球面調和関数表

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。

球面調和関数

l = 0 から l = 5 までは Varshalovich, Moskalev & Khersonskii (1988) を典拠としている。また、l = 0 から l = 3 までの θ 形式での関数は MathWorld でも確認できる。

l = 0

Y 0 0 ( x ) = 1 2 1 π {\displaystyle Y_{0}^{0}(x)={\frac {1}{2}}{\sqrt {\frac {1}{\pi }}}}

l = 1

Y 1 − 1 ( x ) = 1 2 3 2 π ⋅ e − i φ ⋅ sin ⁡ θ = 1 2 3 2 π ⋅ x − i y r {\displaystyle Y_{1}^{-1}(x)={\frac {1}{2}}{\sqrt {\frac {3}{2\pi }}}\cdot e^{-i\varphi }\cdot \sin \theta ={\frac {1}{2}}{\sqrt {\frac {3}{2\pi }}}\cdot {\frac {x-iy}{r}}}
Y 1 0 ( x ) = 1 2 3 π ⋅ cos ⁡ θ = 1 2 3 π ⋅ z r {\displaystyle Y_{1}^{0}(x)={\frac {1}{2}}{\sqrt {\frac {3}{\pi }}}\cdot \cos \theta ={\frac {1}{2}}{\sqrt {\frac {3}{\pi }}}\cdot {\frac {z}{r}}}
Y 1 1 ( x ) = − 1 2 3 2 π ⋅ e i φ ⋅ sin ⁡ θ = − 1 2 3 2 π ⋅ x + i y r {\displaystyle Y_{1}^{1}(x)=-{\frac {1}{2}}{\sqrt {\frac {3}{2\pi }}}\cdot e^{i\varphi }\cdot \sin \theta =-{\frac {1}{2}}{\sqrt {\frac {3}{2\pi }}}\cdot {\frac {x+iy}{r}}}

l = 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 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 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}}}}

l = 3

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 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 3 0 ( x ) = 1 4 7 π ⋅ ( 5 cos 3 ⁡ θ − 3 cos ⁡ θ ) = 1 4 7 π ⋅ − 3 x 2 z − 3 y 2 z + 2 z 3 r 3 {\displaystyle Y_{3}^{0}(x)={\frac {1}{4}}{\sqrt {\frac {7}{\pi }}}\cdot (5\cos ^{3}\theta -3\cos \theta )={\frac {1}{4}}{\sqrt {\frac {7}{\pi }}}\cdot {\frac {-3x^{2}z-3y^{2}z+2z^{3}}{r^{3}}}}
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 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 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}}}}

l = 4

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 }
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 − 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 − 1 ( x ) = 3 8 5 π ⋅ e − i φ ⋅ sin ⁡ θ ⋅ ( 7 cos 3 ⁡ θ − 3 cos ⁡ θ ) {\displaystyle Y_{4}^{-1}(x)={\frac {3}{8}}{\sqrt {\frac {5}{\pi }}}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (7\cos ^{3}\theta -3\cos \theta )}
Y 4 0 ( x ) = 3 16 1 π ⋅ ( 35 cos 4 ⁡ θ − 30 cos 2 ⁡ θ + 3 ) {\displaystyle Y_{4}^{0}(x)={\frac {3}{16}}{\sqrt {\frac {1}{\pi }}}\cdot (35\cos ^{4}\theta -30\cos ^{2}\theta +3)}
Y 4 1 ( x ) = − 3 8 5 π ⋅ e i φ ⋅ sin ⁡ θ ⋅ ( 7 cos 3 ⁡ θ − 3 cos ⁡ θ ) {\displaystyle Y_{4}^{1}(x)=-{\frac {3}{8}}{\sqrt {\frac {5}{\pi }}}\cdot e^{i\varphi }\cdot \sin \theta \cdot (7\cos ^{3}\theta -3\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 }

l = 5

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 5 − 2 ( x ) = 1 8 1155 2 π ⋅ e − 2 i φ ⋅ sin 2 ⁡ θ ⋅ ( 3 cos 3 ⁡ θ − 1 cos ⁡ θ ) {\displaystyle Y_{5}^{-2}(x)={\frac {1}{8}}{\sqrt {\frac {1155}{2\pi }}}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (3\cos ^{3}\theta -1\cos \theta )}
Y 5 − 1 ( x ) = 1 16 165 2 π ⋅ e − i φ ⋅ sin ⁡ θ ⋅ ( 21 cos 4 ⁡ θ − 14 cos 2 ⁡ θ + 1 ) {\displaystyle Y_{5}^{-1}(x)={\frac {1}{16}}{\sqrt {\frac {165}{2\pi }}}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (21\cos ^{4}\theta -14\cos ^{2}\theta +1)}
Y 5 0 ( x ) = 1 16 11 π ⋅ ( 63 cos 5 ⁡ θ − 70 cos 3 ⁡ θ + 15 cos ⁡ θ ) {\displaystyle Y_{5}^{0}(x)={\frac {1}{16}}{\sqrt {\frac {11}{\pi }}}\cdot (63\cos ^{5}\theta -70\cos ^{3}\theta +15\cos \theta )}
Y 5 1 ( x ) = − 1 16 165 2 π ⋅ e i φ ⋅ sin ⁡ θ ⋅ ( 21 cos 4 ⁡ θ − 14 cos 2 ⁡ θ + 1 ) {\displaystyle Y_{5}^{1}(x)=-{\frac {1}{16}}{\sqrt {\frac {165}{2\pi }}}\cdot e^{i\varphi }\cdot \sin \theta \cdot (21\cos ^{4}\theta -14\cos ^{2}\theta +1)}
Y 5 2 ( x ) = 1 8 1155 2 π ⋅ e 2 i φ ⋅ sin 2 ⁡ θ ⋅ ( 3 cos 3 ⁡ θ − 1 cos ⁡ θ ) {\displaystyle Y_{5}^{2}(x)={\frac {1}{8}}{\sqrt {\frac {1155}{2\pi }}}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (3\cos ^{3}\theta -1\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 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 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 }

l = 6

Y 6 − 6 ( x ) = 1 64 3003 π ⋅ e − 6 i φ ⋅ sin 6 ⁡ θ {\displaystyle Y_{6}^{-6}(x)={\frac {1}{64}}{\sqrt {\frac {3003}{\pi }}}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta }
Y 6 − 5 ( x ) = 3 32 1001 π ⋅ e − 5 i φ ⋅ sin 5 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{6}^{-5}(x)={\frac {3}{32}}{\sqrt {\frac {1001}{\pi }}}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot \cos \theta }
Y 6 − 4 ( x ) = 3 32 91 2 π ⋅ e − 4 i φ ⋅ sin 4 ⁡ θ ⋅ ( 11 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{6}^{-4}(x)={\frac {3}{32}}{\sqrt {\frac {91}{2\pi }}}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (11\cos ^{2}\theta -1)}
Y 6 − 3 ( x ) = 1 32 1365 π ⋅ e − 3 i φ ⋅ sin 3 ⁡ θ ⋅ ( 11 cos 3 ⁡ θ − 3 cos ⁡ θ ) {\displaystyle Y_{6}^{-3}(x)={\frac {1}{32}}{\sqrt {\frac {1365}{\pi }}}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (11\cos ^{3}\theta -3\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 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 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 6 3 ( x ) = − 1 32 1365 π ⋅ e 3 i φ ⋅ sin 3 ⁡ θ ⋅ ( 11 cos 3 ⁡ θ − 3 cos ⁡ θ ) {\displaystyle Y_{6}^{3}(x)=-{\frac {1}{32}}{\sqrt {\frac {1365}{\pi }}}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (11\cos ^{3}\theta -3\cos \theta )}
Y 6 4 ( x ) = 3 32 91 2 π ⋅ e 4 i φ ⋅ sin 4 ⁡ θ ⋅ ( 11 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{6}^{4}(x)={\frac {3}{32}}{\sqrt {\frac {91}{2\pi }}}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot (11\cos ^{2}\theta -1)}
Y 6 5 ( x ) = − 3 32 1001 π ⋅ e 5 i φ ⋅ sin 5 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{6}^{5}(x)=-{\frac {3}{32}}{\sqrt {\frac {1001}{\pi }}}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot \cos \theta }
Y 6 6 ( x ) = 1 64 3003 π ⋅ e 6 i φ ⋅ sin 6 ⁡ θ {\displaystyle Y_{6}^{6}(x)={\frac {1}{64}}{\sqrt {\frac {3003}{\pi }}}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta }

l = 7

Y 7 − 7 ( x ) = 3 64 715 2 π ⋅ e − 7 i φ ⋅ sin 7 ⁡ θ {\displaystyle Y_{7}^{-7}(x)={\frac {3}{64}}{\sqrt {\frac {715}{2\pi }}}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta }
Y 7 − 6 ( x ) = 3 64 5005 π ⋅ e − 6 i φ ⋅ sin 6 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{7}^{-6}(x)={\frac {3}{64}}{\sqrt {\frac {5005}{\pi }}}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \cdot \cos \theta }
Y 7 − 5 ( x ) = 3 64 385 2 π ⋅ e − 5 i φ ⋅ sin 5 ⁡ θ ⋅ ( 13 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{7}^{-5}(x)={\frac {3}{64}}{\sqrt {\frac {385}{2\pi }}}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot (13\cos ^{2}\theta -1)}
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 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 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 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 5 ( x ) = − 3 64 385 2 π ⋅ e 5 i φ ⋅ sin 5 ⁡ θ ⋅ ( 13 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{7}^{5}(x)=-{\frac {3}{64}}{\sqrt {\frac {385}{2\pi }}}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot (13\cos ^{2}\theta -1)}
Y 7 6 ( x ) = 3 64 5005 π ⋅ e 6 i φ ⋅ sin 6 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{7}^{6}(x)={\frac {3}{64}}{\sqrt {\frac {5005}{\pi }}}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot \cos \theta }
Y 7 7 ( x ) = − 3 64 715 2 π ⋅ e 7 i φ ⋅ sin 7 ⁡ θ {\displaystyle Y_{7}^{7}(x)=-{\frac {3}{64}}{\sqrt {\frac {715}{2\pi }}}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta }

l = 8

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 }
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 − 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 − 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 )}
Y 8 − 2 ( x ) = 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}(x)={\frac {3}{128}}{\sqrt {\frac {595}{\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 ( x ) = 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}(x)={\frac {3}{64}}{\sqrt {\frac {17}{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 ( x ) = 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}(x)={\frac {1}{256}}{\sqrt {\frac {17}{\pi }}}\cdot (6435\cos ^{8}\theta -12012\cos ^{6}\theta +6930\cos ^{4}\theta -1260\cos ^{2}\theta +35)}
Y 8 1 ( x ) = − 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}(x)=-{\frac {3}{64}}{\sqrt {\frac {17}{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 ( x ) = 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}(x)={\frac {3}{128}}{\sqrt {\frac {595}{\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 ( 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 )}
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 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 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 }

l = 9

Y 9 − 9 ( x ) = 1 512 230945 π ⋅ e − 9 i φ ⋅ sin 9 ⁡ θ {\displaystyle Y_{9}^{-9}(x)={\frac {1}{512}}{\sqrt {\frac {230945}{\pi }}}\cdot e^{-9i\varphi }\cdot \sin ^{9}\theta }
Y 9 − 8 ( x ) = 3 256 230945 2 π ⋅ e − 8 i φ ⋅ sin 8 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{9}^{-8}(x)={\frac {3}{256}}{\sqrt {\frac {230945}{2\pi }}}\cdot e^{-8i\varphi }\cdot \sin ^{8}\theta \cdot \cos \theta }
Y 9 − 7 ( x ) = 3 512 13585 π ⋅ e − 7 i φ ⋅ sin 7 ⁡ θ ⋅ ( 17 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{9}^{-7}(x)={\frac {3}{512}}{\sqrt {\frac {13585}{\pi }}}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta \cdot (17\cos ^{2}\theta -1)}
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 9 − 3 ( x ) = 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}(x)={\frac {1}{256}}{\sqrt {\frac {21945}{\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 ( x ) = 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}(x)={\frac {3}{128}}{\sqrt {\frac {1045}{\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 ( x ) = 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}(x)={\frac {3}{256}}{\sqrt {\frac {95}{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 ( x ) = 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}(x)={\frac {1}{256}}{\sqrt {\frac {19}{\pi }}}\cdot (12155\cos ^{9}\theta -25740\cos ^{7}\theta +18018\cos ^{5}\theta -4620\cos ^{3}\theta +315\cos \theta )}
Y 9 1 ( x ) = − 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}(x)=-{\frac {3}{256}}{\sqrt {\frac {95}{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 ( x ) = 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}(x)={\frac {3}{128}}{\sqrt {\frac {1045}{\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 ( x ) = − 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}(x)=-{\frac {1}{256}}{\sqrt {\frac {21945}{\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 ( 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 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 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 7 ( x ) = − 3 512 13585 π ⋅ e 7 i φ ⋅ sin 7 ⁡ θ ⋅ ( 17 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{9}^{7}(x)=-{\frac {3}{512}}{\sqrt {\frac {13585}{\pi }}}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta \cdot (17\cos ^{2}\theta -1)}
Y 9 8 ( x ) = 3 256 230945 2 π ⋅ e 8 i φ ⋅ sin 8 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{9}^{8}(x)={\frac {3}{256}}{\sqrt {\frac {230945}{2\pi }}}\cdot e^{8i\varphi }\cdot \sin ^{8}\theta \cdot \cos \theta }
Y 9 9 ( x ) = − 1 512 230945 π ⋅ e 9 i φ ⋅ sin 9 ⁡ θ {\displaystyle Y_{9}^{9}(x)=-{\frac {1}{512}}{\sqrt {\frac {230945}{\pi }}}\cdot e^{9i\varphi }\cdot \sin ^{9}\theta }

