_4^3(\theta ,\phi )=\frac{-3}{8}\sqrt{\frac{35}{\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot \cos \theta =\frac{-3}{8}\sqrt{\frac{35}{\pi }}\cdot \frac{(x+iy{)}^{3}z}{r^4}$

$Y_4^4(\theta ,\phi )=\frac{3}{16}\sqrt{\frac{35}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta =\frac{3}{16}\sqrt{\frac{35}{2\pi }}\cdot \frac{(x+iy{)}^4}{r^4}$

$l=5$

$Y_5^{-5}(\theta ,\phi )=\frac{3}{32}\sqrt{\frac{77}{\pi }}\cdot e^{-5i\phi }\cdot {\sin }^5\theta$

$Y_5^{-4}(\theta ,\phi )=\frac{3}{16}\sqrt{\frac{385}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta \cdot \cos \theta$

$Y_5^{-3}(\theta ,\phi )=\frac{1}{32}\sqrt{\frac{385}{\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot (9{\cos }^2\theta -1)$

$Y_5^{-2}(\theta ,\phi )=\frac{1}{8}\sqrt{\frac{1155}{2\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (3{\cos }^3\theta -1\cos \theta )$

$Y_5^{-1}(\theta ,\phi )=\frac{1}{16}\sqrt{\frac{165}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (21{\cos }^4\theta -14{\cos }^2\theta +1)$

$Y_5^0(\theta ,\phi )=\frac{1}{16}\sqrt{\frac{11}{\pi }}\cdot (63{\cos }^5\theta -70{\cos }^3\theta +15\cos \theta )$

$Y_5^1(\theta ,\phi )=\frac{-1}{16}\sqrt{\frac{165}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (21{\cos }^4\theta -14{\cos }^2\theta +1)$

$Y_5^2(\theta ,\phi )=\frac{1}{8}\sqrt{\frac{1155}{2\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (3{\cos }^3\theta -1\cos \theta )$

$Y_5^3(\theta ,\phi )=\frac{-1}{32}\sqrt{\frac{385}{\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot (9{\cos }^2\theta -1)$

$Y_5^4(\theta ,\phi )=\frac{3}{16}\sqrt{\frac{385}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta \cdot \cos \theta$

$Y_5^5(\theta ,\phi )=\frac{-3}{32}\sqrt{\frac{77}{\pi }}\cdot e^{5i\phi }\cdot {\sin }^5\theta$

$l=6$

$Y_6^{-6}(\theta ,\phi )=\frac{1}{64}\sqrt{\frac{3003}{\pi }}\cdot e^{-6i\phi }\cdot {\sin }^6\theta$

$Y_6^{-5}(\theta ,\phi )=\frac{3}{32}\sqrt{\frac{1001}{\pi }}\cdot e^{-5i\phi }\cdot {\sin }^5\theta \cdot \cos \theta$

$Y_6^{-4}(\theta ,\phi )=\frac{3}{32}\sqrt{\frac{91}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta \cdot (11{\cos }^2\theta -1)$

$Y_6^{-3}(\theta ,\phi )=\frac{1}{32}\sqrt{\frac{1365}{\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot (11{\cos }^3\theta -3\cos \theta )$

$Y_6^{-2}(\theta ,\phi )=\frac{1}{64}\sqrt{\frac{1365}{\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (33{\cos }^4\theta -18{\cos }^2\theta +1)$

$Y_6^{-1}(\theta ,\phi )=\frac{1}{16}\sqrt{\frac{273}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (33{\cos }^5\theta -30{\cos }^3\theta +5\cos \theta )$

$Y_6^0(\theta ,\phi )=\frac{1}{32}\sqrt{\frac{13}{\pi }}\cdot (231{\cos }^6\theta -315{\cos }^4\theta +105{\cos }^2\theta -5)$

$Y_6^1(\theta ,\phi )=\frac{-1}{16}\sqrt{\frac{273}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (33{\cos }^5\theta -30{\cos }^3\theta +5\cos \theta )$

$Y_6^2(\theta ,\phi )=\frac{1}{64}\sqrt{\frac{1365}{\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (33{\cos }^4\theta -18{\cos }^2\theta +1)$

