Tavola delle armoniche sferiche

Voce principale: armoniche sferiche.

In questa pagina sono riportate le armoniche sferiche per l = 0 , 1 , … , 10 {\displaystyle l=0,1,\ldots ,10} ed m = − l , … , − 1 , 0 , 1 , … , l {\displaystyle m=-l,\ldots ,-1,0,1,\ldots ,l} .[1] Tra le coordinate sferiche r , θ , φ {\displaystyle r,\theta ,\varphi } e le coordinate cartsiane x , y , z {\displaystyle x,y,z} usate talvolta sussistono le seguenti relazioni:

{ cos ⁡ θ = z r e i φ sin ⁡ θ = x + i y r . {\displaystyle {\begin{cases}\cos \theta ={\frac {z}{r}}\\e^{i\varphi }\sin \theta ={\frac {x+iy}{r}}.\end{cases}}}

Armoniche sferiche con l = 0

Y 0 0 ( x ) = 1 2 1 π {\displaystyle Y_{0}^{0}(x)={\frac {1}{2}}{\sqrt {1 \over \pi }}}

Armoniche sferiche con l = 1

Y 1 − 1 ( θ , φ ) = 1 2 3 2 π e − i φ sin ⁡ θ = 1 2 3 2 π ( x − i y ) r Y 1 0 ( θ , φ ) = 1 2 3 π cos ⁡ θ = 1 2 3 π z r Y 1 1 ( θ , φ ) = − 1 2 3 2 π e i φ sin ⁡ θ = − 1 2 3 2 π ( x + i y ) r {\displaystyle {\begin{aligned}Y_{1}^{-1}(\theta ,\varphi )&=&&{\frac {1}{2}}{\sqrt {3 \over 2\pi }}\,e^{-i\varphi }\,\sin \theta &&=&&{\frac {1}{2}}{\sqrt {3 \over 2\pi }}\,{(x-iy) \over r}\\Y_{1}^{0}(\theta ,\varphi )&=&&{\frac {1}{2}}{\sqrt {3 \over \pi }}\,\cos \theta &&=&&{\frac {1}{2}}{\sqrt {3 \over \pi }}\,{z \over r}\\Y_{1}^{1}(\theta ,\varphi )&=&-&{\frac {1}{2}}{\sqrt {3 \over 2\pi }}\,e^{i\varphi }\,\sin \theta &&=&-&{\frac {1}{2}}{\sqrt {3 \over 2\pi }}\,{(x+iy) \over r}\end{aligned}}}

Armoniche sferiche con l = 2

Y 2 − 2 ( θ , φ ) = 1 4 15 2 π e − 2 i φ sin 2 ⁡ θ = 1 4 15 2 π ( x − i y ) 2 r 2 Y 2 − 1 ( θ , φ ) = 1 2 15 2 π e − i φ sin ⁡ θ cos ⁡ θ = 1 2 15 2 π ( x − i y ) z r 2 Y 2 0 ( θ , φ ) = 1 4 5 π ( 3 cos 2 ⁡ θ − 1 ) = 1 4 5 π ( 2 z 2 − x 2 − y 2 ) r 2 Y 2 1 ( θ , φ ) = − 1 2 15 2 π e i φ sin ⁡ θ cos ⁡ θ = − 1 2 15 2 π ( x + i y ) z r 2 Y 2 2 ( θ , φ ) = 1 4 15 2 π e 2 i φ sin 2 ⁡ θ = 1 4 15 2 π ( x + i y ) 2 r 2 {\displaystyle {\begin{aligned}Y_{2}^{-2}(\theta ,\varphi )&=&&{\frac {1}{4}}{\sqrt {15 \over 2\pi }}\,e^{-2i\varphi }\,\sin ^{2}\theta \quad &&=&&{\frac {1}{4}}{\sqrt {15 \over 2\pi }}\,{(x-iy)^{2} \over r^{2}}&\\Y_{2}^{-1}(\theta ,\varphi )&=&&{\frac {1}{2}}{\sqrt {15 \over 2\pi }}\,e^{-i\varphi }\,\sin \theta \,\cos \theta \quad &&=&&{\frac {1}{2}}{\sqrt {15 \over 2\pi }}\,{(x-iy)z \over r^{2}}&\\Y_{2}^{0}(\theta ,\varphi )&=&&{\frac {1}{4}}{\sqrt {5 \over \pi }}\,(3\cos ^{2}\theta -1)\quad &&=&&{\frac {1}{4}}{\sqrt {5 \over \pi }}\,{(2z^{2}-x^{2}-y^{2}) \over r^{2}}&\\Y_{2}^{1}(\theta ,\varphi )&=&-&{\frac {1}{2}}{\sqrt {15 \over 2\pi }}\,e^{i\varphi }\,\sin \theta \,\cos \theta \quad &&=&-&{\frac {1}{2}}{\sqrt {15 \over 2\pi }}\,{(x+iy)z \over r^{2}}&\\Y_{2}^{2}(\theta ,\varphi )&=&&{\frac {1}{4}}{\sqrt {15 \over 2\pi }}\,e^{2i\varphi }\,\sin ^{2}\theta \quad &&=&&{\frac {1}{4}}{\sqrt {15 \over 2\pi }}\,{(x+iy)^{2} \over r^{2}}&\end{aligned}}}

Armoniche sferiche con l = 3

Y 3 − 3 ( θ , φ ) = 1 8 35 π e − 3 i φ sin 3 ⁡ θ = 1 8 35 π ( x − i y ) 3 r 3 Y 3 − 2 ( θ , φ ) = 1 4 105 2 π e − 2 i φ sin 2 ⁡ θ cos ⁡ θ = 1 4 105 2 π ( x − i y ) 2 z r 3 Y 3 − 1 ( θ , φ ) = 1 8 21 π e − i φ sin ⁡ θ ( 5 cos 2 ⁡ θ − 1 ) = 1 8 21 π ( x − i y ) ( 5 z 2 − r 2 ) r 3 Y 3 0 ( θ , φ ) = 1 4 7 π ( 5 cos 3 ⁡ θ − 3 cos ⁡ θ ) = 1 4 7 π z ( 5 z 2 − 3 r 2 ) r 3 Y 3 1 ( θ , φ ) = − 1 8 21 π e i φ sin ⁡ θ ( 5 cos 2 ⁡ θ − 1 ) = − 1 8 21 π ( x + i y ) ( 5 z 2 − r 2 ) r 3 Y 3 2 ( θ , φ ) = 1 4 105 2 π e 2 i φ sin 2 ⁡ θ cos ⁡ θ = 1 4 105 2 π ( x + i y ) 2 z r 3 Y 3 3 ( θ , φ ) = − 1 8 35 π e 3 i φ sin 3 ⁡ θ = − 1 8 35 π ( x + i y ) 3 r 3 {\displaystyle {\begin{aligned}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 )&=&&{\frac {1}{4}}{\sqrt {105 \over 2\pi }}\,e^{-2i\varphi }\,\sin ^{2}\theta \,\cos \theta \quad &&=&&{\frac {1}{4}}{\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 )&=&&{\frac {1}{4}}{\sqrt {7 \over \pi }}\,(5\cos ^{3}\theta -3\cos \theta )\quad &&=&&{\frac {1}{4}}{\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 )&=&&{\frac {1}{4}}{\sqrt {105 \over 2\pi }}\,e^{2i\varphi }\,\sin ^{2}\theta \,\cos \theta \quad &&=&&{\frac {1}{4}}{\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{aligned}}}

