Tavola delle armoniche sferiche

Template:Torna a In questa pagina sono riportate le armoniche sferiche per $l=0,1,\dots ,10$ ed $m=-l,\dots ,-1,0,1,\dots ,l$. Tra le coordinate sferiche $r,\theta ,\phi$ e le coordinate cartsiane $x,y,z$ usate talvolta sussistono le seguenti relazioni:

$\{\begin{array}{c}\cos \theta =\frac{z}{r}\\ e^{i\phi }\sin \theta =\frac{x+iy}{r}.\end{array}$

Armoniche sferiche con l = 0

$Y_0^0(x)=\frac{1}{2}\sqrt{\frac{1}{\pi }}$

Armoniche sferiche con l = 1

$\begin{array}{cccccccc}{Y}_1^{-1}(\theta ,\phi )& =& & \frac{1}{2}\sqrt{\frac{3}{2\pi }}{e}^{-i\phi }\sin \theta & & =& & \frac{1}{2}\sqrt{\frac{3}{2\pi }}\frac{(x-iy)}{r}\\ Y_1^0(\theta ,\phi )& =& & \frac{1}{2}\sqrt{\frac{3}{\pi }}\cos \theta & & =& & \frac{1}{2}\sqrt{\frac{3}{\pi }}\frac{z}{r}\\ Y_1^1(\theta ,\phi )& =& -& \frac{1}{2}\sqrt{\frac{3}{2\pi }}{e}^{i\phi }\sin \theta & & =& -& \frac{1}{2}\sqrt{\frac{3}{2\pi }}\frac{(x+iy)}{r}\end{array}$

Armoniche sferiche con l = 2

$\begin{array}{ccccccccc}{Y}_2^{-2}(\theta ,\phi )& =& & \frac{1}{4}\sqrt{\frac{15}{2\pi }}{e}^{-2i\phi }{\sin }^2\theta & & =& & \frac{1}{4}\sqrt{\frac{15}{2\pi }}\frac{(x-iy{)}^2}{r^2}& \\ Y_2^{-1}(\theta ,\phi )& =& & \frac{1}{2}\sqrt{\frac{15}{2\pi }}{e}^{-i\phi }\sin \theta \cos \theta & & =& & \frac{1}{2}\sqrt{\frac{15}{2\pi }}\frac{(x-iy)z}{r^2}& \\ Y_2^0(\theta ,\phi )& =& & \frac{1}{4}\sqrt{\frac{5}{\pi }}(3{\cos }^2\theta -1)& & =& & \frac{1}{4}\sqrt{\frac{5}{\pi }}\frac{(2z^2-x^2-y^2)}{r^2}& \\ Y_2^1(\theta ,\phi )& =& -& \frac{1}{2}\sqrt{\frac{15}{2\pi }}{e}^{i\phi }\sin \theta \cos \theta & & =& -& \frac{1}{2}\sqrt{\frac{15}{2\pi }}\frac{(x+iy)z}{r^2}& \\ Y_2^2(\theta ,\phi )& =& & \frac{1}{4}\sqrt{\frac{15}{2\pi }}{e}^{2i\phi }{\sin }^2\theta & & =& & \frac{1}{4}\sqrt{\frac{15}{2\pi }}\frac{(x+iy{)}^2}{r^2}& \end{array}$

