Acesta este tabelul armonicelor sferice orthonormalizate care implică faza Condon-Shortley până la gradul I=10. Unele din aceste formule sunt date și în versiunea "carteziană", coordonatele x, y și z fiind legate de θ {\displaystyle \theta } φ {\displaystyle \varphi \,}
x = r sin θ cos φ y = r sin θ sin φ z = r cos θ {\displaystyle {\begin{aligned}x&=r\sin \theta \cos \varphi \\y&=r\sin \theta \sin \varphi \\z&=r\cos \theta \end{aligned}}}
l = 0 Y 0 0 ( θ , φ ) = 1 2 1 π {\displaystyle Y_{0}^{0}(\theta ,\varphi )={1 \over 2}{\sqrt {1 \over \pi }}}
l = 0 s = Y 0 0 = 1 2 1 π {\displaystyle s=Y_{0}^{0}={\frac {1}{2}}{\sqrt {\frac {1}{\pi }}}}
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 )&={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}(\theta ,\varphi )&={1 \over 2}{\sqrt {3 \over \pi }}\cdot \cos \theta \quad \quad ={1 \over 2}{\sqrt {3 \over \pi }}\cdot {z \over r}\\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}\end{aligned}}}
l = 1 p x = 1 2 ( Y 1 − 1 − Y 1 1 ) = 3 4 π ⋅ x r p y = i 1 2 ( Y 1 − 1 + Y 1 1 ) = 3 4 π ⋅ y r p z = Y 1 0 = 3 4 π ⋅ z r {\displaystyle {\begin{aligned}p_{x}&={\sqrt {\frac {1}{2}}}\left(Y_{1}^{-1}-Y_{1}^{1}\right)={\sqrt {\frac {3}{4\pi }}}\cdot {\frac {x}{r}}\\p_{y}&=i{\sqrt {\frac {1}{2}}}\left(Y_{1}^{-1}+Y_{1}^{1}\right)={\sqrt {\frac {3}{4\pi }}}\cdot {\frac {y}{r}}\\p_{z}&=Y_{1}^{0}={\sqrt {\frac {3}{4\pi }}}\cdot {\frac {z}{r}}\end{aligned}}}
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 {\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}}}
l = 2 d z 2 = Y 2 0 = 1 4 5 π ⋅ − x 2 − y 2 + 2 z 2 r 2 {\displaystyle d_{z^{2}}=Y_{2}^{0}={\frac {1}{4}}{\sqrt {\frac {5}{\pi }}}\cdot {\frac {-x^{2}-y^{2}+2z^{2}}{r^{2}}}} d y z = i 1 2 ( Y 2 − 1 + Y 2 1 ) = 1 2 15 π ⋅ y z r 2 {\displaystyle d_{yz}=i{\sqrt {\frac {1}{2}}}\left(Y_{2}^{-1}+Y_{2}^{1}\right)={\frac {1}{2}}{\sqrt {\frac {15}{\pi }}}\cdot {\frac {yz}{r^{2}}}} d x z = 1 2 ( Y 2 − 1 − Y 2 1 ) = 1 2 15 π ⋅ z x r 2 {\displaystyle d_{xz}={\sqrt {\frac {1}{2}}}\left(Y_{2}^{-1}-Y_{2}^{1}\right)={\frac {1}{2}}{\sqrt {\frac {15}{\pi }}}\cdot {\frac {zx}{r^{2}}}} d x y = i 1 2 ( Y 2 − 2 − Y 2 2 ) = 1 2 15 π ⋅ x y r 2 {\displaystyle d_{xy}=i{\sqrt {\frac {1}{2}}}\left(Y_{2}^{-2}-Y_{2}^{2}\right)={\frac {1}{2}}{\sqrt {\frac {15}{\pi }}}\cdot {\frac {xy}{r^{2}}}} d x 2 − y 2 = 1 2 ( Y 2 − 2 + Y 2 2 ) = 1 4 15 π ⋅ x 2 − y 2 r 2 {\displaystyle d_{x^{2}-y^{2}}={\sqrt {\frac {1}{2}}}\left(Y_{2}^{-2}+Y_{2}^{2}\right)={\frac {1}{4}}{\sqrt {\frac {15}{\pi }}}\cdot {\frac {x^{2}-y^{2}}{r^{2}}}}