Armoniche sferiche con l = 4

Y 4 − 4 ( θ , φ ) = 3 16 35 2 π e − 4 i φ sin 4 ⁡ θ = 3 16 35 2 π ( x − i y ) 4 r 4 Y 4 − 3 ( θ , φ ) = 3 8 35 π e − 3 i φ sin 3 ⁡ θ cos ⁡ θ = 3 8 35 π ( x − i y ) 3 z r 4 Y 4 − 2 ( θ , φ ) = 3 8 5 2 π e − 2 i φ sin 2 ⁡ θ ( 7 cos 2 ⁡ θ − 1 ) = 3 8 5 2 π ( x − i y ) 2 ( 7 z 2 − r 2 ) r 4 Y 4 − 1 ( θ , φ ) = 3 8 5 π e − i φ sin ⁡ θ ( 7 cos 3 ⁡ θ − 3 cos ⁡ θ ) = 3 8 5 π ( x − i y ) z ( 7 z 2 − 3 r 2 ) r 4 Y 4 0 ( θ , φ ) = 3 16 1 π ( 35 cos 4 ⁡ θ − 30 cos 2 ⁡ θ + 3 ) = 3 16 1 π ( 35 z 4 − 30 z 2 r 2 + 3 r 4 ) r 4 Y 4 1 ( θ , φ ) = − 3 8 5 π e i φ sin ⁡ θ ( 7 cos 3 ⁡ θ − 3 cos ⁡ θ ) = − 3 8 5 π ( x + i y ) z ( 7 z 2 − 3 r 2 ) r 4 Y 4 2 ( θ , φ ) = 3 8 5 2 π e 2 i φ sin 2 ⁡ θ ( 7 cos 2 ⁡ θ − 1 ) = 3 8 5 2 π ( x + i y ) 2 ( 7 z 2 − r 2 ) r 4 Y 4 3 ( θ , φ ) = − 3 8 35 π e 3 i φ sin 3 ⁡ θ cos ⁡ θ = − 3 8 35 π ( x + i y ) 3 z r 4 Y 4 4 ( θ , φ ) = 3 16 35 2 π e 4 i φ sin 4 ⁡ θ = 3 16 35 2 π ( x + i y ) 4 r 4 {\displaystyle {\begin{aligned}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-iy)^{4}}{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-iy)^{3}z}{r^{4}}}&\\Y_{4}^{-2}(\theta ,\varphi )&=&&{\frac {3}{8}}{\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-iy)^{2}\,(7z^{2}-r^{2})}{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-iy)\,z\,(7z^{2}-3r^{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 {(35z^{4}-30z^{2}r^{2}+3r^{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+iy)\,z\,(7z^{2}-3r^{2})}{r^{4}}}&\\Y_{4}^{2}(\theta ,\varphi )&=&&{\frac {3}{8}}{\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+iy)^{2}\,(7z^{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+iy)^{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+iy)^{4}}{r^{4}}}&\\\end{aligned}}}

Armoniche sferiche con l = 5

Y 5 − 5 ( θ , φ ) = 3 32 77 π e − 5 i φ sin 5 ⁡ θ Y 5 − 4 ( θ , φ ) = 3 16 385 2 π e − 4 i φ sin 4 ⁡ θ cos ⁡ θ Y 5 − 3 ( θ , φ ) = 1 32 385 π e − 3 i φ sin 3 ⁡ θ ( 9 cos 2 ⁡ θ − 1 ) Y 5 − 2 ( θ , φ ) = 1 8 1155 2 π e − 2 i φ sin 2 ⁡ θ ( 3 cos 3 ⁡ θ − cos ⁡ θ ) Y 5 − 1 ( θ , φ ) = 1 16 165 2 π e − i φ sin ⁡ θ ( 21 cos 4 ⁡ θ − 14 cos 2 ⁡ θ + 1 ) Y 5 0 ( θ , φ ) = 1 16 11 π ( 63 cos 5 ⁡ θ − 70 cos 3 ⁡ θ + 15 cos ⁡ θ ) Y 5 1 ( θ , φ ) = − 1 16 165 2 π e i φ sin ⁡ θ ( 21 cos 4 ⁡ θ − 14 cos 2 ⁡ θ + 1 ) Y 5 2 ( θ , φ ) = 1 8 1155 2 π e 2 i φ sin 2 ⁡ θ ( 3 cos 3 ⁡ θ − cos ⁡ θ ) Y 5 3 ( θ , φ ) = − 1 32 385 π e 3 i φ sin 3 ⁡ θ ( 9 cos 2 ⁡ θ − 1 ) Y 5 4 ( θ , φ ) = 3 16 385 2 π e 4 i φ sin 4 ⁡ θ cos ⁡ θ Y 5 5 ( θ , φ ) = − 3 32 77 π e 5 i φ sin 5 ⁡ θ {\displaystyle {\begin{aligned}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{aligned}}}

