Visualizza wikitesto di Tavola delle armoniche sferiche

{{torna a|armoniche sferiche}}
In questa pagina sono riportate le [[armoniche sferiche]] per <math>l=0,1,\ldots,10</math> ed <math>m=-l,\ldots,-1,0,1,\ldots,l</math>.<ref name="Varshalovich1988">{{cita libro|titolo=Quantum theory of angular momentum : irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols|anno=1988|editore=World Scientific Pub.|città=Singapore|isbn=9971-50-107-4|autore1=D. A. Varshalovich|autore2=A. N. Moskalev|autore3=V. K. Khersonskii|edizione=prima ristampa|pp=155-156}}</ref> Tra le [[coordinate sferiche]] <math>r, \theta,\varphi</math> e le coordinate cartsiane <math>x,y,z</math> usate talvolta sussistono le seguenti relazioni:

:<math>\begin{cases}
\cos\theta = \frac{z}{r} \\
e^{i\varphi} \sin\theta = \frac{x+iy}{r}.
\end{cases}</math>

== Armoniche sferiche con ''l'' = 0 ==

:<math>Y_{0}^{0}(x)=\frac12\sqrt{1\over \pi}</math>

== Armoniche sferiche con ''l'' = 1 ==

:<math>\begin{align}
Y_{1}^{-1}(\theta,\varphi) &= & & \frac12\sqrt{3\over 2\pi}\, e^{-i\varphi}\,\sin\theta & &= & &\frac12\sqrt{3\over 2\pi} \,{(x-iy)\over r} \\
Y_{1}^{ 0}(\theta,\varphi) &= & & \frac12\sqrt{3\over  \pi}\,                   \cos\theta & &= & &\frac12\sqrt{3\over  \pi} \,{z\over r} \\
Y_{1}^{ 1}(\theta,\varphi) &= &-& \frac12\sqrt{3\over 2\pi}\, e^{i\varphi}\,  \sin\theta & &= &-&\frac12\sqrt{3\over 2\pi} \, {(x+iy)\over r}
\end{align} </math>

== Armoniche sferiche con ''l'' = 2 ==

:<math>  \begin{align}
Y_{2}^{-2}(\theta,\varphi)&=& &\frac14\sqrt{15\over 2\pi}\,  e^{-2i\varphi}\, \sin^{2}\theta\quad                        &&=& &\frac14\sqrt{15\over 2\pi}\, {(x - iy)^2 \over r^{2}}&\\
Y_{2}^{-1}(\theta,\varphi)&=& &\frac12\sqrt{15\over 2\pi}\,  e^{- i\varphi}\, \sin    \theta\,   \cos\theta\quad       &&=& &\frac12\sqrt{15\over 2\pi}\, {(x - iy)z \over r^{2}}&\\
Y_{2}^{ 0}(\theta,\varphi)&=& &\frac14\sqrt{ 5\over  \pi}\,                                        (3\cos^{2}\theta-1)\quad&&=& &\frac14\sqrt{ 5\over  \pi}\, {(2z^{2}-x^{2}-y^{2})\over r^{2}}&\\
Y_{2}^{ 1}(\theta,\varphi)&=&-&\frac12\sqrt{15\over 2\pi}\,  e^{  i\varphi}\, \sin    \theta\,   \cos\theta\quad       &&=&-&\frac12\sqrt{15\over 2\pi}\, {(x + iy)z \over r^{2}}&\\
Y_{2}^{ 2}(\theta,\varphi)&=& &\frac14\sqrt{15\over 2\pi}\,  e^{ 2i\varphi}\, \sin^{2}\theta\quad                        &&=& &\frac14\sqrt{15\over 2\pi}\, {(x + iy)^2 \over r^{2}}&
   \end{align}</math>