$Y_6^3(\theta ,\phi )=\frac{-1}{32}\sqrt{\frac{1365}{\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot (11{\cos }^3\theta -3\cos \theta )$

$Y_6^4(\theta ,\phi )=\frac{3}{32}\sqrt{\frac{91}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta \cdot (11{\cos }^2\theta -1)$

$Y_6^5(\theta ,\phi )=\frac{-3}{32}\sqrt{\frac{1001}{\pi }}\cdot e^{5i\phi }\cdot {\sin }^5\theta \cdot \cos \theta$

$Y_6^6(\theta ,\phi )=\frac{1}{64}\sqrt{\frac{3003}{\pi }}\cdot e^{6i\phi }\cdot {\sin }^6\theta$

$l=7$

$Y_7^{-7}(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{715}{2\pi }}\cdot e^{-7i\phi }\cdot {\sin }^7\theta$

$Y_7^{-6}(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{5005}{\pi }}\cdot e^{-6i\phi }\cdot {\sin }^6\theta \cdot \cos \theta$

$Y_7^{-5}(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{385}{2\pi }}\cdot e^{-5i\phi }\cdot {\sin }^5\theta \cdot (13{\cos }^2\theta -1)$

$Y_7^{-4}(\theta ,\phi )=\frac{3}{32}\sqrt{\frac{385}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta \cdot (13{\cos }^3\theta -3\cos \theta )$

$Y_7^{-3}(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{35}{2\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot (143{\cos }^4\theta -66{\cos }^2\theta +3)$

$Y_7^{-2}(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{35}{\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (143{\cos }^5\theta -110{\cos }^3\theta +15\cos \theta )$

$Y_7^{-1}(\theta ,\phi )=\frac{1}{64}\sqrt{\frac{105}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (429{\cos }^6\theta -495{\cos }^4\theta +135{\cos }^2\theta -5)$

$Y_7^0(\theta ,\phi )=\frac{1}{32}\sqrt{\frac{15}{\pi }}\cdot (429{\cos }^7\theta -693{\cos }^5\theta +315{\cos }^3\theta -35\cos \theta )$

$Y_7^1(\theta ,\phi )=\frac{-1}{64}\sqrt{\frac{105}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (429{\cos }^6\theta -495{\cos }^4\theta +135{\cos }^2\theta -5)$

$Y_7^2(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{35}{\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (143{\cos }^5\theta -110{\cos }^3\theta +15\cos \theta )$

$Y_7^3(\theta ,\phi )=\frac{-3}{64}\sqrt{\frac{35}{2\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot (143{\cos }^4\theta -66{\cos }^2\theta +3)$

$Y_7^4(\theta ,\phi )=\frac{3}{32}\sqrt{\frac{385}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta \cdot (13{\cos }^3\theta -3\cos \theta )$

$Y_7^5(\theta ,\phi )=\frac{-3}{64}\sqrt{\frac{385}{2\pi }}\cdot e^{5i\phi }\cdot {\sin }^5\theta \cdot (13{\cos }^2\theta -1)$

$Y_7^6(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{5005}{\pi }}\cdot e^{6i\phi }\cdot {\sin }^6\theta \cdot \cos \theta$

$Y_7^7(\theta ,\phi )=\frac{-3}{64}\sqrt{\frac{715}{2\pi }}\cdot e^{7i\phi }\cdot {\sin }^7\theta$

$l=8$

$Y_8^{-8}(\theta ,\phi )=\frac{3}{256}\sqrt{\frac{12155}{2\pi }}\cdot e^{-8i\phi }\cdot {\sin }^8\theta$

$Y_8^{-7}(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{12155}{2\pi }}\cdot e^{-7i\phi }\cdot {\sin }^7\theta \cdot \cos \theta$

$Y_8^{-6}(\theta ,\phi )=\frac{1}{128}\sqrt{\frac{7293}{\pi }}\cdot e^{-6i\phi }\cdot {\sin }^6\theta \cdot (15{\cos }^2\theta -1)$

$Y_8^{-5}(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{17017}{2\pi }}\cdot e^{-5i\phi }\cdot {\sin }^5\theta \cdot (5{\cos }^3\theta -1\cos \theta )$

$Y_8^{-4}(\theta ,\phi )=\frac{3}{128}\sqrt{\frac{1309}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta \cdot (65{\cos }^4\theta -26{\cos }^2\theta +1)$

$Y_8^{-3}(\theta ,\phi )=\frac{1}{64}\sqrt{\frac{19635}{2\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot (39{\cos }^5\theta -26{\cos }^3\theta +3\cos \theta )$

$Y_8^{-2}(\theta ,\phi )=\frac{3}{128}\sqrt{\frac{595}{\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (143{\cos }^6\theta -143{\cos }^4\theta +33{\cos }^2\theta -1)$

$Y_8^{-1}(\theta ,\phi )=\frac{3}{64}\sqrt{\frac{17}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (715{\cos }^7\theta -1001{\cos }^5\theta +385{\cos }^3\theta -35\cos \theta )$

$Y_8^0(\theta ,\phi )=\frac{1}{256}\sqrt{\frac{17}{\pi }}\cdot (6435{\cos }^8\theta -12012{\cos }^6\theta +6930{\cos }^4\theta -1260{\cos }^2\theta +35)$

$Y_8^1(\theta ,\phi )=\frac{-3}{64}\sqrt{\frac{17}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (715{\cos }^7\theta -1001{\cos }^5\theta +385{\cos }^3\theta -35\cos \theta )$

$Y_8^2(\theta ,\phi )=\frac{3}{128}\sqrt{\frac{595}{\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (143{\cos }^6\theta -143{\cos }^4\theta +33{\cos }^2\theta -1)$

$Y_8^3(\theta ,\phi )=\frac{-1}{64}\sqrt{\frac{19635}{2\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot (39{\cos }^5\theta -26{\cos }^3\theta +3\cos \theta )$

$Y_8^4(\theta ,\phi )=\frac{3}{128}\sqrt{\frac{1309}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta \cdot (65{\cos }^4\theta -26{\cos }^2\theta +1)$

$Y_8^5(\theta ,\phi )=\frac{-3}{64}\sqrt{\frac{17017}{2\pi }}\cdot e^{5i\phi }\cdot {\sin }^5\theta \cdot (5{\cos }^3\theta -1\cos \theta )$

$Y_8^6(\theta ,\phi )=\frac{1}{128}\sqrt{\frac{7293}{\pi }}\cdot e^{6i\phi }\cdot {\sin }^6\theta \cdot (15{\cos }^2\theta -1)$

$Y_8^7(\theta ,\phi )=\frac{-3}{64}\sqrt{\frac{12155}{2\pi }}\cdot e^{7i\phi }\cdot {\sin }^7\theta \cdot \cos \theta$

$Y_8^8(\theta ,\phi )=\frac{3}{256}\sqrt{\frac{12155}{2\pi }}\cdot e^{8i\phi }\cdot {\sin }^8\theta$

$l=9$

$Y_9^{-9}(\theta ,\phi )=\frac{1}{512}\sqrt{\frac{230945}{\pi }}\cdot e^{-9i\phi }\cdot {\sin }^9\theta$

$Y_9^{-8}(\theta ,\phi )=\frac{3}{256}\sqrt{\frac{230945}{2\pi }}\cdot e^{-8i\phi }\cdot {\sin }^8\theta \cdot \cos \theta$

$Y_9^{-7}(\theta ,\phi )=\frac{3}{512}\sqrt{\frac{13585}{\pi }}\cdot e^{-7i\phi }\cdot {\sin }^7\theta \cdot (17{\cos }^2\theta -1)$

$Y_9^{-6}(\theta ,\phi )=\frac{1}{128}\sqrt{\frac{40755}{\pi }}\cdot e^{-6i\phi }\cdot {\sin }^6\theta \cdot (17{\cos }^3\theta -3\cos \theta )$

