球面調和関数表
Y 5 − 5 ( x ) = 3 32 77 π ⋅ e − 5 i φ ⋅ sin 5  θ {\displaystyle Y_{5}^{-5}(x)={\frac {3}{32}}{\sqrt {\frac {77}{\pi }}}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta }
Y 5 − 4 ( x ) = 3 16 385 2 π ⋅ e − 4 i φ ⋅ sin 4  θ ⋅ cos  θ {\displaystyle Y_{5}^{-4}(x)={\frac {3}{16}}{\sqrt {\frac {385}{2\pi }}}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot \cos \theta }
Y 5 − 3 ( x ) = 1 32 385 π ⋅ e − 3 i φ ⋅ sin 3  θ ⋅ ( 9 cos 2  θ − 1 ) {\displaystyle Y_{5}^{-3}(x)={\frac {1}{32}}{\sqrt {\frac {385}{\pi }}}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (9\cos ^{2}\theta -1)}
Y 3 , − 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 Y_{3,-3}=f_{y(3x^{2}-y^{2})}=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}}}}
Y 3 , − 2 = f x y z = i 1 2 ( Y 3 − 2 − Y 3 2 ) = 1 2 105 π ⋅ x y z r 3 {\displaystyle Y_{3,-2}=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}}}}
以下は球面調和関数の表である。ただし、x, y, z と r, θ, φ との関係としては
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}}}
である。
Y 7 2 ( x ) = 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}(x)={\frac {3}{64}}{\sqrt {\frac {35}{\pi }}}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (143\cos ^{5}\theta -110\cos ^{3}\theta +15\cos \theta )}
Y 7 3 ( x ) = − 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}(x)=-{\frac {3}{64}}{\sqrt {\frac {35}{2\pi }}}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (143\cos ^{4}\theta -66\cos ^{2}\theta +3)}
Y 8 6 ( x ) = 1 128 7293 π ⋅ e 6 i φ ⋅ sin 6  θ ⋅ ( 15 cos 2  θ − 1 ) {\displaystyle Y_{8}^{6}(x)={\frac {1}{128}}{\sqrt {\frac {7293}{\pi }}}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot (15\cos ^{2}\theta -1)}
Y 8 7 ( x ) = − 3 64 12155 2 π ⋅ e 7 i φ ⋅ sin 7  θ ⋅ cos  θ {\displaystyle Y_{8}^{7}(x)=-{\frac {3}{64}}{\sqrt {\frac {12155}{2\pi }}}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta \cdot \cos \theta }
Y 8 8 ( x ) = 3 256 12155 2 π ⋅ e 8 i φ ⋅ sin 8  θ {\displaystyle Y_{8}^{8}(x)={\frac {3}{256}}{\sqrt {\frac {12155}{2\pi }}}\cdot e^{8i\varphi }\cdot \sin ^{8}\theta }