球面調和関数表
Y 3 , − 1 = 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 Y_{3,-1}=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}}}}
Y 30 = f z 3 = Y 3 0 = 1 4 7 π ⋅ z ( 2 z 2 − 3 x 2 − 3 y 2 ) r 3 {\displaystyle Y_{30}=f_{z^{3}}=Y_{3}^{0}={\frac {1}{4}}{\sqrt {\frac {7}{\pi }}}\cdot {\frac {z(2z^{2}-3x^{2}-3y^{2})}{r^{3}}}}
Y 31 = 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 Y_{31}=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}}}}
Y 7 − 4 ( x ) = 3 32 385 2 π ⋅ e − 4 i φ ⋅ sin 4  θ ⋅ ( 13 cos 3  θ − 3 cos  θ ) {\displaystyle Y_{7}^{-4}(x)={\frac {3}{32}}{\sqrt {\frac {385}{2\pi }}}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (13\cos ^{3}\theta -3\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 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 4 2 ( x ) = 3 8 5 2 π ⋅ e 2 i φ ⋅ sin 2  θ ⋅ ( 7 cos 2  θ − 1 ) {\displaystyle Y_{4}^{2}(x)={\frac {3}{8}}{\sqrt {\frac {5}{2\pi }}}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (7\cos ^{2}\theta -1)}
Y 4 3 ( x ) = − 3 8 35 π ⋅ e 3 i φ ⋅ sin 3  θ ⋅ cos  θ {\displaystyle Y_{4}^{3}(x)=-{\frac {3}{8}}{\sqrt {\frac {35}{\pi }}}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot \cos \theta }
Y 4 4 ( x ) = 3 16 35 2 π ⋅ e 4 i φ ⋅ sin 4  θ {\displaystyle Y_{4}^{4}(x)={\frac {3}{16}}{\sqrt {\frac {35}{2\pi }}}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta }