球諧函數表

這裏列出球諧函數 Y l m {\displaystyle Y_{l}^{m}} ，以方程式表示為 Y ℓ m ( θ , φ ) = ( 2 ℓ + 1 ) 4 π ( ℓ − m ) ! ( ℓ + m ) ! P ℓ m ( cos ⁡ θ ) e i m φ {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )={\sqrt {{(2\ell +1) \over 4\pi }{(\ell -m)! \over (\ell +m)!}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\varphi }} ； 其中， l {\displaystyle l} 為正值整數， m {\displaystyle m} 為小於或等於 l {\displaystyle l} 的正值整數， P ℓ m {\displaystyle P_{\ell }^{m}} 是伴隨勒讓德多項式，以方程式表示為 P ℓ m ( x ) = ( − 1 ) m ( 1 − x 2 ) m / 2 d m P ℓ ( x ) d x m {\displaystyle 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 ⁡ θ cos ⁡ φ {\displaystyle x=r\sin \theta \cos \varphi \,} 、 y = r sin ⁡ θ sin ⁡ φ {\displaystyle y=r\sin \theta \sin \varphi \,} 、 z = r cos ⁡ θ {\displaystyle z=r\cos \theta \,} 。 Y 0 0 ( θ , φ ) = 1 2 1 π {\displaystyle Y_{0}^{0}(\theta ,\varphi )={1 \over 2}{\sqrt {1 \over \pi }}} 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 Y_{1}^{-1}(\theta ,\varphi )={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 ( θ , φ ) = 1 2 3 π ⋅ cos ⁡ θ = 1 2 3 π ⋅ z r {\displaystyle Y_{1}^{0}(\theta ,\varphi )={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 ( θ , φ ) = − 1 2 3 2 π ⋅ e i φ ⋅ sin ⁡ θ = − 1 2 3 2 π ⋅ ( x + i y ) r {\displaystyle Y_{1}^{1}(\theta ,\varphi )={-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 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 Y_{2}^{-2}(\theta ,\varphi )={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-iy)^{2} \over 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 {\displaystyle Y_{2}^{-1}(\theta ,\varphi )={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 {(x-iy)z \over r^{2}}} Y 2 0 ( θ , φ ) = 1 4 5 π ⋅ ( 3 cos 2 ⁡ θ − 1 ) = 1 4 5 π ⋅ ( − x 2 − y 2 + 2 z 2 ) r 2 {\displaystyle Y_{2}^{0}(\theta ,\varphi )={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 ( 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 {\displaystyle Y_{2}^{1}(\theta ,\varphi )={-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 {(x+iy)z \over 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 Y_{2}^{2}(\theta ,\varphi )={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+iy)^{2} \over r^{2}}} 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 Y_{3}^{-3}(\theta ,\varphi )={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-iy)^{3} \over 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 {\displaystyle Y_{3}^{-2}(\theta ,\varphi )={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-iy)^{2}z \over 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 ) ( 4 z 2 − x 2 − y 2 ) r 3 {\displaystyle Y_{3}^{-1}(\theta ,\varphi )={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-iy)(4z^{2}-x^{2}-y^{2}) \over r^{3}}} Y 3 0 ( θ , φ ) = 1 4 7 π ⋅ ( 5 cos 3 ⁡ θ − 3 cos ⁡ θ ) = 1 4 7 π ⋅ z ( 2 z 2 − 3 x 2 − 3 y 2 ) r 3 {\displaystyle Y_{3}^{0}(\theta ,\varphi )={1 \over 4}{\sqrt {7 \over \pi }}\cdot (5\cos ^{3}\theta -3\cos \theta )\quad ={1 \over 4}{\sqrt {7 \over \pi }}\cdot {z(2z^{2}-3x^{2}-3y^{2}) \over 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 ) ( 4 z 2 − x 2 − y 2 ) r 3 {\displaystyle Y_{3}^{1}(\theta ,\varphi )={-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+iy)(4z^{2}-x^{2}-y^{2}) \over 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 {\displaystyle Y_{3}^{2}(\theta ,\varphi )={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+iy)^{2}z \over 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 Y_{3}^{3}(\theta ,\varphi )={-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+iy)^{3} \over r^{3}}} 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 Y_{4}^{-4}(\theta ,\varphi )={3 \over 16}{\sqrt {35 \over 2\pi }}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta ={\frac {3}{16}}{\sqrt {\frac {35}{2\pi }}}\cdot {\frac {(x-iy)^{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 {\displaystyle Y_{4}^{-3}(\theta ,\varphi )={3 \over 8}{\sqrt {35 \over \pi }}\cdot e^{-3i\varphi }\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 ( 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 {\displaystyle Y_{4}^{-2}(\theta ,\varphi )={3 \over 8}{\sqrt {5 \over 2\pi }}\cdot e^{-2i\varphi }\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 ( 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 {\displaystyle Y_{4}^{-1}(\theta ,\varphi )={3 \over 8}{\sqrt {5 \over \pi }}\cdot e^{-i\varphi }\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 ( 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 {\displaystyle Y_{4}^{0}(\theta ,\varphi )={3 \over 16}{\sqrt {1 \over \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 ( 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 {\displaystyle Y_{4}^{1}(\theta ,\varphi )={-3 \over 8}{\sqrt {5 \over \pi }}\cdot e^{i\varphi }\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 ( 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 {\displaystyle Y_{4}^{2}(\theta ,\varphi )={3 \over 8}{\sqrt {5 \over 2\pi }}\cdot e^{2i\varphi }\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 ( 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 {\displaystyle Y_{4}^{3}(\theta ,\varphi )={-3 \over 8}{\sqrt {35 \over \pi }}\cdot e^{3i\varphi }\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 ( 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 Y_{4}^{4}(\theta ,\varphi )={3 \over 16}{\sqrt {35 \over 2\pi }}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta ={\frac {3}{16}}{\sqrt {\frac {35}{2\pi }}}\cdot {\frac {(x+iy)^{4}}{r^{4}}}} Y 5 − 5 ( θ , φ ) = 3 32 77 π ⋅ e − 5 i φ ⋅ sin 5 ⁡ θ {\displaystyle Y_{5}^{-5}(\theta ,\varphi )={3 \over 32}{\sqrt {77 \over \pi }}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta } Y 5 − 4 ( θ , φ ) = 3 16 385 2 π ⋅ e − 4 i φ ⋅ sin 4 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{5}^{-4}(\theta ,\varphi )={3 \over 16}{\sqrt {385 \over 2\pi }}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot \cos \theta } Y 5 − 3 ( θ , φ ) = 1 32 385 π ⋅ e − 3 i φ ⋅ sin 3 ⁡ θ ⋅ ( 9 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{5}^{-3}(\theta ,\varphi )={1 \over 32}{\sqrt {385 \over \pi }}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (9\cos ^{2}\theta -1)} Y 5 − 2 ( θ , φ ) = 1 8 1155 2 π ⋅ e − 2 i φ ⋅ sin 2 ⁡ θ ⋅ ( 3 cos 3 ⁡ θ − 1 cos ⁡ θ ) {\displaystyle Y_{5}^{-2}(\theta ,\varphi )={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 ( θ , φ ) = 1 16 165 2 π ⋅ e − i φ ⋅ sin ⁡ θ ⋅ ( 21 cos 4 ⁡ θ − 14 cos 2 ⁡ θ + 1 ) {\displaystyle Y_{5}^{-1}(\theta ,\varphi )={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 ( θ , φ ) = 1 16 11 π ⋅ ( 63 cos 5 ⁡ θ − 70 cos 3 ⁡ θ + 15 cos ⁡ θ ) {\displaystyle Y_{5}^{0}(\theta ,\varphi )={1 \over 16}{\sqrt {11 \over \pi }}\cdot (63\cos ^{5}\theta -70\cos ^{3}\theta +15\cos \theta )} Y 5 1 ( θ , φ ) = − 1 16 165 2 π ⋅ e i φ ⋅ sin ⁡ θ ⋅ ( 21 cos 4 ⁡ θ − 14 cos 2 ⁡ θ + 1 ) {\displaystyle Y_{5}^{1}(\theta ,\varphi )={-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 ( θ , φ ) = 1 8 1155 2 π ⋅ e 2 i φ ⋅ sin 2 ⁡ θ ⋅ ( 3 cos 3 ⁡ θ − 1 cos ⁡ θ ) {\displaystyle Y_{5}^{2}(\theta ,\varphi )={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 ( θ , φ ) = − 1 32 385 π ⋅ e 3 i φ ⋅ sin 3 ⁡ θ ⋅ ( 9 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{5}^{3}(\theta ,\varphi )={-1 \over 32}{\sqrt {385 \over \pi }}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (9\cos ^{2}\theta -1)} Y 5 4 ( θ , φ ) = 3 16 385 2 π ⋅ e 4 i φ ⋅ sin 4 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{5}^{4}(\theta ,\varphi )={3 \over 16}{\sqrt {385 \over 2\pi }}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot \cos \theta } Y 5 5 ( θ , φ ) = − 3 32 77 π ⋅ e 5 i φ ⋅ sin 5 ⁡ θ {\displaystyle Y_{5}^{5}(\theta ,\varphi )={-3 \over 32}{\sqrt {77 \over \pi }}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta } Y 6 − 6 ( θ , φ ) = 1 64 3003 π ⋅ e − 6 i φ ⋅ sin 6 ⁡ θ {\displaystyle Y_{6}^{-6}(\theta ,\varphi )={1 \over 64}{\sqrt {3003 \over \pi }}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta } Y 6 − 5 ( θ , φ ) = 3 32 1001 π ⋅ e − 5 i φ ⋅ sin 5 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{6}^{-5}(\theta ,\varphi )={3 \over 32}{\sqrt {1001 \over \pi }}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot \cos \theta } Y 6 − 4 ( θ , φ ) = 3 32 91 2 π ⋅ e − 4 i φ ⋅ sin 4 ⁡ θ ⋅ ( 11 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{6}^{-4}(\theta ,\varphi )={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 ( θ , φ ) = 1 32 1365 π ⋅ e − 3 i φ ⋅ sin 3 ⁡ θ ⋅ ( 11 cos 3 ⁡ θ − 3 cos ⁡ θ ) {\displaystyle Y_{6}^{-3}(\theta ,\varphi )={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 ( θ , φ ) = 1 64 1365 π ⋅ e − 2 i φ ⋅ sin 2 ⁡ θ ⋅ ( 33 cos 4 ⁡ θ − 18 cos 2 ⁡ θ + 1 ) {\displaystyle Y_{6}^{-2}(\theta ,\varphi )={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 ( θ , φ ) = 1 16 273 2 π ⋅ e − i φ ⋅ sin ⁡ θ ⋅ ( 33 cos 5 ⁡ θ − 30 cos 3 ⁡ θ + 5 cos ⁡ θ ) {\displaystyle Y_{6}^{-1}(\theta ,\varphi )={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 ( θ , φ ) = 1 32 13 π ⋅ ( 231 cos 6 ⁡ θ − 315 cos 4 ⁡ θ + 105 cos 2 ⁡ θ − 5 ) {\displaystyle Y_{6}^{0}(\theta ,\varphi )={1 \over 32}{\sqrt {13 \over \pi }}\cdot (231\cos ^{6}\theta -315\cos ^{4}\theta +105\cos ^{2}\theta -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 ) {\displaystyle Y_{6}^{1}(\theta ,\varphi )={-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 ( θ , φ ) = 1 64 1365 π ⋅ e 2 i φ ⋅ sin 2 ⁡ θ ⋅ ( 33 cos 4 ⁡ θ − 18 cos 2 ⁡ θ + 1 ) {\displaystyle Y_{6}^{2}(\theta ,\varphi )={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 ( θ , φ ) = − 1 32 1365 π ⋅ e 3 i φ ⋅ sin 3 ⁡ θ ⋅ ( 11 cos 3 ⁡ θ − 3 cos ⁡ θ ) {\displaystyle Y_{6}^{3}(\theta ,\varphi )={-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 ( θ , φ ) = 3 32 91 2 π ⋅ e 4 i φ ⋅ sin 4 ⁡ θ ⋅ ( 11 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{6}^{4}(\theta ,\varphi )={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 ( θ , φ ) = − 3 32 1001 π ⋅ e 5 i φ ⋅ sin 5 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{6}^{5}(\theta ,\varphi )={-3 \over 32}{\sqrt {1001 \over \pi }}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot \cos \theta } Y 6 6 ( θ , φ ) = 1 64 3003 π ⋅ e 6 i φ ⋅ sin 6 ⁡ θ {\displaystyle Y_{6}^{6}(\theta ,\varphi )={1 \over 64}{\sqrt {3003 \over \pi }}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta } Y 7 − 7 ( θ , φ ) = 3 64 715 2 π ⋅ e − 7 i φ ⋅ sin 7 ⁡ θ {\displaystyle Y_{7}^{-7}(\theta ,\varphi )={3 \over 64}{\sqrt {715 \over 2\pi }}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta } Y 7 − 6 ( θ , φ ) = 3 64 5005 π ⋅ e − 6 i φ ⋅ sin 6 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{7}^{-6}(\theta ,\varphi )={3 \over 64}{\sqrt {5005 \over \pi }}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \cdot \cos \theta } Y 7 − 5 ( θ , φ ) = 3 64 385 2 π ⋅ e − 5 i φ ⋅ sin 5 ⁡ θ ⋅ ( 13 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{7}^{-5}(\theta ,\varphi )={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 ( θ , φ ) = 3 32 385 2 π ⋅ e − 4 i φ ⋅ sin 4 ⁡ θ ⋅ ( 13 cos 3 ⁡ θ − 3 cos ⁡ θ ) {\displaystyle Y_{7}^{-4}(\theta ,\varphi )={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 ( θ , φ ) = 3 64 35 2 π ⋅ e − 3 i φ ⋅ sin 3 ⁡ θ ⋅ ( 143 cos 4 ⁡ θ − 66 cos 2 ⁡ θ + 3 ) {\displaystyle Y_{7}^{-3}(\theta ,\varphi )={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 ( θ , φ ) = 3 64 35 π ⋅ e − 2 i φ ⋅ sin 2 ⁡ θ ⋅ ( 143 cos 5 ⁡ θ − 110 cos 3 ⁡ θ + 15 cos ⁡ θ ) {\displaystyle Y_{7}^{-2}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{7}^{-1}(\theta ,\varphi )={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 ( θ , φ ) = 1 32 15 π ⋅ ( 429 cos 7 ⁡ θ − 693 cos 5 ⁡ θ + 315 cos 3 ⁡ θ − 35 cos ⁡ θ ) {\displaystyle Y_{7}^{0}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{7}^{1}(\theta ,\varphi )={-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 ( θ , φ ) = 3 64 35 π ⋅ e 2 i φ ⋅ sin 2 ⁡ θ ⋅ ( 143 cos 5 ⁡ θ − 110 cos 3 ⁡ θ + 15 cos ⁡ θ ) {\displaystyle Y_{7}^{2}(\theta ,\varphi )={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 ( θ , φ ) = − 3 64 35 2 π ⋅ e 3 i φ ⋅ sin 3 ⁡ θ ⋅ ( 143 cos 4 ⁡ θ − 66 cos 2 ⁡ θ + 3 ) {\displaystyle Y_{7}^{3}(\theta ,\varphi )={-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 ( θ , φ ) = 3 32 385 2 π ⋅ e 4 i φ ⋅ sin 4 ⁡ θ ⋅ ( 13 cos 3 ⁡ θ − 3 cos ⁡ θ ) {\displaystyle Y_{7}^{4}(\theta ,\varphi )={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 ( θ , φ ) = − 3 64 385 2 π ⋅ e 5 i φ ⋅ sin 5 ⁡ θ ⋅ ( 13 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{7}^{5}(\theta ,\varphi )={-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 ( θ , φ ) = 3 64 5005 π ⋅ e 6 i φ ⋅ sin 