== Armoniche sferiche con ''l'' = 3 ==

:<math>  \begin{align}
Y_{3}^{-3}(\theta,\varphi)
  &=& &{1\over 8}\sqrt{ 35\over  \pi}\,  e^{-3i\varphi}\, \sin^{3}\theta\quad&
  &=& & {1\over 8}\sqrt{35\over \pi}\, {(x - iy)^{3}\over r^{3}}&\\
Y_{3}^{-2}(\theta,\varphi)
  &=& &\frac14\sqrt{105\over 2\pi}\,  e^{-2i\varphi}\, \sin^{2}\theta\, \cos\theta\quad&
  &=& & \frac14\sqrt{105\over 2\pi}\, {(x- iy)^2 z \over r^{3}}&\\
Y_{3}^{-1}(\theta,\varphi)
  &=& &{1\over 8}\sqrt{ 21\over  \pi}\,  e^{-i\varphi}\, \sin\theta\, (5\cos^{2}\theta-1)\quad&
  &=& &{1\over 8}\sqrt{21\over \pi}\, {(x - iy)(5z^2- r^2)\over r^{3}}&\\
Y_{3}^{ 0}(\theta,\varphi)
  &=& &\frac14\sqrt{  7\over  \pi}\, (5\cos^{3}\theta-3\cos\theta)\quad&
  &=& &\frac14\sqrt{7\over \pi}\, {z(5z^2 - 3r^2)\over r^{3}}&\\
Y_{3}^{ 1}(\theta,\varphi)
  &=&-&{1\over 8}\sqrt{ 21\over  \pi}\,  e^ { i\varphi}\, \sin\theta\, (5\cos^{2}\theta-1)\quad&
  &=& &{-1\over 8}\sqrt{21\over \pi}\, {(x + iy) (5z^2 - r^2) \over r^{3}}&\\
Y_{3}^{ 2}(\theta,\varphi)
  &=& &\frac14\sqrt{105\over 2\pi}\,  e^ {2i\varphi}\, \sin^{2}\theta\, \cos\theta\quad&
  &=& &\frac14\sqrt{105\over 2\pi}\, {(x + iy)^2 z \over r^{3}}&\\
Y_{3}^{ 3}(\theta,\varphi)
  &=&-&{1\over 8}\sqrt{ 35\over  \pi}\,  e^ {3i\varphi}\, \sin^{3}\theta\quad&
  &=& &{-1\over 8}\sqrt{35\over \pi}\, {(x + iy)^3\over r^{3}}&
   \end{align}</math>