Armoniche sferiche con l = 6

Y 6 − 6 ( θ , φ ) = 1 64 3003 π e − 6 i φ sin 6 ⁡ θ Y 6 − 5 ( θ , φ ) = 3 32 1001 π e − 5 i φ sin 5 ⁡ θ cos ⁡ θ Y 6 − 4 ( θ , φ ) = 3 32 91 2 π e − 4 i φ sin 4 ⁡ θ ( 11 cos 2 ⁡ θ − 1 ) Y 6 − 3 ( θ , φ ) = 1 32 1365 π e − 3 i φ sin 3 ⁡ θ ( 11 cos 3 ⁡ θ − 3 cos ⁡ θ ) Y 6 − 2 ( θ , φ ) = 1 64 1365 π e − 2 i φ sin 2 ⁡ θ ( 33 cos 4 ⁡ θ − 18 cos 2 ⁡ θ + 1 ) Y 6 − 1 ( θ , φ ) = 1 16 273 2 π e − i φ sin ⁡ θ ( 33 cos 5 ⁡ θ − 30 cos 3 ⁡ θ + 5 cos ⁡ θ ) Y 6 0 ( θ , φ ) = 1 32 13 π ( 231 cos 6 ⁡ θ − 315 cos 4 ⁡ θ + 105 cos 2 ⁡ θ − 5 ) Y 6 1 ( θ , φ ) = − 1 16 273 2 π e i φ sin ⁡ θ ( 33 cos 5 ⁡ θ − 30 cos 3 ⁡ θ + 5 cos ⁡ θ ) Y 6 2 ( θ , φ ) = 1 64 1365 π e 2 i φ sin 2 ⁡ θ ( 33 cos 4 ⁡ θ − 18 cos 2 ⁡ θ + 1 ) Y 6 3 ( θ , φ ) = − 1 32 1365 π e 3 i φ sin 3 ⁡ θ ( 11 cos 3 ⁡ θ − 3 cos ⁡ θ ) Y 6 4 ( θ , φ ) = 3 32 91 2 π e 4 i φ sin 4 ⁡ θ ( 11 cos 2 ⁡ θ − 1 ) Y 6 5 ( θ , φ ) = − 3 32 1001 π e 5 i φ sin 5 ⁡ θ cos ⁡ θ Y 6 6 ( θ , φ ) = 1 64 3003 π e 6 i φ sin 6 ⁡ θ {\displaystyle {\begin{aligned}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{aligned}}}

Armoniche sferiche con l = 7

Y 7 − 7 ( θ , φ ) = 3 64 715 2 π e − 7 i φ sin 7 ⁡ θ Y 7 − 6 ( θ , φ ) = 3 64 5005 π e − 6 i φ sin 6 ⁡ θ cos ⁡ θ Y 7 − 5 ( θ , φ ) = 3 64 385 2 π e − 5 i φ sin 5 ⁡ θ ( 13 cos 2 ⁡ θ − 1 ) Y 7 − 4 ( θ , φ ) = 3 32 385 2 π e − 4 i φ sin 4 ⁡ θ ( 13 cos 3 ⁡ θ − 3 cos ⁡ θ ) Y 7 − 3 ( θ , φ ) = 3 64 35 2 π e − 3 i φ sin 3 ⁡ θ ( 143 cos 4 ⁡ θ − 66 cos 2 ⁡ θ + 3 ) Y 7 − 2 ( θ , φ ) = 3 64 35 π e − 2 i φ sin 2 ⁡ θ ( 143 cos 5 ⁡ θ − 110 cos 3 ⁡ θ + 15 cos ⁡ θ ) Y 7 − 1 ( θ , φ ) = 1 64 105 2 π e − i φ sin ⁡ θ ( 429 cos 6 ⁡ θ − 495 cos 4 ⁡ θ + 135 cos 2 ⁡ θ − 5 ) Y 7 0 ( θ , φ ) = 1 32 15 π ( 429 cos 7 ⁡ θ − 693 cos 5 ⁡ θ + 315 cos 3 ⁡ θ − 35 cos ⁡ θ ) Y 7 1 ( θ , φ ) = − 1 64 105 2 π e i φ sin ⁡ θ ( 429 cos 6 ⁡ θ − 495 cos 4 ⁡ θ + 135 cos 2 ⁡ θ − 5 ) Y 7 2 ( θ , φ ) = 3 64 35 π e 2 i φ sin 2 ⁡ θ ( 143 cos 5 ⁡ θ − 110 cos 3 ⁡ θ + 15 cos ⁡ θ ) Y 7 3 ( θ , φ ) = − 3 64 35 2 π e 3 i φ sin 3 ⁡ θ ( 143 cos 4 ⁡ θ − 66 cos 2 ⁡ θ + 3 ) Y 7 4 ( θ , φ ) = 3 32 385 2 π e 4 i φ sin 4 ⁡ θ ( 13 cos 3 ⁡ θ − 3 cos ⁡ θ ) Y 7 5 ( θ , φ ) = − 3 64 385 2 π e 5 i φ sin 5 ⁡ θ ( 13 cos 2 ⁡ θ − 1 ) Y 7 6 ( θ , φ ) = 3 64 5005 π e 6 i φ sin 6 ⁡ θ cos ⁡ θ Y 7 7 ( θ , φ ) = − 3 64 715 2 π e 7 i φ sin 7 ⁡ θ {\displaystyle {\begin{aligned}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{aligned}}}

