球面調和関数表
Y 3 − 1 ( x ) = 1 8 21 π ⋅ e − i φ ⋅ sin  θ ⋅ ( 5 cos 2  θ − 1 ) = 1 8 21 π ⋅ − x 3 + i x 2 y − x y 2 + 4 x z 2 + i y 3 − 4 i y z 2 r 3 {\displaystyle Y_{3}^{-1}(x)={\frac {1}{8}}{\sqrt {\frac {21}{\pi }}}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (5\cos ^{2}\theta -1)={\frac {1}{8}}{\sqrt {\frac {21}{\pi }}}\cdot {\frac {-x^{3}+ix^{2}y-xy^{2}+4xz^{2}+iy^{3}-4iyz^{2}}{r^{3}}}}
Y 10 − 7 ( x ) = 3 512 85085 π ⋅ e − 7 i φ ⋅ sin 7  θ ⋅ ( 19 cos 3  θ − 3 cos  θ ) {\displaystyle Y_{10}^{-7}(x)={\frac {3}{512}}{\sqrt {\frac {85085}{\pi }}}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta \cdot (19\cos ^{3}\theta -3\cos \theta )}
Y 10 − 6 ( x ) = 3 1024 5005 π ⋅ e − 6 i φ ⋅ sin 6  θ ⋅ ( 323 cos 4  θ − 102 cos 2  θ + 3 ) {\displaystyle Y_{10}^{-6}(x)={\frac {3}{1024}}{\sqrt {\frac {5005}{\pi }}}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \cdot (323\cos ^{4}\theta -102\cos ^{2}\theta +3)}
Y 10 − 5 ( x ) = 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}(x)={\frac {3}{256}}{\sqrt {\frac {1001}{\pi }}}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot (323\cos ^{5}\theta -170\cos ^{3}\theta +15\cos \theta )}
Y 7 − 1 ( x ) = 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}(x)={\frac {1}{64}}{\sqrt {\frac {105}{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 ( x ) = 1 32 15 π ⋅ ( 429 cos 7  θ − 693 cos 5  θ + 315 cos 3  θ − 35 cos  θ ) {\displaystyle Y_{7}^{0}(x)={\frac {1}{32}}{\sqrt {\frac {15}{\pi }}}\cdot (429\cos ^{7}\theta -693\cos ^{5}\theta +315\cos ^{3}\theta -35\cos \theta )}
Y 7 1 ( x ) = − 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}(x)=-{\frac {1}{64}}{\sqrt {\frac {105}{2\pi }}}\cdot e^{i\varphi }\cdot \sin \theta \cdot (429\cos ^{6}\theta -495\cos ^{4}\theta +135\cos ^{2}\theta -5)}
Y 10 − 2 ( x ) = 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}(x)={\frac {3}{512}}{\sqrt {\frac {385}{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 ( x ) = 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}(x)={\frac {1}{256}}{\sqrt {\frac {1155}{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 2 0 ( x ) = 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}(x)={\frac {1}{4}}{\sqrt {\frac {5}{\pi }}}\cdot (3\cos ^{2}\theta -1)={\frac {1}{4}}{\sqrt {\frac {5}{\pi }}}\cdot {\frac {-x^{2}-y^{2}+2z^{2}}{r^{2}}}}
Y 2 1 ( x ) = − 1 2 15 2 π ⋅ e i φ ⋅ sin  θ ⋅ cos  θ = − 1 2 15 2 π ⋅ x z + i y z r 2 {\displaystyle Y_{2}^{1}(x)=-{\frac {1}{2}}{\sqrt {\frac {15}{2\pi }}}\cdot e^{i\varphi }\cdot \sin \theta \cdot \cos \theta =-{\frac {1}{2}}{\sqrt {\frac {15}{2\pi }}}\cdot {\frac {xz+iyz}{r^{2}}}}
Y 2 2 ( x ) = 1 4 15 2 π ⋅ e 2 i φ ⋅ sin 2  θ = 1 4 15 2 π ⋅ x 2 + 2 i x y − y 2 r 2 {\displaystyle Y_{2}^{2}(x)={\frac {1}{4}}{\sqrt {\frac {15}{2\pi }}}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta ={\frac {1}{4}}{\sqrt {\frac {15}{2\pi }}}\cdot {\frac {x^{2}+2ixy-y^{2}}{r^{2}}}}
Y 8 − 5 ( x ) = 3 64 17017 2 π ⋅ e − 5 i φ ⋅ sin 5  θ ⋅ ( 5 cos 3  θ − 1 cos  θ ) {\displaystyle Y_{8}^{-5}(x)={\frac {3}{64}}{\sqrt {\frac {17017}{2\pi }}}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot (5\cos ^{3}\theta -1\cos \theta )}
Y 8 − 4 ( x ) = 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}(x)={\frac {3}{128}}{\sqrt {\frac {1309}{2\pi }}}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (65\cos ^{4}\theta -26\cos ^{2}\theta +1)}
Y 8 − 3 ( x ) = 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}(x)={\frac {1}{64}}{\sqrt {\frac {19635}{2\pi }}}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (39\cos ^{5}\theta -26\cos ^{3}\theta +3\cos \theta )}