Armoniche sferiche con l = 3

$\begin{array}{ccccccccc}{Y}_3^{-3}(\theta ,\phi )& =& & \frac{1}{8}\sqrt{\frac{35}{\pi }}{e}^{-3i\phi }{\sin }^3\theta & & =& & \frac{1}{8}\sqrt{\frac{35}{\pi }}\frac{(x-iy{)}^3}{r^3}& \\ Y_3^{-2}(\theta ,\phi )& =& & \frac{1}{4}\sqrt{\frac{105}{2\pi }}{e}^{-2i\phi }{\sin }^2\theta \cos \theta & & =& & \frac{1}{4}\sqrt{\frac{105}{2\pi }}\frac{(x-iy{)}^{2}z}{r^3}& \\ Y_3^{-1}(\theta ,\phi )& =& & \frac{1}{8}\sqrt{\frac{21}{\pi }}{e}^{-i\phi }\sin \theta (5{\cos }^2\theta -1)& & =& & \frac{1}{8}\sqrt{\frac{21}{\pi }}\frac{(x-iy)(5z^2-r^2)}{r^3}& \\ Y_3^0(\theta ,\phi )& =& & \frac{1}{4}\sqrt{\frac{7}{\pi }}(5{\cos }^3\theta -3\cos \theta )& & =& & \frac{1}{4}\sqrt{\frac{7}{\pi }}\frac{z(5z^2-3r^2)}{r^3}& \\ Y_3^1(\theta ,\phi )& =& -& \frac{1}{8}\sqrt{\frac{21}{\pi }}{e}^{i\phi }\sin \theta (5{\cos }^2\theta -1)& & =& & \frac{-1}{8}\sqrt{\frac{21}{\pi }}\frac{(x+iy)(5z^2-r^2)}{r^3}& \\ Y_3^2(\theta ,\phi )& =& & \frac{1}{4}\sqrt{\frac{105}{2\pi }}{e}^{2i\phi }{\sin }^2\theta \cos \theta & & =& & \frac{1}{4}\sqrt{\frac{105}{2\pi }}\frac{(x+iy{)}^{2}z}{r^3}& \\ Y_3^3(\theta ,\phi )& =& -& \frac{1}{8}\sqrt{\frac{35}{\pi }}{e}^{3i\phi }{\sin }^3\theta & & =& & \frac{-1}{8}\sqrt{\frac{35}{\pi }}\frac{(x+iy{)}^3}{r^3}& \end{array}$

Armoniche sferiche con l = 4

$\begin{array}{ccccccccc}{Y}_4^{-4}(\theta ,\phi )& =& & \frac{3}{16}\sqrt{\frac{35}{2\pi }}{e}^{-4i\phi }{\sin }^4\theta & & =& & \frac{3}{16}\sqrt{\frac{35}{2\pi }}\frac{(x-iy{)}^4}{r^4}& \\ Y_4^{-3}(\theta ,\phi )& =& & \frac{3}{8}\sqrt{\frac{35}{\pi }}{e}^{-3i\phi }{\sin }^3\theta \cos \theta & & =& & \frac{3}{8}\sqrt{\frac{35}{\pi }}\frac{(x-iy{)}^{3}z}{r^4}& \\ Y_4^{-2}(\theta ,\phi )& =& & \frac{3}{8}\sqrt{\frac{5}{2\pi }}{e}^{-2i\phi }{\sin }^2\theta (7{\cos }^2\theta -1)& & =& & \frac{3}{8}\sqrt{\frac{5}{2\pi }}\frac{(x-iy{)}^2(7z^2-r^2)}{r^4}& \\ Y_4^{-1}(\theta ,\phi )& =& & \frac{3}{8}\sqrt{\frac{5}{\pi }}{e}^{-i\phi }\sin \theta (7{\cos }^3\theta -3\cos \theta )& & =& & \frac{3}{8}\sqrt{\frac{5}{\pi }}\frac{(x-iy)z(7z^2-3r^2)}{r^4}& \\ Y_4^0(\theta ,\phi )& =& & \frac{3}{16}\sqrt{\frac{1}{\pi }}(35{\cos }^4\theta -30{\cos }^2\theta +3)& & =& & \frac{3}{16}\sqrt{\frac{1}{\pi }}\frac{(35z^4-30z^2{r}^2+3r^4)}{r^4}& \\ Y_4^1(\theta ,\phi )& =& & \frac{-3}{8}\sqrt{\frac{5}{\pi }}{e}^{i\phi }\sin \theta (7{\cos }^3\theta -3\cos \theta )& & =& & \frac{-3}{8}\sqrt{\frac{5}{\pi }}\frac{(x+iy)z(7z^2-3r^2)}{r^4}& \\ Y_4^2(\theta ,\phi )& =& & \frac{3}{8}\sqrt{\frac{5}{2\pi }}{e}^{2i\phi }{\sin }^2\theta (7{\cos }^2\theta -1)& & =& & \frac{3}{8}\sqrt{\frac{5}{2\pi }}\frac{(x+iy{)}^2(7z^2-r^2)}{r^4}& \\ Y_4^3(\theta ,\phi )& =& & \frac{-3}{8}\sqrt{\frac{35}{\pi }}{e}^{3i\phi }{\sin }^3\theta \cos \theta & & =& & \frac{-3}{8}\sqrt{\frac{35}{\pi }}\frac{(x+iy{)}^{3}z}{r^4}& \\ Y_4^4(\theta ,\phi )& =& & \frac{3}{16}\sqrt{\frac{35}{2\pi }}{e}^{4i\phi }{\sin }^4\theta & & =& & \frac{3}{16}\sqrt{\frac{35}{2\pi }}\frac{(x+iy{)}^4}{r^4}& \\ \end{array}$

Armoniche sferiche con l = 5

$\begin{array}{cc}{Y}_5^{-5}(\theta ,\phi )& =\frac{3}{32}\sqrt{\frac{77}{\pi }}{e}^{-5i\phi }{\sin }^5\theta \\ Y_5^{-4}(\theta ,\phi )& =\frac{3}{16}\sqrt{\frac{385}{2\pi }}{e}^{-4i\phi }{\sin }^4\theta \cos \theta \\ Y_5^{-3}(\theta ,\phi )& =\frac{1}{32}\sqrt{\frac{385}{\pi }}{e}^{-3i\phi }{\sin }^3\theta (9{\cos }^2\theta -1)\\ Y_5^{-2}(\theta ,\phi )& =\frac{1}{8}\sqrt{\frac{1155}{2\pi }}{e}^{-2i\phi }{\sin }^2\theta (3{\cos }^3\theta -\cos \theta )\\ Y_5^{-1}(\theta ,\phi )& =\frac{1}{16}\sqrt{\frac{165}{2\pi }}{e}^{-i\phi }\sin \theta (21{\cos }^4\theta -14{\cos }^2\theta +1)\\ Y_5^0(\theta ,\phi )& =\frac{1}{16}\sqrt{\frac{11}{\pi }}(63{\cos }^5\theta -70{\cos }^3\theta +15\cos \theta )\\ Y_5^1(\theta ,\phi )& =\frac{-1}{16}\sqrt{\frac{165}{2\pi }}{e}^{i\phi }\sin \theta (21{\cos }^4\theta -14{\cos }^2\theta +1)\\ Y_5^2(\theta ,\phi )& =\frac{1}{8}\sqrt{\frac{1155}{2\pi }}{e}^{2i\phi }{\sin }^2\theta (3{\cos }^3\theta -\cos \theta )\\ Y_5^3(\theta ,\phi )& =\frac{-1}{32}\sqrt{\frac{385}{\pi }}{e}^{3i\phi }{\sin }^3\theta (9{\cos }^2\theta -1)\\ Y_5^4(\theta ,\phi )& =\frac{3}{16}\sqrt{\frac{385}{2\pi }}{e}^{4i\phi }{\sin }^4\theta \cos \theta \\ Y_5^5(\theta ,\phi )& =\frac{-3}{32}\sqrt{\frac{77}{\pi }}{e}^{5i\phi }{\sin }^5\theta \end{array}$

Armoniche sferiche con l = 6

$\begin{array}{cc}{Y}_6^{-6}(\theta ,\phi )& =\frac{1}{64}\sqrt{\frac{3003}{\pi }}{e}^{-6i\phi }{\sin }^6\theta \\ Y_6^{-5}(\theta ,\phi )& =\frac{3}{32}\sqrt{\frac{1001}{\pi }}{e}^{-5i\phi }{\sin }^5\theta \cos \theta \\ Y_6^{-4}(\theta ,\phi )& =\frac{3}{32}\sqrt{\frac{91}{2\pi }}{e}^{-4i\phi }{\sin }^4\theta (11{\cos }^2\theta -1)\\ Y_6^{-3}(\theta ,\phi )& =\frac{1}{32}\sqrt{\frac{1365}{\pi }}{e}^{-3i\phi }{\sin }^3\theta (11{\cos }^3\theta -3\cos \theta )\\ Y_6^{-2}(\theta ,\phi )& =\frac{1}{64}\sqrt{\frac{1365}{\pi }}{e}^{-2i\phi }{\sin }^2\theta (33{\cos }^4\theta -18{\cos }^2\theta +1)\\ Y_6^{-1}(\theta ,\phi )& =\frac{1}{16}\sqrt{\frac{273}{2\pi }}{e}^{-i\phi }\sin \theta (33{\cos }^5\theta -30{\cos }^3\theta +5\cos \theta )\\ Y_6^0(\theta ,\phi )& =\frac{1}{32}\sqrt{\frac{13}{\pi }}(231{\cos }^6\theta -315{\cos }^4\theta +105{\cos }^2\theta -5)\\ Y_6^1(\theta ,\phi )& =-\frac{1}{16}\sqrt{\frac{273}{2\pi }}{e}^{i\phi }\sin \theta (33{\cos }^5\theta -30{\cos }^3\theta +5\cos \theta )\\ Y_6^2(\theta ,\phi )& =\frac{1}{64}\sqrt{\frac{1365}{\pi }}{e}^{2i\phi }{\sin }^2\theta (33{\cos }^4\theta -18{\cos }^2\theta +1)\\ Y_6^3(\theta ,\phi )& =-\frac{1}{32}\sqrt{\frac{1365}{\pi }}{e}^{3i\phi }{\sin }^3\theta (11{\cos }^3\theta -3\cos \theta )\\ Y_6^4(\theta ,\phi )& =\frac{3}{32}\sqrt{\frac{91}{2\pi }}{e}^{4i\phi }{\sin }^4\theta (11{\cos }^2\theta -1)\\ Y_6^5(\theta ,\phi )& =-\frac{3}{32}\sqrt{\frac{1001}{\pi }}{e}^{5i\phi }{\sin }^5\theta \cos \theta \\ Y_6^6(\theta ,\phi )& =\frac{1}{64}\sqrt{\frac{3003}{\pi }}{e}^{6i\phi }{\sin }^6\theta \end{array}$

Armoniche sferiche con l = 7

$\begin{array}{cc}{Y}_7^{-7}(\theta ,\phi )& =\frac{3}{64}\sqrt{\frac{715}{2\pi }}{e}^{-7i\phi }{\sin }^7\theta \\ Y_7^{-6}(\theta ,\phi )& =\frac{3}{64}\sqrt{\frac{5005}{\pi }}{e}^{-6i\phi }{\sin }^6\theta \cos \theta \\ Y_7^{-5}(\theta ,\phi )& =\frac{3}{64}\sqrt{\frac{385}{2\pi }}{e}^{-5i\phi }{\sin }^5\theta (13{\cos }^2\theta -1)\\ Y_7^{-4}(\theta ,\phi )& =\frac{3}{32}\sqrt{\frac{385}{2\pi }}{e}^{-4i\phi }{\sin }^4\theta (13{\cos }^3\theta -3\cos \theta )\\ Y_7^{-3}(\theta ,\phi )& =\frac{3}{64}\sqrt{\frac{35}{2\pi }}{e}^{-3i\phi }{\sin }^3\theta (143{\cos }^4\theta -66{\cos }^2\theta +3)\\ Y_7^{-2}(\theta ,\phi )& =\frac{3}{64}\sqrt{\frac{35}{\pi }}{e}^{-2i\phi }{\sin }^2\theta (143{\cos }^5\theta -110{\cos }^3\theta +15\cos \theta )\\ Y_7^{-1}(\theta ,\phi )& =\frac{1}{64}\sqrt{\frac{105}{2\pi }}{e}^{-i\phi }\sin \theta (429{\cos }^6\theta -495{\cos }^4\theta +135{\cos }^2\theta -5)\\ Y_7^0(\theta ,\phi )& =\frac{1}{32}\sqrt{\frac{15}{\pi }}(429{\cos }^7\theta -693{\cos }^5\theta +315{\cos }^3\theta -35\cos \theta )\\ Y_7^1(\theta ,\phi )& =-\frac{1}{64}\sqrt{\frac{105}{2\pi }}{e}^{i\phi }\sin \theta (429{\cos }^6\theta -495{\cos }^4\theta +135{\cos }^2\theta -5)\\ Y_7^2(\theta ,\phi )& =\frac{3}{64}\sqrt{\frac{35}{\pi }}{e}^{2i\phi }{\sin }^2\theta (143{\cos }^5\theta -110{\cos }^3\theta +15\cos \theta )\\ Y_7^3(\theta ,\phi )& =-\frac{3}{64}\sqrt{\frac{35}{2\pi }}{e}^{3i\phi }{\sin }^3\theta (143{\cos }^4\theta -66{\cos }^2\theta +3)\\ Y_7^4(\theta ,\phi )& =\frac{3}{32}\sqrt{\frac{385}{2\pi }}{e}^{4i\phi }{\sin }^4\theta (13{\cos }^3\theta -3\cos \theta )\\ Y_7^5(\theta ,\phi )& =-\frac{3}{64}\sqrt{\frac{385}{2\pi }}{e}^{5i\phi }{\sin }^5\theta (13{\cos }^2\theta -1)\\ Y_7^6(\theta ,\phi )& =\frac{3}{64}\sqrt{\frac{5005}{\pi }}{e}^{6i\phi }{\sin }^6\theta \cos \theta \\ Y_7^7(\theta ,\phi )& =-\frac{3}{64}\sqrt{\frac{715}{2\pi }}{e}^{7i\phi }{\sin }^7\theta \end{array}$

Armoniche sferiche con l = 8

$\begin{array}{cc}{Y}_8^{-8}(\theta ,\phi )& =\frac{3}{256}\sqrt{\frac{12155}{2\pi }}{e}^{-8i\phi }{\sin }^8\theta \\ Y_8^{-7}(\theta ,\phi )& =\frac{3}{64}\sqrt{\frac{12155}{2\pi }}{e}^{-7i\phi }{\sin }^7\theta \cos \theta \\ Y_8^{-6}(\theta ,\phi )& =\frac{1}{128}\sqrt{\frac{7293}{\pi }}{e}^{-6i\phi }{\sin }^6\theta (15{\cos }^2\theta -1)\\ Y_8^{-5}(\theta ,\phi )& =\frac{3}{64}\sqrt{\frac{17017}{2\pi }}{e}^{-5i\phi }{\sin }^5\theta (5{\cos }^3\theta -\cos \theta )\\ Y_8^{-4}(\theta ,\phi )& =\frac{3}{128}\sqrt{\frac{1309}{2\pi }}{e}^{-4i\phi }{\sin }^4\theta (65{\cos }^4\theta -26{\cos }^2\theta +1)\\ Y_8^{-3}(\theta ,\phi )& =\frac{1}{64}\sqrt{\frac{19635}{2\pi }}{e}^{-3i\phi }{\sin }^3\theta (39{\cos }^5\theta -26{\cos }^3\theta +3\cos \theta )\\ Y_8^{-2}(\theta ,\phi )& =\frac{3}{128}\sqrt{\frac{595}{\pi }}{e}^{-2i\phi }{\sin }^2\theta (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 }}{e}^{-i\phi }\sin \theta (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 }}(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 }}{e}^{i\phi }\sin \theta (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 }}{e}^{2i\phi }{\sin }^2\theta (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 }}{e}^{3i\phi }{\sin }^3\theta (39{\cos }^5\theta -26{\cos }^3\theta +3\cos \theta )\\ Y_8^4(\theta ,\phi )& =\frac{3}{128}\sqrt{\frac{1309}{2\pi }}{e}^{4i\phi }{\sin }^4\theta (65{\cos }^4\theta -26{\cos }^2\theta +1)\\ Y_8^5(\theta ,\phi )& =\frac{-3}{64}\sqrt{\frac{17017}{2\pi }}{e}^{5i\phi }{\sin }^5\theta (5{\cos }^3\theta -\cos \theta )\\ Y_8^6(\theta ,\phi )& =\frac{1}{128}\sqrt{\frac{7293}{\pi }}{e}^{6i\phi }{\sin }^6\theta (15{\cos }^2\theta -1)\\ Y_8^7(\theta ,\phi )& =\frac{-3}{64}\sqrt{\frac{12155}{2\pi }}{e}^{7i\phi }{\sin }^7\theta \cos \theta \\ Y_8^8(\theta ,\phi )& =\frac{3}{256}\sqrt{\frac{12155}{2\pi }}{e}^{8i\phi }{\sin }^8\theta \end{array}$

Armoniche sferiche con l = 9

$\begin{array}{cc}{Y}_9^{-9}(\theta ,\phi )& =\frac{1}{512}\sqrt{\frac{230945}{\pi }}{e}^{-9i\phi }{\sin }^9\theta \\ Y_9^{-8}(\theta ,\phi )& =\frac{3}{256}\sqrt{\frac{230945}{2\pi }}{e}^{-8i\phi }{\sin }^8\theta \cos \theta \\ Y_9^{-7}(\theta ,\phi )& =\frac{3}{512}\sqrt{\frac{13585}{\pi }}{e}^{-7i\phi }{\sin }^7\theta (17{\cos }^2\theta -1)\\ Y_9^{-6}(\theta ,\phi )& =\frac{1}{128}\sqrt{\frac{40755}{\pi }}{e}^{-6i\phi }{\sin }^6\theta (17{\cos }^3\theta -3\cos \theta )\\ Y_9^{-5}(\theta ,\phi )& =\frac{3}{256}\sqrt{\frac{2717}{\pi }}{e}^{-5i\phi }{\sin }^5\theta (85{\cos }^4\theta -30{\cos }^2\theta +1)\\ Y_9^{-4}(\theta ,\phi )& =\frac{3}{128}\sqrt{\frac{95095}{2\pi }}{e}^{-4i\phi }{\sin }^4\theta (17{\cos }^5\theta -10{\cos }^3\theta +\cos \theta )\\ Y_9^{-3}(\theta ,\phi )& =\frac{1}{256}\sqrt{\frac{21945}{\pi }}{e}^{-3i\phi }{\sin }^3\theta (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 }}{e}^{-2i\phi }{\sin }^2\theta (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 }}{e}^{-i\phi }\sin \theta (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 }}(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 }}{e}^{i\phi }\sin \theta (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 }}{e}^{2i\phi }{\sin }^2\theta (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 }}{e}^{3i\phi }{\sin }^3\theta (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 }}{e}^{4i\phi }{\sin }^4\theta (17{\cos }^5\theta -10{\cos }^3\theta +\cos \theta )\\ Y_9^5(\theta ,\phi )& =\frac{-3}{256}\sqrt{\frac{2717}{\pi }}{e}^{5i\phi }{\sin }^5\theta (85{\cos }^4\theta -30{\cos }^2\theta +1)\\ Y_9^6(\theta ,\phi )& =\frac{1}{128}\sqrt{\frac{40755}{\pi }}{e}^{6i\phi }{\sin }^6\theta (17{\cos }^3\theta -3\cos \theta )\\ Y_9^7(\theta ,\phi )& =\frac{-3}{512}\sqrt{\frac{13585}{\pi }}{e}^{7i\phi }{\sin }^7\theta (17{\cos }^2\theta -1)\\ Y_9^8(\theta ,\phi )& =\frac{3}{256}\sqrt{\frac{230945}{2\pi }}{e}^{8i\phi }{\sin }^8\theta \cos \theta \\ Y_9^9(\theta ,\phi )& =\frac{-1}{512}\sqrt{\frac{230945}{\pi }}{e}^{9i\phi }{\sin }^9\theta \end{array}$