l = 10

Y 10 − 10 ( x ) = 1 1024 969969 π ⋅ e − 10 i φ ⋅ sin 10 ⁡ θ {\displaystyle Y_{10}^{-10}(x)={\frac {1}{1024}}{\sqrt {\frac {969969}{\pi }}}\cdot e^{-10i\varphi }\cdot \sin ^{10}\theta }
Y 10 − 9 ( x ) = 1 512 4849845 π ⋅ e − 9 i φ ⋅ sin 9 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{10}^{-9}(x)={\frac {1}{512}}{\sqrt {\frac {4849845}{\pi }}}\cdot e^{-9i\varphi }\cdot \sin ^{9}\theta \cdot \cos \theta }
Y 10 − 8 ( x ) = 1 512 255255 2 π ⋅ e − 8 i φ ⋅ sin 8 ⁡ θ ⋅ ( 19 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{10}^{-8}(x)={\frac {1}{512}}{\sqrt {\frac {255255}{2\pi }}}\cdot e^{-8i\varphi }\cdot \sin ^{8}\theta \cdot (19\cos ^{2}\theta -1)}
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 10 − 4 ( x ) = 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}(x)={\frac {3}{256}}{\sqrt {\frac {5005}{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 ( x ) = 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}(x)={\frac {3}{256}}{\sqrt {\frac {5005}{\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 ( 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 10 0 ( x ) = 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}(x)={\frac {1}{512}}{\sqrt {\frac {21}{\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 ( 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 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 3 ( x ) = − 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}(x)=-{\frac {3}{256}}{\sqrt {\frac {5005}{\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 ( x ) = 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}(x)={\frac {3}{256}}{\sqrt {\frac {5005}{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 ( 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 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 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 8 ( x ) = 1 512 255255 2 π ⋅ e 8 i φ ⋅ sin 8 ⁡ θ ⋅ ( 19 cos 2 ⁡ θ − 1 ) {\displaystyle Y_{10}^{8}(x)={\frac {1}{512}}{\sqrt {\frac {255255}{2\pi }}}\cdot e^{8i\varphi }\cdot \sin ^{8}\theta \cdot (19\cos ^{2}\theta -1)}
Y 10 9 ( x ) = − 1 512 4849845 π ⋅ e 9 i φ ⋅ sin 9 ⁡ θ ⋅ cos ⁡ θ {\displaystyle Y_{10}^{9}(x)=-{\frac {1}{512}}{\sqrt {\frac {4849845}{\pi }}}\cdot e^{9i\varphi }\cdot \sin ^{9}\theta \cdot \cos \theta }
Y 10 10 ( x ) = 1 1024 969969 π ⋅ e 10 i φ ⋅ sin 10 ⁡ θ {\displaystyle Y_{10}^{10}(x)={\frac {1}{1024}}{\sqrt {\frac {969969}{\pi }}}\cdot e^{10i\varphi }\cdot \sin ^{10}\theta }