Armoniche sferiche con l = 8

Y 8 − 8 ( θ , φ ) = 3 256 12155 2 π e − 8 i φ sin 8 ⁡ θ Y 8 − 7 ( θ , φ ) = 3 64 12155 2 π e − 7 i φ sin 7 ⁡ θ cos ⁡ θ Y 8 − 6 ( θ , φ ) = 1 128 7293 π e − 6 i φ sin 6 ⁡ θ ( 15 cos 2 ⁡ θ − 1 ) Y 8 − 5 ( θ , φ ) = 3 64 17017 2 π e − 5 i φ sin 5 ⁡ θ ( 5 cos 3 ⁡ θ − cos ⁡ θ ) Y 8 − 4 ( θ , φ ) = 3 128 1309 2 π e − 4 i φ sin 4 ⁡ θ ( 65 cos 4 ⁡ θ − 26 cos 2 ⁡ θ + 1 ) Y 8 − 3 ( θ , φ ) = 1 64 19635 2 π e − 3 i φ sin 3 ⁡ θ ( 39 cos 5 ⁡ θ − 26 cos 3 ⁡ θ + 3 cos ⁡ θ ) Y 8 − 2 ( θ , φ ) = 3 128 595 π e − 2 i φ sin 2 ⁡ θ ( 143 cos 6 ⁡ θ − 143 cos 4 ⁡ θ + 33 cos 2 ⁡ θ − 1 ) Y 8 − 1 ( θ , φ ) = 3 64 17 2 π e − i φ sin ⁡ θ ( 715 cos 7 ⁡ θ − 1001 cos 5 ⁡ θ + 385 cos 3 ⁡ θ − 35 cos ⁡ θ ) Y 8 0 ( θ , φ ) = 1 256 17 π ( 6435 cos 8 ⁡ θ − 12012 cos 6 ⁡ θ + 6930 cos 4 ⁡ θ − 1260 cos 2 ⁡ θ + 35 ) Y 8 1 ( θ , φ ) = − 3 64 17 2 π e i φ sin ⁡ θ ( 715 cos 7 ⁡ θ − 1001 cos 5 ⁡ θ + 385 cos 3 ⁡ θ − 35 cos ⁡ θ ) Y 8 2 ( θ , φ ) = 3 128 595 π e 2 i φ sin 2 ⁡ θ ( 143 cos 6 ⁡ θ − 143 cos 4 ⁡ θ + 33 cos 2 ⁡ θ − 1 ) Y 8 3 ( θ , φ ) = − 1 64 19635 2 π e 3 i φ sin 3 ⁡ θ ( 39 cos 5 ⁡ θ − 26 cos 3 ⁡ θ + 3 cos ⁡ θ ) Y 8 4 ( θ , φ ) = 3 128 1309 2 π e 4 i φ sin 4 ⁡ θ ( 65 cos 4 ⁡ θ − 26 cos 2 ⁡ θ + 1 ) Y 8 5 ( θ , φ ) = − 3 64 17017 2 π e 5 i φ sin 5 ⁡ θ ( 5 cos 3 ⁡ θ − cos ⁡ θ ) Y 8 6 ( θ , φ ) = 1 128 7293 π e 6 i φ sin 6 ⁡ θ ( 15 cos 2 ⁡ θ − 1 ) Y 8 7 ( θ , φ ) = − 3 64 12155 2 π e 7 i φ sin 7 ⁡ θ cos ⁡ θ Y 8 8 ( θ , φ ) = 3 256 12155 2 π e 8 i φ sin 8 ⁡ θ {\displaystyle {\begin{aligned}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{aligned}}}