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 {\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}}}
l = 3 f z 3 = Y 3 0 = 1 4 7 π ⋅ z ( 2 z 2 − 3 x 2 − 3 y 2 ) r 3 {\displaystyle f_{z^{3}}=Y_{3}^{0}={\frac {1}{4}}{\sqrt {\frac {7}{\pi }}}\cdot {\frac {z(2z^{2}-3x^{2}-3y^{2})}{r^{3}}}} f y ( 3 x 2 − y 2 ) = i 1 2 ( Y 3 − 3 + Y 3 3 ) = 1 4 35 2 π ⋅ ( 3 x 2 − y 2 ) y r 3 {\displaystyle f_{y\left(3x^{2}-y^{2}\right)}=i{\sqrt {\frac {1}{2}}}\left(Y_{3}^{-3}+Y_{3}^{3}\right)={\frac {1}{4}}{\sqrt {\frac {35}{2\pi }}}\cdot {\frac {\left(3x^{2}-y^{2}\right)y}{r^{3}}}} f x ( x 2 − 3 y 2 ) = 1 2 ( Y 3 − 3 − Y 3 3 ) = 1 4 35 2 π ⋅ ( x 2 − 3 y 2 ) x r 3 {\displaystyle f_{x\left(x^{2}-3y^{2}\right)}={\sqrt {\frac {1}{2}}}\left(Y_{3}^{-3}-Y_{3}^{3}\right)={\frac {1}{4}}{\sqrt {\frac {35}{2\pi }}}\cdot {\frac {\left(x^{2}-3y^{2}\right)x}{r^{3}}}} f z ( x 2 − y 2 ) = 1 2 ( Y 3 − 2 + Y 3 2 ) = 1 4 105 π ⋅ ( x 2 − y 2 ) z r 3 {\displaystyle f_{z\left(x^{2}-y^{2}\right)}={\sqrt {\frac {1}{2}}}\left(Y_{3}^{-2}+Y_{3}^{2}\right)={\frac {1}{4}}{\sqrt {\frac {105}{\pi }}}\cdot {\frac {\left(x^{2}-y^{2}\right)z}{r^{3}}}} f x y z = i 1 2 ( Y 3 − 2 − Y 3 2 ) = 1 2 105 π ⋅ x y z r 3 {\displaystyle f_{xyz}=i{\sqrt {\frac {1}{2}}}\left(Y_{3}^{-2}-Y_{3}^{2}\right)={\frac {1}{2}}{\sqrt {\frac {105}{\pi }}}\cdot {\frac {xyz}{r^{3}}}} f y z 2 = i 1 2 ( Y 3 − 1 + Y 3 1 ) = 1 4 21 2 π ⋅ y ( 4 z 2 − x 2 − y 2 ) r 3 {\displaystyle f_{yz^{2}}=i{\sqrt {\frac {1}{2}}}\left(Y_{3}^{-1}+Y_{3}^{1}\right)={\frac {1}{4}}{\sqrt {\frac {21}{2\pi }}}\cdot {\frac {y(4z^{2}-x^{2}-y^{2})}{r^{3}}}} f x z 2 = 1 2 ( Y 3 − 1 − Y 3 1 ) = 1 4 21 2 π ⋅ x ( 4 z 2 − x 2 − y 2 ) r 3 {\displaystyle f_{xz^{2}}={\sqrt {\frac {1}{2}}}\left(Y_{3}^{-1}-Y_{3}^{1}\right)={\frac {1}{4}}{\sqrt {\frac {21}{2\pi }}}\cdot {\frac {x(4z^{2}-x^{2}-y^{2})}{r^{3}}}}
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 {\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}}}}
l = 4 g z 4 = Y 4 0 = 3 16 1 π ⋅ ( 35 z 4 − 30 z 2 r 2 + 3 r 4 ) r 4 {\displaystyle g_{z^{4}}=Y_{4}^{0}={\frac {3}{16}}{\sqrt {\frac {1}{\pi }}}\cdot {\frac {(35z^{4}-30z^{2}r^{2}+3r^{4})}{r^{4}}}} g z 3 x = 1 2 ( Y 4 − 1 − Y 4 1 ) = 3 4 5 2 π ⋅ x z ⋅ ( 7 z 2 − 3 r 2 ) r 4 {\displaystyle g_{z^{3}x}={\sqrt {\frac {1}{2}}}\left(Y_{4}^{-1}-Y_{4}^{1}\right)={\frac {3}{4}}{\sqrt {\frac {5}{2\pi }}}\cdot {\frac {xz\cdot (7z^{2}-3r^{2})}{r^{4}}}} g z 3 y = i 1 2 ( Y 4 − 1 + Y 4 1 ) = 3 4 5 2 π ⋅ y z ⋅ ( 7 z 2 − 3 r 2 ) r 4 {\displaystyle g_{z^{3}y}=i{\sqrt {\frac {1}{2}}}\left(Y_{4}^{-1}+Y_{4}^{1}\right)={\frac {3}{4}}{\sqrt {\frac {5}{2\pi }}}\cdot {\frac {yz\cdot (7z^{2}-3r^{2})}{r^{4}}}} g z 2 ( x 2 − y 2 ) = 1 2 ( Y 4 − 2 + Y 4 2 ) = 3 8 5 π ⋅ ( x 2 − y 2 ) ⋅ ( 7 z 2 − r 2 ) r 4 {\displaystyle g_{z^{2}\left(x^{2}-y^{2}\right)}={\sqrt {\frac {1}{2}}}\left(Y_{4}^{-2}+Y_{4}^{2}\right)={\frac {3}{8}}{\sqrt {\frac {5}{\pi }}}\cdot {\frac {(x^{2}-y^{2})\cdot (7z^{2}-r^{2})}{r^{4}}}} g z 2 x y = i 1 2 ( Y 4 − 2 − Y 4 2 ) = 3 4 5 π ⋅ x y ⋅ ( 7 z 2 − r 2 ) r 4 {\displaystyle g_{z^{2}{xy}}=i{\sqrt {\frac {1}{2}}}\left(Y_{4}^{-2}-Y_{4}^{2}\right)={\frac {3}{4}}{\sqrt {\frac {5}{\pi }}}\cdot {\frac {xy\cdot (7z^{2}-r^{2})}{r^{4}}}} g z x 3 = 1 2 ( Y 4 − 3 − Y 4 3 ) = 3 4 35 2 π ⋅ ( x 2 − 3 y 2 ) x z r 4 {\displaystyle g_{zx^{3}}={\sqrt {\frac {1}{2}}}\left(Y_{4}^{-3}-Y_{4}^{3}\right)={\frac {3}{4}}{\sqrt {\frac {35}{2\pi }}}\cdot {\frac {(x^{2}-3y^{2})xz}{r^{4}}}} g z y 3 = i 1 2 ( Y 4 − 3 + Y 4 3 ) = 3 4 35 2 π ⋅ ( 3 x 2 − y 2 ) y z r 4 {\displaystyle g_{zy^{3}}=i{\sqrt {\frac {1}{2}}}\left(Y_{4}^{-3}+Y_{4}^{3}\right)={\frac {3}{4}}{\sqrt {\frac {35}{2\pi }}}\cdot {\frac {(3x^{2}-y^{2})yz}{r^{4}}}} g x 4 + y 4 = 1 2 ( Y 4 − 4 + Y 4 4 ) = 3 16 35 π ⋅ x 2 ( x 2 − 3 y 2 ) − y 2 ( 3 x 2 − y 2 ) r 4 {\displaystyle g_{x^{4}+y^{4}}={\sqrt {\frac {1}{2}}}\left(Y_{4}^{-4}+Y_{4}^{4}\right)={\frac {3}{16}}{\sqrt {\frac {35}{\pi }}}\cdot {\frac {x^{2}\left(x^{2}-3y^{2}\right)-y^{2}\left(3x^{2}-y^{2}\right)}{r^{4}}}} g x y ( x 2 − y 2 ) = i 1 2 ( Y 4 − 4 − Y 4 4 ) = 3 4 35 π ⋅ x y ( x 2 − y 2 ) r 4 {\displaystyle g_{xy\left(x^{2}-y^{2}\right)}=i{\sqrt {\frac {1}{2}}}\left(Y_{4}^{-4}-Y_{4}^{4}\right)={\frac {3}{4}}{\sqrt {\frac {35}{\pi }}}\cdot {\frac {xy\left(x^{2}-y^{2}\right)}{r^{4}}}}
l = 5 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 }
l = 6 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 }
l = 7 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 }
l = 8 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 }
l = 9 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 }
l = 10 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 }
Note Mathar, R. J. (2009 ). „Zernike basis to cartesian transformations”. Serbian Astronomical Journal . 179 : 107–120. Bibcode:2009SerAj.179..107M. doi:10.2298/SAJ0979107M. 