== Armoniche sferiche con ''l'' = 4 ==

:<math>\begin{align}
Y_{4}^{-4}(\theta,\varphi)&=& &\frac{3}{16}\sqrt{35\over 2\pi}\,  e^{-4i\varphi}\, \sin^{4}\theta&&=& &\frac{3}{16} \sqrt{\frac{35}{2 \pi}} \,  \frac{(x - i y)^4}{r^4}&\\
Y_{4}^{-3}(\theta,\varphi)&=& &\frac38\sqrt{35\over  \pi}\,  e^{-3i\varphi}\, \sin^{3}\theta\, \cos\theta&&=& &\frac{3}{8} \sqrt{\frac{35}{\pi}} \,  \frac{(x - i y)^3 z}{r^4}&\\
Y_{4}^{-2}(\theta,\varphi)&=& &\frac38\sqrt{ 5\over 2\pi}\,  e^{-2i\varphi}\, \sin^{2}\theta\, (7\cos^{2}\theta-1)&&=& &\frac{3}{8} \sqrt{\frac{5}{2 \pi}} \,  \frac{(x - i y)^2 \,  (7 z^2 - r^2)}{r^4}&\\
Y_{4}^{-1}(\theta,\varphi)&=& &\frac38\sqrt{ 5\over  \pi}\,  e^{- i\varphi}\, \sin\theta\, (7\cos^{3}\theta-3\cos\theta)&&=& &\frac{3}{8} \sqrt{\frac{5}{\pi}} \,  \frac{(x - i y) \,  z \,  (7 z^2 - 3 r^2)}{r^4}&\\
Y_{4}^{ 0}(\theta,\varphi)&=& &\frac{3}{16}\sqrt{ 1\over  \pi}\, (35\cos^{4}\theta-30\cos^{2}\theta+3)&&=& &\frac{3}{16} \sqrt{\frac{1}{\pi}} \,  \frac{(35 z^4 - 30 z^2 r^2 + 3 r^4)}{r^4}&\\
Y_{4}^{ 1}(\theta,\varphi)&=& &\frac{-3}8\sqrt{ 5\over  \pi}\,  e^{  i\varphi}\, \sin\theta\, (7\cos^{3}\theta-3\cos\theta)&&=& &\frac{- 3}{8} \sqrt{\frac{5}{\pi}} \,  \frac{(x + i y) \,  z \,  (7 z^2 - 3 r^2)}{r^4}&\\
Y_{4}^{ 2}(\theta,\varphi)&=& &\frac38\sqrt{ 5\over 2\pi}\,  e^{ 2i\varphi}\, \sin^{2}\theta\, (7\cos^{2}\theta-1)&&=& &\frac{3}{8} \sqrt{\frac{5}{2 \pi}} \,  \frac{(x + i y)^2 \,  (7 z^2 - r^2)}{r^4}&\\
Y_{4}^{ 3}(\theta,\varphi)&=& &\frac{-3}8\sqrt{35\over  \pi}\,  e^{ 3i\varphi}\, \sin^{3}\theta\, \cos\theta&&=& &\frac{- 3}{8} \sqrt{\frac{35}{\pi}} \,  \frac{(x + i y)^3 z}{r^4}&\\
Y_{4}^{ 4}(\theta,\varphi)&=& &\frac{3}{16}\sqrt{35\over 2\pi}\,  e^{ 4i\varphi}\, \sin^{4}\theta&&=& &\frac{3}{16} \sqrt{\frac{35}{2 \pi}} \,  \frac{(x + i y)^4}{r^4}&\\
\end{align}</math>

== Armoniche sferiche con ''l'' = 5 ==

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

== Armoniche sferiche con ''l'' = 6 ==

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

== Armoniche sferiche con ''l'' = 7 ==

:<math>  \begin{align}
Y_{7}^{-7}(\theta,\varphi)&= {3\over 64}\sqrt{ 715\over 2\pi}\,  e^{-7i\varphi}\, \sin^{7}\theta\\
Y_{7}^{-6}(\theta,\varphi)&= {3\over 64}\sqrt{5005\over  \pi}\,  e^{-6i\varphi}\, \sin^{6}\theta\, \cos\theta\\
Y_{7}^{-5}(\theta,\varphi)&= {3\over 64}\sqrt{ 385\over 2\pi}\,  e^{-5i\varphi}\, \sin^{5}\theta\, (13\cos^{2}\theta-1)\\
Y_{7}^{-4}(\theta,\varphi)&= {3\over 32}\sqrt{ 385\over 2\pi}\,  e^{-4i\varphi}\, \sin^{4}\theta\, (13\cos^{3}\theta-3\cos\theta)\\
Y_{7}^{-3}(\theta,\varphi)&= {3\over 64}\sqrt{  35\over 2\pi}\,  e^{-3i\varphi}\, \sin^{3}\theta\, (143\cos^{4}\theta-66\cos^{2}\theta+3)\\
Y_{7}^{-2}(\theta,\varphi)&= {3\over 64}\sqrt{  35\over  \pi}\,  e^{-2i\varphi}\, \sin^{2}\theta\, (143\cos^{5}\theta-110\cos^{3}\theta+15\cos\theta)\\
Y_{7}^{-1}(\theta,\varphi)&= {1\over 64}\sqrt{ 105\over 2\pi}\,  e^{- i\varphi}\, \sin    \theta\, (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}\,                                        (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}\,  e^{  i\varphi}\, \sin    \theta\, (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}\,  e^{ 2i\varphi}\, \sin^{2}\theta\, (143\cos^{5}\theta-110\cos^{3}\theta+15\cos\theta)\\
Y_{7}^{ 3}(\theta,\varphi)&=-{3\over 64}\sqrt{  35\over 2\pi}\,  e^{ 3i\varphi}\, \sin^{3}\theta\, (143\cos^{4}\theta-66\cos^{2}\theta+3)\\
Y_{7}^{ 4}(\theta,\varphi)&= {3\over 32}\sqrt{ 385\over 2\pi}\,  e^{ 4i\varphi}\, \sin^{4}\theta\, (13\cos^{3}\theta-3\cos\theta)\\
Y_{7}^{ 5}(\theta,\varphi)&=-{3\over 64}\sqrt{ 385\over 2\pi}\,  e^{ 5i\varphi}\, \sin^{5}\theta\, (13\cos^{2}\theta-1)\\
Y_{7}^{ 6}(\theta,\varphi)&= {3\over 64}\sqrt{5005\over  \pi}\,  e^{ 6i\varphi}\, \sin^{6}\theta\, \cos\theta\\
Y_{7}^{ 7}(\theta,\varphi)&=-{3\over 64}\sqrt{ 715\over 2\pi}\,  e^{ 7i\varphi}\, \sin^{7}\theta
\end{align}</math>