6 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{7}^{6}(\theta ,\varphi )={3 \over 64}{\sqrt {5005 \over \pi }}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot \cos \theta } Y 7 7 ( θ , φ ) = − 3 64 715 2 π ⋅ e 7 i φ ⋅ sin 7 ⁡ θ {\displaystyle Y_{7}^{7}(\theta ,\varphi )={-3 \over 64}{\sqrt {715 \over 2\pi }}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta } Y 8 − 8 ( θ , φ ) = 3 256 12155 2 π ⋅ e − 8 i φ ⋅ sin 8 ⁡ θ {\displaystyle Y_{8}^{-8}(\theta ,\varphi )={3 \over 256}{\sqrt {12155 \over 2\pi }}\cdot e^{-8i\varphi }\cdot \sin ^{8}\theta } Y 8 − 7 ( θ , φ ) = 3 64 12155 2 π ⋅ e − 7 i φ ⋅ sin 7 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{8}^{-7}(\theta ,\varphi )={3 \over 64}{\sqrt {12155 \over 2\pi }}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta \cdot \cos \theta } Y 8 − 6 ( θ , φ ) = 1 128 7293 π ⋅ e − 6 i φ ⋅ sin 6 ⁡ θ ⋅ ( 15 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{8}^{-6}(\theta ,\varphi )={1 \over 128}{\sqrt {7293 \over \pi }}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \cdot (15\cos ^{2}\theta -1)} Y 8 − 5 ( θ , φ ) = 3 64 17017 2 π ⋅ e − 5 i φ ⋅ sin 5 ⁡ θ ⋅ ( 5 cos 3 ⁡ θ − 1 cos ⁡ θ ) {\displaystyle Y_{8}^{-5}(\theta ,\varphi )={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 ( θ , φ ) = 3 128 1309 2 π ⋅ e − 4 i φ ⋅ sin 4 ⁡ θ ⋅ ( 65 cos 4 ⁡ θ − 26 cos 2 ⁡ θ + 1 ) {\displaystyle Y_{8}^{-4}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{8}^{-3}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{8}^{-2}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{8}^{-1}(\theta ,\varphi )={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 ( θ , φ ) = 1 256 17 π ⋅ ( 6435 cos 8 ⁡ θ − 12012 cos 6 ⁡ θ + 6930 cos 4 ⁡ θ − 1260 cos 2 ⁡ θ + 35 ) {\displaystyle Y_{8}^{0}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{8}^{1}(\theta ,\varphi )={-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 ( 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 ) {\displaystyle Y_{8}^{2}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{8}^{3}(\theta ,\varphi )={-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 ( θ , φ ) = 3 128 1309 2 π ⋅ e 4 i φ ⋅ sin 4 ⁡ θ ⋅ ( 65 cos 4 ⁡ θ − 26 cos 2 ⁡ θ + 1 ) {\displaystyle Y_{8}^{4}(\theta ,\varphi )={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 ( θ , φ ) = − 3 64 17017 2 π ⋅ e 5 i φ ⋅ sin 5 ⁡ θ ⋅ ( 5 cos 3 ⁡ θ − 1 cos ⁡ θ ) {\displaystyle Y_{8}^{5}(\theta ,\varphi )={-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 ( θ , φ ) = 1 128 7293 π ⋅ e 6 i φ ⋅ sin 6 ⁡ θ ⋅ ( 15 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{8}^{6}(\theta ,\varphi )={1 \over 128}{\sqrt {7293 \over \pi }}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot (15\cos ^{2}\theta -1)} Y 8 7 ( θ , φ ) = − 3 64 12155 2 π ⋅ e 7 i φ ⋅ sin 7 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{8}^{7}(\theta ,\varphi )={-3 \over 64}{\sqrt {12155 \over 2\pi }}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta \cdot \cos \theta } Y 8 8 ( θ , φ ) = 3 256 12155 2 π ⋅ e 8 i φ ⋅ sin 8 ⁡ θ {\displaystyle Y_{8}^{8}(\theta ,\varphi )={3 \over 256}{\sqrt {12155 \over 2\pi }}\cdot e^{8i\varphi }\cdot \sin ^{8}\theta } Y 9 − 9 ( θ , φ ) = 1 512 230945 π ⋅ e − 9 i φ ⋅ sin 9 ⁡ θ {\displaystyle Y_{9}^{-9}(\theta ,\varphi )={1 \over 512}{\sqrt {230945 \over \pi }}\cdot e^{-9i\varphi }\cdot \sin ^{9}\theta } Y 9 − 8 ( θ , φ ) = 3 256 230945 2 π ⋅ e − 8 i φ ⋅ sin 8 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{9}^{-8}(\theta ,\varphi )={3 \over 256}{\sqrt {230945 \over 2\pi }}\cdot e^{-8i\varphi }\cdot \sin ^{8}\theta \cdot \cos \theta } Y 9 − 7 ( θ , φ ) = 3 512 13585 π ⋅ e − 7 i φ ⋅ sin 7 ⁡ θ ⋅ ( 17 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{9}^{-7}(\theta ,\varphi )={3 \over 512}{\sqrt {13585 \over \pi }}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta \cdot (17\cos ^{2}\theta -1)} Y 9 − 6 ( θ , φ ) = 1 128 40755 π ⋅ e − 6 i φ ⋅ sin 6 ⁡ θ ⋅ ( 17 cos 3 ⁡ θ − 3 cos ⁡ θ ) {\displaystyle Y_{9}^{-6}(\theta ,\varphi )={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 ( θ , φ ) = 3 256 2717 π ⋅ e − 5 i φ ⋅ sin 5 ⁡ θ ⋅ ( 85 cos 4 ⁡ θ − 30 cos 2 ⁡ θ + 1 ) {\displaystyle Y_{9}^{-5}(\theta ,\varphi )={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 ( θ , φ ) = 3 128 95095 2 π ⋅ e − 4 i φ ⋅ sin 4 ⁡ θ ⋅ ( 17 cos 5 ⁡ θ − 10 cos 3 ⁡ θ + 1 cos ⁡ θ ) {\displaystyle Y_{9}^{-4}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{9}^{-3}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{9}^{-2}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{9}^{-1}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{9}^{0}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{9}^{1}(\theta ,\varphi )={-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 ( 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 ) {\displaystyle Y_{9}^{2}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{9}^{3}(\theta ,\varphi )={-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 ( θ , φ ) = 3 128 95095 2 π ⋅ e 4 i φ ⋅ sin 4 ⁡ θ ⋅ ( 17 cos 5 ⁡ θ − 10 cos 3 ⁡ θ + 1 cos ⁡ θ ) {\displaystyle Y_{9}^{4}(\theta ,\varphi )={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 ( θ , φ ) = − 3 256 2717 π ⋅ e 5 i φ ⋅ sin 5 ⁡ θ ⋅ ( 85 cos 4 ⁡ θ − 30 cos 2 ⁡ θ + 1 ) {\displaystyle Y_{9}^{5}(\theta ,\varphi )={-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 ( θ , φ ) = 1 128 40755 π ⋅ e 6 i φ ⋅ sin 6 ⁡ θ ⋅ ( 17 cos 3 ⁡ θ − 3 cos ⁡ θ ) {\displaystyle Y_{9}^{6}(\theta ,\varphi )={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 ( θ , φ ) = − 3 512 13585 π ⋅ e 7 i φ ⋅ sin 7 ⁡ θ ⋅ ( 17 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{9}^{7}(\theta ,\varphi )={-3 \over 512}{\sqrt {13585 \over \pi }}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta \cdot (17\cos ^{2}\theta -1)} Y 9 8 ( θ , φ ) = 3 256 230945 2 π ⋅ e 8 i φ ⋅ sin 8 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{9}^{8}(\theta ,\varphi )={3 \over 256}{\sqrt {230945 \over 2\pi }}\cdot e^{8i\varphi }\cdot \sin ^{8}\theta \cdot \cos \theta } Y 9 9 ( θ , φ ) = − 1 512 230945 π ⋅ e 9 i φ ⋅ sin 9 ⁡ θ {\displaystyle Y_{9}^{9}(\theta ,\varphi )={-1 \over 512}{\sqrt {230945 \over \pi }}\cdot e^{9i\varphi }\cdot \sin ^{9}\theta } Y 10 − 10 ( θ , φ ) = 1 1024 969969 π ⋅ e − 10 i φ ⋅ sin 10 ⁡ θ {\displaystyle Y_{10}^{-10}(\theta ,\varphi )={1 \over 1024}{\sqrt {969969 \over \pi }}\cdot e^{-10i\varphi }\cdot \sin ^{10}\theta } Y 10 − 9 ( θ , φ ) = 1 512 4849845 π ⋅ e − 9 i φ ⋅ sin 9 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{10}^{-9}(\theta ,\varphi )={1 \over 512}{\sqrt {4849845 \over \pi }}\cdot e^{-9i\varphi }\cdot \sin ^{9}\theta \cdot \cos \theta } Y 10 − 8 ( θ , φ ) = 1 512 255255 2 π ⋅ e − 8 i φ ⋅ sin 8 ⁡ θ ⋅ ( 19 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{10}^{-8}(\theta ,\varphi )={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 ( θ , φ ) = 3 512 85085 π ⋅ e − 7 i φ ⋅ sin 7 ⁡ θ ⋅ ( 19 cos 3 ⁡ θ − 3 cos ⁡ θ ) {\displaystyle Y_{10}^{-7}(\theta ,\varphi )={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 ( θ , φ ) = 3 1024 5005 π ⋅ e − 6 i φ ⋅ sin 6 ⁡ θ ⋅ ( 323 cos 4 ⁡ θ − 102 cos 2 ⁡ θ + 3 ) {\displaystyle Y_{10}^{-6}(\theta ,\varphi )={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 ( θ , φ ) = 3 256 1001 π ⋅ e − 5 i φ ⋅ sin 5 ⁡ θ ⋅ ( 323 cos 5 ⁡ θ − 170 cos 3 ⁡ θ + 15 cos ⁡ θ ) {\displaystyle Y_{10}^{-5}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{10}^{-4}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{10}^{-3}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{10}^{-2}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{10}^{-1}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{10}^{0}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{10}^{1}(\theta ,\varphi )={-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 ( 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 ) {\displaystyle Y_{10}^{2}(\theta ,\varphi )={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 ( 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 ) {\displaystyle Y_{10}^{3}(\theta ,\varphi )={-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 ( 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 ) {\displaystyle Y_{10}^{4}(\theta ,\varphi )={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 ( θ , φ ) = − 3 256 1001 π ⋅ e 5 i φ ⋅ sin 5 ⁡ θ ⋅ ( 323 cos 5 ⁡ θ − 170 cos 3 ⁡ θ + 15 cos ⁡ θ ) {\displaystyle Y_{10}^{5}(\theta ,\varphi )={-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 ( θ , φ ) = 3 1024 5005 π ⋅ e 6 i φ ⋅ sin 6 ⁡ θ ⋅ ( 323 cos 4 ⁡ θ − 102 cos 2 ⁡ θ + 3 ) {\displaystyle Y_{10}^{6}(\theta ,\varphi )={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 ( θ , φ ) = − 3 512 85085 π ⋅ e 7 i φ ⋅ sin 7 ⁡ θ ⋅ ( 19 cos 3 ⁡ θ − 3 cos ⁡ θ ) {\displaystyle Y_{10}^{7}(\theta ,\varphi )={-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 ( θ , φ ) = 1 512 255255 2 π ⋅ e 8 i φ ⋅ sin 8 ⁡ θ ⋅ ( 19 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{10}^{8}(\theta ,\varphi )={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 ( θ , φ ) = − 1 512 4849845 π ⋅ e 9 i φ ⋅ sin 9 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{10}^{9}(\theta ,\varphi )={-1 \over 512}{\sqrt {4849845 \over \pi }}\cdot e^{9i\varphi }\cdot \sin ^{9}\theta \cdot \cos \theta } Y 10 10 ( θ , φ ) = 1 1024 969969 π ⋅ e 10 i φ ⋅ sin 10 ⁡ θ {\displaystyle Y_{10}^{10}(\theta ,\varphi )={1 \over 1024}{\sqrt {969969 \over \pi }}\cdot e^{10i\varphi }\cdot \sin ^{10}\theta }