Armoniche sferiche con l = 9

Y 9 − 9 ( θ , φ ) = 1 512 230945 π e − 9 i φ sin 9 ⁡ θ Y 9 − 8 ( θ , φ ) = 3 256 230945 2 π e − 8 i φ sin 8 ⁡ θ cos ⁡ θ Y 9 − 7 ( θ , φ ) = 3 512 13585 π e − 7 i φ sin 7 ⁡ θ ( 17 cos 2 ⁡ θ − 1 ) Y 9 − 6 ( θ , φ ) = 1 128 40755 π e − 6 i φ sin 6 ⁡ θ ( 17 cos 3 ⁡ θ − 3 cos ⁡ θ ) Y 9 − 5 ( θ , φ ) = 3 256 2717 π e − 5 i φ sin 5 ⁡ θ ( 85 cos 4 ⁡ θ − 30 cos 2 ⁡ θ + 1 ) Y 9 − 4 ( θ , φ ) = 3 128 95095 2 π e − 4 i φ sin 4 ⁡ θ ( 17 cos 5 ⁡ θ − 10 cos 3 ⁡ θ + cos ⁡ θ ) Y 9 − 3 ( θ , φ ) = 1 256 21945 π e − 3 i φ sin 3 ⁡ θ ( 221 cos 6 ⁡ θ − 195 cos 4 ⁡ θ + 39 cos 2 ⁡ θ − 1 ) Y 9 − 2 ( θ , φ ) = 3 128 1045 π e − 2 i φ sin 2 ⁡ θ ( 221 cos 7 ⁡ θ − 273 cos 5 ⁡ θ + 91 cos 3 ⁡ θ − 7 cos ⁡ θ ) Y 9 − 1 ( θ , φ ) = 3 256 95 2 π e − i φ sin ⁡ θ ( 2431 cos 8 ⁡ θ − 4004 cos 6 ⁡ θ + 2002 cos 4 ⁡ θ − 308 cos 2 ⁡ θ + 7 ) Y 9 0 ( θ , φ ) = 1 256 19 π ( 12155 cos 9 ⁡ θ − 25740 cos 7 ⁡ θ + 18018 cos 5 ⁡ θ − 4620 cos 3 ⁡ θ + 315 cos ⁡ θ ) Y 9 1 ( θ , φ ) = − 3 256 95 2 π e i φ sin ⁡ θ ( 2431 cos 8 ⁡ θ − 4004 cos 6 ⁡ θ + 2002 cos 4 ⁡ θ − 308 cos 2 ⁡ θ + 7 ) Y 9 2 ( θ , φ ) = 3 128 1045 π e 2 i φ sin 2 ⁡ θ ( 221 cos 7 ⁡ θ − 273 cos 5 ⁡ θ + 91 cos 3 ⁡ θ − 7 cos ⁡ θ ) Y 9 3 ( θ , φ ) = − 1 256 21945 π e 3 i φ sin 3 ⁡ θ ( 221 cos 6 ⁡ θ − 195 cos 4 ⁡ θ + 39 cos 2 ⁡ θ − 1 ) Y 9 4 ( θ , φ ) = 3 128 95095 2 π e 4 i φ sin 4 ⁡ θ ( 17 cos 5 ⁡ θ − 10 cos 3 ⁡ θ + cos ⁡ θ ) Y 9 5 ( θ , φ ) = − 3 256 2717 π e 5 i φ sin 5 ⁡ θ ( 85 cos 4 ⁡ θ − 30 cos 2 ⁡ θ + 1 ) Y 9 6 ( θ , φ ) = 1 128 40755 π e 6 i φ sin 6 ⁡ θ ( 17 cos 3 ⁡ θ − 3 cos ⁡ θ ) Y 9 7 ( θ , φ ) = − 3 512 13585 π e 7 i φ sin 7 ⁡ θ ( 17 cos 2 ⁡ θ − 1 ) Y 9 8 ( θ , φ ) = 3 256 230945 2 π e 8 i φ sin 8 ⁡ θ cos ⁡ θ Y 9 9 ( θ , φ ) = − 1 512 230945 π e 9 i φ sin 9 ⁡ θ {\displaystyle {\begin{aligned}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{aligned}}}