Armoniche sferiche con l = 10

$\begin{array}{cc}{Y}_{10}^{-10}(\theta ,\phi )& =\frac{1}{1024}\sqrt{\frac{969969}{\pi }}{e}^{-10i\phi }{\sin }^{10}\theta \\ Y_{10}^{-9}(\theta ,\phi )& =\frac{1}{512}\sqrt{\frac{4849845}{\pi }}{e}^{-9i\phi }{\sin }^9\theta \cos \theta \\ Y_{10}^{-8}(\theta ,\phi )& =\frac{1}{512}\sqrt{\frac{255255}{2\pi }}{e}^{-8i\phi }{\sin }^8\theta (19{\cos }^2\theta -1)\\ Y_{10}^{-7}(\theta ,\phi )& =\frac{3}{512}\sqrt{\frac{85085}{\pi }}{e}^{-7i\phi }{\sin }^7\theta (19{\cos }^3\theta -3\cos \theta )\\ Y_{10}^{-6}(\theta ,\phi )& =\frac{3}{1024}\sqrt{\frac{5005}{\pi }}{e}^{-6i\phi }{\sin }^6\theta (323{\cos }^4\theta -102{\cos }^2\theta +3)\\ Y_{10}^{-5}(\theta ,\phi )& =\frac{3}{256}\sqrt{\frac{1001}{\pi }}{e}^{-5i\phi }{\sin }^5\theta (323{\cos }^5\theta -170{\cos }^3\theta +15\cos \theta )\\ Y_{10}^{-4}(\theta ,\phi )& =\frac{3}{256}\sqrt{\frac{5005}{2\pi }}{e}^{-4i\phi }{\sin }^4\theta (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 }}{e}^{-3i\phi }{\sin }^3\theta (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 }}{e}^{-2i\phi }{\sin }^2\theta (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 }}{e}^{-i\phi }\sin \theta (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 }}(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 }}{e}^{i\phi }\sin \theta (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 }}{e}^{2i\phi }{\sin }^2\theta (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 }}{e}^{3i\phi }{\sin }^3\theta (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 }}{e}^{4i\phi }{\sin }^4\theta (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 }}{e}^{5i\phi }{\sin }^5\theta (323{\cos }^5\theta -170{\cos }^3\theta +15\cos \theta )\\ Y_{10}^6(\theta ,\phi )& =\frac{3}{1024}\sqrt{\frac{5005}{\pi }}{e}^{6i\phi }{\sin }^6\theta (323{\cos }^4\theta -102{\cos }^2\theta +3)\\ Y_{10}^7(\theta ,\phi )& =\frac{-3}{512}\sqrt{\frac{85085}{\pi }}{e}^{7i\phi }{\sin }^7\theta (19{\cos }^3\theta -3\cos \theta )\\ Y_{10}^8(\theta ,\phi )& =\frac{1}{512}\sqrt{\frac{255255}{2\pi }}{e}^{8i\phi }{\sin }^8\theta (19{\cos }^2\theta -1)\\ Y_{10}^9(\theta ,\phi )& =\frac{-1}{512}\sqrt{\frac{4849845}{\pi }}{e}^{9i\phi }{\sin }^9\theta \cos \theta \\ Y_{10}^{10}(\theta ,\phi )& =\frac{1}{1024}\sqrt{\frac{969969}{\pi }}{e}^{10i\phi }{\sin }^{10}\theta \end{array}$

Voci correlate

• Armoniche sferiche