線型結合された球面調和関数

線型結合により導出される実際の電子軌道の球面調和関数。l = 0 から l = 2 までは Chisholm (1976) 及び Blanco, Flórez & Bermejo (1996) を、l = 3 は Chisholm (1976) のみを典拠としている。

l = 0

Y 00 = s = Y 0 0 = 1 2 1 π {\displaystyle Y_{00}=s=Y_{0}^{0}={\frac {1}{2}}{\sqrt {\frac {1}{\pi }}}}

l = 1

Y 1 , − 1 = p y = i 1 2 ( Y 1 − 1 + Y 1 1 ) = 3 4 π ⋅ y r {\displaystyle Y_{1,-1}=p_{y}=i{\sqrt {\frac {1}{2}}}\left(Y_{1}^{-1}+Y_{1}^{1}\right)={\sqrt {\frac {3}{4\pi }}}\cdot {\frac {y}{r}}}
Y 10 = p z = Y 1 0 = 3 4 π ⋅ z r {\displaystyle Y_{10}=p_{z}=Y_{1}^{0}={\sqrt {\frac {3}{4\pi }}}\cdot {\frac {z}{r}}}
Y 11 = p x = 1 2 ( Y 1 − 1 − Y 1 1 ) = 3 4 π ⋅ x r {\displaystyle Y_{11}=p_{x}={\sqrt {\frac {1}{2}}}\left(Y_{1}^{-1}-Y_{1}^{1}\right)={\sqrt {\frac {3}{4\pi }}}\cdot {\frac {x}{r}}}

l = 2

Y 2 , − 2 = d x y = i 1 2 ( Y 2 − 2 − Y 2 2 ) = 1 2 15 π ⋅ x y r 2 {\displaystyle Y_{2,-2}=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}}}}
Y 2 , − 1 = d y z = i 1 2 ( Y 2 − 1 + Y 2 1 ) = 1 2 15 π ⋅ y z r 2 {\displaystyle Y_{2,-1}=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}}}}
Y 20 = d z 2 = Y 2 0 = 1 4 5 π ⋅ − x 2 − y 2 + 2 z 2 r 2 {\displaystyle Y_{20}=d_{z^{2}}=Y_{2}^{0}={\frac {1}{4}}{\sqrt {\frac {5}{\pi }}}\cdot {\frac {-x^{2}-y^{2}+2z^{2}}{r^{2}}}}
Y 21 = d x z = 1 2 ( Y 2 − 1 − Y 2 1 ) = 1 2 15 π ⋅ z x r 2 {\displaystyle Y_{21}=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}}}}
Y 22 = d x 2 − y 2 = 1 2 ( Y 2 − 2 + Y 2 2 ) = 1 4 15 π ⋅ x 2 − y 2 r 2 {\displaystyle Y_{22}=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 = 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}}}}
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 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}}}}

l = 4

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}}}}
Y 4 , − 2 = 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 Y_{4,-2}=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}}}}
Y 4 , − 1 = 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 Y_{4,-1}=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}}}}
Y 40 = g z 4 = Y 4 0 = 3 16 1 π ⋅ ( 35 z 4 − 30 z 2 r 2 + 3 r 4 ) r 4 {\displaystyle Y_{40}=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}}}}
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 43 = 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 Y_{43}=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}}}}
Y 44 = 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 Y_{44}=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}}}}

参考文献

原論文

  • Blanco, Miguel A.; Flórez, M.; Bermejo, M. (1 November 1996). “Evaluation of the rotation matrices in the basis of real spherical harmonics” (PDF). Journal of Molecular Structure: THEOCHEM (Amsterdam: Elsevier ScienceDirect) 419 (1–3): 19–27. doi:10.1016/S0166-1280(97)00185-1. ISSN 0022-2860. OCLC 224506237. http://ac.els-cdn.com/S0166128097001851/1-s2.0-S0166128097001851-main.pdf?_tid=9b5146ea-a25c-11e6-9353-00000aab0f02&acdnat=1478243110_840f06f35087d21e1c1809eb2de68a1e. 

書籍

  • Chisholm, C.D.H. (March 8, 1976). Group theoretical techniques in quantum chemistry. Theoretical chemistry: a series of monographs. 5 (1st ed.). New York: Academic Press. ASIN 0121729508. ISBN 0-12-172950-8. NCID BA03187896. LCCN 75-27326. OCLC 3104116 
  • Varshalovich, D. A.; Moskalev, A. N.; Khersonskii, V. K. (October 1, 1988). Quantum theory of angular momentum : irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols (1. repr. ed.). Singapore: World Scientific Pub.. pp. 155–156. ASIN 9971501074. ISBN 9971-50-107-4. NCID BA04808445. LCCN 86-9279. OCLC 13525826 

関連項目

外部リンク

  • Weisstein, Eric W. "Spherical Harmonic". mathworld.wolfram.com (英語).