Armoniche sferiche con l = 10

Y 10 − 10 ( θ , φ ) = 1 1024 969969 π e − 10 i φ sin 10 ⁡ θ Y 10 − 9 ( θ , φ ) = 1 512 4849845 π e − 9 i φ sin 9 ⁡ θ cos ⁡ θ Y 10 − 8 ( θ , φ ) = 1 512 255255 2 π e − 8 i φ sin 8 ⁡ θ ( 19 cos 2 ⁡ θ − 1 ) Y 10 − 7 ( θ , φ ) = 3 512 85085 π e − 7 i φ sin 7 ⁡ θ ( 19 cos 3 ⁡ θ − 3 cos ⁡ θ ) Y 10 − 6 ( θ , φ ) = 3 1024 5005 π e − 6 i φ sin 6 ⁡ θ ( 323 cos 4 ⁡ θ − 102 cos 2 ⁡ θ + 3 ) Y 10 − 5 ( θ , φ ) = 3 256 1001 π e − 5 i φ sin 5 ⁡ θ ( 323 cos 5 ⁡ θ − 170 cos 3 ⁡ θ + 15 cos ⁡ θ ) Y 10 − 4 ( θ , φ ) = 3 256 5005 2 π e − 4 i φ sin 4 ⁡ θ ( 323 cos 6 ⁡ θ − 255 cos 4 ⁡ θ + 45 cos 2 ⁡ θ − 1 ) Y 10 − 3 ( θ , φ ) = 3 256 5005 π e − 3 i φ sin 3 ⁡ θ ( 323 cos 7 ⁡ θ − 357 cos 5 ⁡ θ + 105 cos 3 ⁡ θ − 7 cos ⁡ θ ) Y 10 − 2 ( θ , φ ) = 3 512 385 2 π e − 2 i φ sin 2 ⁡ θ ( 4199 cos 8 ⁡ θ − 6188 cos 6 ⁡ θ + 2730 cos 4 ⁡ θ − 364 cos 2 ⁡ θ + 7 ) Y 10 − 1 ( θ , φ ) = 1 256 1155 2 π e − i φ sin ⁡ θ ( 4199 cos 9 ⁡ θ − 7956 cos 7 ⁡ θ + 4914 cos 5 ⁡ θ − 1092 cos 3 ⁡ θ + 63 cos ⁡ θ ) Y 10 0 ( θ , φ ) = 1 512 21 π ( 46189 cos 10 ⁡ θ − 109395 cos 8 ⁡ θ + 90090 cos 6 ⁡ θ − 30030 cos 4 ⁡ θ + 3465 cos 2 ⁡ θ − 63 ) Y 10 1 ( θ , φ ) = − 1 256 1155 2 π e i φ sin ⁡ θ ( 4199 cos 9 ⁡ θ − 7956 cos 7 ⁡ θ + 4914 cos 5 ⁡ θ − 1092 cos 3 ⁡ θ + 63 cos ⁡ θ ) Y 10 2 ( θ , φ ) = 3 512 385 2 π e 2 i φ sin 2 ⁡ θ ( 4199 cos 8 ⁡ θ − 6188 cos 6 ⁡ θ + 2730 cos 4 ⁡ θ − 364 cos 2 ⁡ θ + 7 ) Y 10 3 ( θ , φ ) = − 3 256 5005 π e 3 i φ sin 3 ⁡ θ ( 323 cos 7 ⁡ θ − 357 cos 5 ⁡ θ + 105 cos 3 ⁡ θ − 7 cos ⁡ θ ) Y 10 4 ( θ , φ ) = 3 256 5005 2 π e 4 i φ sin 4 ⁡ θ ( 323 cos 6 ⁡ θ − 255 cos 4 ⁡ θ + 45 cos 2 ⁡ θ − 1 ) Y 10 5 ( θ , φ ) = − 3 256 1001 π e 5 i φ sin 5 ⁡ θ ( 323 cos 5 ⁡ θ − 170 cos 3 ⁡ θ + 15 cos ⁡ θ ) Y 10 6 ( θ , φ ) = 3 1024 5005 π e 6 i φ sin 6 ⁡ θ ( 323 cos 4 ⁡ θ − 102 cos 2 ⁡ θ + 3 ) Y 10 7 ( θ , φ ) = − 3 512 85085 π e 7 i φ sin 7 ⁡ θ ( 19 cos 3 ⁡ θ − 3 cos ⁡ θ ) Y 10 8 ( θ , φ ) = 1 512 255255 2 π e 8 i φ sin 8 ⁡ θ ( 19 cos 2 ⁡ θ − 1 ) Y 10 9 ( θ , φ ) = − 1 512 4849845 π e 9 i φ sin 9 ⁡ θ cos ⁡ θ Y 10 10 ( θ , φ ) = 1 1024 969969 π e 10 i φ sin 10 ⁡ θ {\displaystyle {\begin{aligned}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{aligned}}}

Note

  1. ^ D. A. Varshalovich, A. N. Moskalev e V. K. Khersonskii, Quantum theory of angular momentum : irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols, prima ristampa, Singapore, World Scientific Pub., 1988, pp. 155–156, ISBN 9971-50-107-4.

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