球面調和関数表
Y 6 0 ( x ) = 1 32 13 π ⋅ ( 231 cos 6  θ − 315 cos 4  θ + 105 cos 2  θ − 5 ) {\displaystyle Y_{6}^{0}(x)={\frac {1}{32}}{\sqrt {\frac {13}{\pi }}}\cdot (231\cos ^{6}\theta -315\cos ^{4}\theta +105\cos ^{2}\theta -5)}
Y 6 1 ( x ) = − 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}(x)=-{\frac {1}{16}}{\sqrt {\frac {273}{2\pi }}}\cdot e^{i\varphi }\cdot \sin \theta \cdot (33\cos ^{5}\theta -30\cos ^{3}\theta +5\cos \theta )}
Y 6 2 ( x ) = 1 64 1365 π ⋅ e 2 i φ ⋅ sin 2  θ ⋅ ( 33 cos 4  θ − 18 cos 2  θ + 1 ) {\displaystyle Y_{6}^{2}(x)={\frac {1}{64}}{\sqrt {\frac {1365}{\pi }}}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (33\cos ^{4}\theta -18\cos ^{2}\theta +1)}
Y 32 = 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 Y_{32}=f_{z(x^{2}-y^{2})}={\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}}}}
Y 33 = 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 Y_{33}=f_{x(x^{2}-3y^{2})}={\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}}}}
Y 9 − 6 ( x ) = 1 128 40755 π ⋅ e − 6 i φ ⋅ sin 6  θ ⋅ ( 17 cos 3  θ − 3 cos  θ ) {\displaystyle Y_{9}^{-6}(x)={\frac {1}{128}}{\sqrt {\frac {40755}{\pi }}}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \cdot (17\cos ^{3}\theta -3\cos \theta )}
Y 9 − 5 ( x ) = 3 256 2717 π ⋅ e − 5 i φ ⋅ sin 5  θ ⋅ ( 85 cos 4  θ − 30 cos 2  θ + 1 ) {\displaystyle Y_{9}^{-5}(x)={\frac {3}{256}}{\sqrt {\frac {2717}{\pi }}}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot (85\cos ^{4}\theta -30\cos ^{2}\theta +1)}
Y 9 − 4 ( x ) = 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}(x)={\frac {3}{128}}{\sqrt {\frac {95095}{2\pi }}}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (17\cos ^{5}\theta -10\cos ^{3}\theta +1\cos \theta )}
Y 41 = 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 Y_{41}=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}}}}
Y 42 = g z 2 x y = 1 2 ( Y 4 − 2 + Y 4 2 ) = 3 8 5 π ⋅ ( x 2 − y 2 ) ⋅ ( 7 z 2 − r 2 ) r 4 {\displaystyle Y_{42}=g_{z^{2}xy}={\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}}}}
Y 3 − 3 ( x ) = 1 8 35 π ⋅ e − 3 i φ ⋅ sin 3  θ = 1 8 35 π ⋅ x 3 − 3 i x 2 y − 3 x y 2 + i y 3 r 3 {\displaystyle Y_{3}^{-3}(x)={\frac {1}{8}}{\sqrt {\frac {35}{\pi }}}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta ={\frac {1}{8}}{\sqrt {\frac {35}{\pi }}}\cdot {\frac {x^{3}-3ix^{2}y-3xy^{2}+iy^{3}}{r^{3}}}}
Y 3 − 2 ( x ) = 1 4 105 2 π ⋅ e − 2 i φ ⋅ sin 2  θ ⋅ cos  θ = 1 4 105 2 π ⋅ x 2 z − 2 i x y z − y 2 z r 3 {\displaystyle Y_{3}^{-2}(x)={\frac {1}{4}}{\sqrt {\frac {105}{2\pi }}}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot \cos \theta ={\frac {1}{4}}{\sqrt {\frac {105}{2\pi }}}\cdot {\frac {x^{2}z-2ixyz-y^{2}z}{r^{3}}}}
Y 4 , − 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 Y_{4,-4}=g_{xy(x^{2}-y^{2})}=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}}}}
Y 4 , − 3 = 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 Y_{4,-3}=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}}}}