== Armoniche sferiche con ''l'' = 8 ==

:<math>  \begin{align}
Y_{8}^{-8}(\theta,\varphi)&={ 3\over 256}\sqrt{12155\over 2\pi}\,  e^{-8i\varphi}\, \sin^{8}\theta\\
Y_{8}^{-7}(\theta,\varphi)&={ 3\over  64}\sqrt{12155\over 2\pi}\,  e^{-7i\varphi}\, \sin^{7}\theta\, \cos\theta\\
Y_{8}^{-6}(\theta,\varphi)&={ 1\over 128}\sqrt{7293\over \pi}\,  e^{-6i\varphi}\, \sin^{6}\theta\, (15\cos^{2}\theta-1)\\
Y_{8}^{-5}(\theta,\varphi)&={ 3\over  64}\sqrt{17017\over 2\pi}\,  e^{-5i\varphi}\, \sin^{5}\theta\, (5\cos^{3}\theta-\cos\theta)\\
Y_{8}^{-4}(\theta,\varphi)&={ 3\over 128}\sqrt{1309\over 2\pi}\,  e^{-4i\varphi}\, \sin^{4}\theta\, (65\cos^{4}\theta-26\cos^{2}\theta+1)\\
Y_{8}^{-3}(\theta,\varphi)&={ 1\over  64}\sqrt{19635\over 2\pi}\,  e^{-3i\varphi}\, \sin^{3}\theta\, (39\cos^{5}\theta-26\cos^{3}\theta+3\cos\theta)\\
Y_{8}^{-2}(\theta,\varphi)&={ 3\over 128}\sqrt{595\over \pi}\,  e^{-2i\varphi}\, \sin^{2}\theta\, (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}\,  e^{-i\varphi}\, \sin\theta\, (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}\, (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}\,  e^{i\varphi}\, \sin\theta\, (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}\,  e^{2i\varphi}\, \sin^{2}\theta\, (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}\,  e^{3i\varphi}\, \sin^{3}\theta\, (39\cos^{5}\theta-26\cos^{3}\theta+3\cos\theta)\\
Y_{8}^{ 4}(\theta,\varphi)&={ 3\over 128}\sqrt{1309\over 2\pi}\,  e^{4i\varphi}\, \sin^{4}\theta\, (65\cos^{4}\theta-26\cos^{2}\theta+1)\\
Y_{8}^{ 5}(\theta,\varphi)&={-3\over  64}\sqrt{17017\over 2\pi}\,  e^{5i\varphi}\, \sin^{5}\theta\, (5\cos^{3}\theta-\cos\theta)\\
Y_{8}^{ 6}(\theta,\varphi)&={ 1\over 128}\sqrt{7293\over \pi}\,  e^{6i\varphi}\, \sin^{6}\theta\, (15\cos^{2}\theta-1)\\
Y_{8}^{ 7}(\theta,\varphi)&={-3\over  64}\sqrt{12155\over 2\pi}\,  e^{7i\varphi}\, \sin^{7}\theta\, \cos\theta\\
Y_{8}^{ 8}(\theta,\varphi)&={ 3\over 256}\sqrt{12155\over 2\pi}\,  e^{8i\varphi}\, \sin^{8}\theta
   \end{align}</math>

== Armoniche sferiche con ''l'' = 9 ==

:<math>  \begin{align}
Y_{9}^{-9}(\theta,\varphi)&={ 1\over 512}\sqrt{230945\over  \pi}\,  e^{-9i\varphi}\, \sin^{9}\theta\\
Y_{9}^{-8}(\theta,\varphi)&={ 3\over 256}\sqrt{230945\over 2\pi}\,  e^{-8i\varphi}\, \sin^{8}\theta\, \cos\theta\\
Y_{9}^{-7}(\theta,\varphi)&={ 3\over 512}\sqrt{ 13585\over  \pi}\,  e^{-7i\varphi}\, \sin^{7}\theta\, (17\cos^{2}\theta-1)\\
Y_{9}^{-6}(\theta,\varphi)&={ 1\over 128}\sqrt{ 40755\over  \pi}\,  e^{-6i\varphi}\, \sin^{6}\theta\, (17\cos^{3}\theta-3\cos\theta)\\
Y_{9}^{-5}(\theta,\varphi)&={ 3\over 256}\sqrt{  2717\over  \pi}\,  e^{-5i\varphi}\, \sin^{5}\theta\, (85\cos^{4}\theta-30\cos^{2}\theta+1)\\
Y_{9}^{-4}(\theta,\varphi)&={ 3\over 128}\sqrt{ 95095\over 2\pi}\,  e^{-4i\varphi}\, \sin^{4}\theta\, (17\cos^{5}\theta-10\cos^{3}\theta+\cos\theta)\\
Y_{9}^{-3}(\theta,\varphi)&={ 1\over 256}\sqrt{ 21945\over  \pi}\,  e^{-3i\varphi}\, \sin^{3}\theta\, (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}\,  e^{-2i\varphi}\, \sin^{2}\theta\, (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}\,  e^{- i\varphi}\, \sin    \theta\, (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}\,                                        (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}\,  e^{  i\varphi}\, \sin    \theta\, (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}\,  e^{ 2i\varphi}\, \sin^{2}\theta\, (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}\,  e^{ 3i\varphi}\, \sin^{3}\theta\, (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}\,  e^{ 4i\varphi}\, \sin^{4}\theta\, (17\cos^{5}\theta-10\cos^{3}\theta+\cos\theta)\\
Y_{9}^{ 5}(\theta,\varphi)&={-3\over 256}\sqrt{  2717\over  \pi}\,  e^{ 5i\varphi}\, \sin^{5}\theta\, (85\cos^{4}\theta-30\cos^{2}\theta+1)\\
Y_{9}^{ 6}(\theta,\varphi)&={ 1\over 128}\sqrt{ 40755\over  \pi}\,  e^{ 6i\varphi}\, \sin^{6}\theta\, (17\cos^{3}\theta-3\cos\theta)\\
Y_{9}^{ 7}(\theta,\varphi)&={-3\over 512}\sqrt{ 13585\over  \pi}\,  e^{ 7i\varphi}\, \sin^{7}\theta\, (17\cos^{2}\theta-1)\\
Y_{9}^{ 8}(\theta,\varphi)&={ 3\over 256}\sqrt{230945\over 2\pi}\,  e^{ 8i\varphi}\, \sin^{8}\theta\, \cos\theta\\
Y_{9}^{ 9}(\theta,\varphi)&={-1\over 512}\sqrt{230945\over  \pi}\,  e^{ 9i\varphi}\, \sin^{9}\theta
   \end{align}</math>