$Y_9^{-5}(\theta ,\phi )=\frac{3}{256}\sqrt{\frac{2717}{\pi }}\cdot e^{-5i\phi }\cdot {\sin }^5\theta \cdot (85{\cos }^4\theta -30{\cos }^2\theta +1)$

$Y_9^{-4}(\theta ,\phi )=\frac{3}{128}\sqrt{\frac{95095}{2\pi }}\cdot e^{-4i\phi }\cdot {\sin }^4\theta \cdot (17{\cos }^5\theta -10{\cos }^3\theta +1\cos \theta )$

$Y_9^{-3}(\theta ,\phi )=\frac{1}{256}\sqrt{\frac{21945}{\pi }}\cdot e^{-3i\phi }\cdot {\sin }^3\theta \cdot (221{\cos }^6\theta -195{\cos }^4\theta +39{\cos }^2\theta -1)$

$Y_9^{-2}(\theta ,\phi )=\frac{3}{128}\sqrt{\frac{1045}{\pi }}\cdot e^{-2i\phi }\cdot {\sin }^2\theta \cdot (221{\cos }^7\theta -273{\cos }^5\theta +91{\cos }^3\theta -7\cos \theta )$

$Y_9^{-1}(\theta ,\phi )=\frac{3}{256}\sqrt{\frac{95}{2\pi }}\cdot e^{-i\phi }\cdot \sin \theta \cdot (2431{\cos }^8\theta -4004{\cos }^6\theta +2002{\cos }^4\theta -308{\cos }^2\theta +7)$

$Y_9^0(\theta ,\phi )=\frac{1}{256}\sqrt{\frac{19}{\pi }}\cdot (12155{\cos }^9\theta -25740{\cos }^7\theta +18018{\cos }^5\theta -4620{\cos }^3\theta +315\cos \theta )$

$Y_9^1(\theta ,\phi )=\frac{-3}{256}\sqrt{\frac{95}{2\pi }}\cdot e^{i\phi }\cdot \sin \theta \cdot (2431{\cos }^8\theta -4004{\cos }^6\theta +2002{\cos }^4\theta -308{\cos }^2\theta +7)$

$Y_9^2(\theta ,\phi )=\frac{3}{128}\sqrt{\frac{1045}{\pi }}\cdot e^{2i\phi }\cdot {\sin }^2\theta \cdot (221{\cos }^7\theta -273{\cos }^5\theta +91{\cos }^3\theta -7\cos \theta )$

$Y_9^3(\theta ,\phi )=\frac{-1}{256}\sqrt{\frac{21945}{\pi }}\cdot e^{3i\phi }\cdot {\sin }^3\theta \cdot (221{\cos }^6\theta -195{\cos }^4\theta +39{\cos }^2\theta -1)$

$Y_9^4(\theta ,\phi )=\frac{3}{128}\sqrt{\frac{95095}{2\pi }}\cdot e^{4i\phi }\cdot {\sin }^4\theta \cdot (17{\cos }^5\theta -10{\cos }^3\theta +1\cos \theta )$

$Y_9^5(\theta ,\phi )=\frac{-3}{256}\sqrt{\frac{2717}{\pi }}\cdot e^{5i\phi }\cdot {\sin }^5\theta \cdot (85{\cos }^4\theta -30{\cos }^2\theta +1)$

$Y_9^6(\theta ,\phi )=\frac{1}{128}\sqrt{\frac{40755}{\pi }}\cdot e^{6i\phi }\cdot {\sin }^6\theta \cdot (17{\cos }^3\theta -3\cos \theta )$

$Y_9^7(\theta ,\phi )=\frac{-3}{512}\sqrt{\frac{13585}{\pi }}\cdot e^{7i\phi }\cdot {\sin }^7\theta \cdot (17{\cos }^2\theta -1)$

$Y_9^8(\theta ,\phi )=\frac{3}{256}\sqrt{\frac{230945}{2\pi }}\cdot e^{8i\phi }\cdot {\sin }^8\theta \cdot \cos \theta$

$Y_9^9(\theta ,\phi )=\frac{-1}{512}\sqrt{\frac{230945}{\pi }}\cdot e^{9i\phi }\cdot {\sin }^9\theta$

$l=10$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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