== Armoniche sferiche con ''l'' = 10 ==

:<math>  \begin{align}
Y_{10}^{-10}(\theta,\varphi)&={1\over 1024}\sqrt{969969\over \pi}\,  e^{-10i\varphi}\, \sin^{10}\theta\\
Y_{10}^{- 9}(\theta,\varphi)&={1\over 512}\sqrt{4849845\over \pi}\,  e^{-9i\varphi}\, \sin^{9}\theta\, \cos\theta\\
Y_{10}^{- 8}(\theta,\varphi)&={1\over 512}\sqrt{255255\over 2\pi}\,  e^{-8i\varphi}\, \sin^{8}\theta\, (19\cos^{2}\theta-1)\\
Y_{10}^{- 7}(\theta,\varphi)&={3\over 512}\sqrt{85085\over \pi}\,  e^{-7i\varphi}\, \sin^{7}\theta\, (19\cos^{3}\theta-3\cos\theta)\\
Y_{10}^{- 6}(\theta,\varphi)&={3\over 1024}\sqrt{5005\over \pi}\,  e^{-6i\varphi}\, \sin^{6}\theta\, (323\cos^{4}\theta-102\cos^{2}\theta+3)\\
Y_{10}^{- 5}(\theta,\varphi)&={3\over 256}\sqrt{1001\over \pi}\,  e^{-5i\varphi}\, \sin^{5}\theta\, (323\cos^{5}\theta-170\cos^{3}\theta+15\cos\theta)\\
Y_{10}^{- 4}(\theta,\varphi)&={3\over 256}\sqrt{5005\over 2\pi}\,  e^{-4i\varphi}\, \sin^{4}\theta\, (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}\,  e^{-3i\varphi}\, \sin^{3}\theta\, (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}\,  e^{-2i\varphi}\, \sin^{2}\theta\, (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}\,  e^{-i\varphi}\, \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,\varphi)&={1\over 512}\sqrt{21\over \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,\varphi)&={-1\over 256}\sqrt{1155\over 2\pi}\,  e^{i\varphi}\, \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,\varphi)&={3\over 512}\sqrt{385\over 2\pi}\,  e^{2i\varphi}\, \sin^{2}\theta\, (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}\,  e^{3i\varphi}\, \sin^{3}\theta\, (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}\,  e^{4i\varphi}\, \sin^{4}\theta\, (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}\,  e^{5i\varphi}\, \sin^{5}\theta\, (323\cos^{5}\theta-170\cos^{3}\theta+15\cos\theta)\\
Y_{10}^{  6}(\theta,\varphi)&={3\over 1024}\sqrt{5005\over \pi}\,  e^{6i\varphi}\, \sin^{6}\theta\, (323\cos^{4}\theta-102\cos^{2}\theta+3)\\
Y_{10}^{  7}(\theta,\varphi)&={-3\over 512}\sqrt{85085\over \pi}\,  e^{7i\varphi}\, \sin^{7}\theta\, (19\cos^{3}\theta-3\cos\theta)\\
Y_{10}^{  8}(\theta,\varphi)&={1\over 512}\sqrt{255255\over 2\pi}\,  e^{8i\varphi}\, \sin^{8}\theta\, (19\cos^{2}\theta-1)\\
Y_{10}^{  9}(\theta,\varphi)&={-1\over 512}\sqrt{4849845\over \pi}\,  e^{9i\varphi}\, \sin^{9}\theta\, \cos\theta\\
Y_{10}^{ 10}(\theta,\varphi)&={1\over 1024}\sqrt{969969\over \pi}\,  e^{10i\varphi}\, \sin^{10}\theta
   \end{align}</math>

== Note ==
<references />
== Voci correlate ==
*[[Armoniche sferiche]]

{{Portale|matematica}}

[[Categoria:Funzioni ipergeometriche speciali]]