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
![{\displaystyle {\begin{aligned}x&=r\sin \theta \cos \varphi \\y&=r\sin \theta \sin \varphi \\z&=r\cos \theta \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b647866393c668647b969ff35e551e8f92c03a16)
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
![{\displaystyle Y_{0}^{0}(x)={\frac {1}{2}}{\sqrt {\frac {1}{\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79fc6c99cc4d03a8081c6524b8e880c982163b15)
l = 1
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6d03e434b1ba1bb89f1b1d9a7df759ed1c01aa0)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/687be6186af31003abe0f9725d786c23c6aacfda)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/170b410835365e57d714850ac1cbc1472d1049d5)
l = 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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae74e58c4c3f7d4c0dad8efe6ba20dc9086f23e3)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff53724e9bb1d7f574ff4b1058bb01dafa31b064)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76c9d5b0342591015d87478549dcaeee9eb01246)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/333d8eecfcae58448d69d55362b3342f9442d839)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7cb914a527e8408035ce3eb7224782f97cc0d74)
l = 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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/080ee4f770f07058c967dd25f35cffe65ab108c6)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6a8d20823a3642a5a402d58bab13e973ccb838a)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08d587462da5a66bc600e523655dfdb1b6911f6b)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00e5347cd25b6bdef024531e97df6067e568df5c)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47b137c5d03f075a967d8eab3519491890d116f7)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6317125e0bcca92dfb7b1680ec445a1db1d4448)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52f34c3aca71b95d69e164dad21025da4ba9bacb)
l = 4
![{\displaystyle Y_{4}^{-4}(x)={\frac {3}{16}}{\sqrt {\frac {35}{2\pi }}}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/522f3a2226acb31803266efe41fbe106449927b5)
![{\displaystyle Y_{4}^{-3}(x)={\frac {3}{8}}{\sqrt {\frac {35}{\pi }}}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot \cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/95995586faecd82a8567d6bef3c31d55f6ba4f4c)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b84a2cbeda676db51b5b45d4a9788cd868ecf0b)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f9c62d498234b9ee015d45ec0f946fc66cc51a1)
![{\displaystyle Y_{4}^{0}(x)={\frac {3}{16}}{\sqrt {\frac {1}{\pi }}}\cdot (35\cos ^{4}\theta -30\cos ^{2}\theta +3)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb1328256558784589072ba124f9e9f32845439)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79c1f66166268cff70834e9d22ea7b06e593e93a)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad21ef2665d4382d42bd3f1018afe3ec74a0d4e6)
![{\displaystyle Y_{4}^{3}(x)=-{\frac {3}{8}}{\sqrt {\frac {35}{\pi }}}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot \cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/60ba9de5450016aef1d470196029f099b878c98f)
![{\displaystyle Y_{4}^{4}(x)={\frac {3}{16}}{\sqrt {\frac {35}{2\pi }}}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/49d27cb3d83c18a82c4833fe88e9ae111d86846e)
l = 5
![{\displaystyle Y_{5}^{-5}(x)={\frac {3}{32}}{\sqrt {\frac {77}{\pi }}}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a205a52cf889a337398528b001d775a9cef0d112)
![{\displaystyle Y_{5}^{-4}(x)={\frac {3}{16}}{\sqrt {\frac {385}{2\pi }}}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot \cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ea2676fb085d7ddd8a569073dad37f50adcd801)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae3d807c02fde9a4a9685326c9239730ec5ede4)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b2f2d8574c2b3c9e0c1d12008f74140cbfdb9ec)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a4e2718727bed98a3a5614a905c6dc3ac64882b)
![{\displaystyle Y_{5}^{0}(x)={\frac {1}{16}}{\sqrt {\frac {11}{\pi }}}\cdot (63\cos ^{5}\theta -70\cos ^{3}\theta +15\cos \theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/293f1377229df05206c67d73ad83e7a6489866b4)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c19c3fd1c2363bbbe87066a3d5dcb4811adc53d8)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5cef9da0fbcaca72694994a0cff40b078de1a40)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9142ae768aba2b6741c34b0cff44b467739fbb1)
![{\displaystyle Y_{5}^{4}(x)={\frac {3}{16}}{\sqrt {\frac {385}{2\pi }}}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot \cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca139d7f4f1f6bfece5fa4d33159bdcb2f103abd)
![{\displaystyle Y_{5}^{5}(x)=-{\frac {3}{32}}{\sqrt {\frac {77}{\pi }}}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e53977364de9f88dc2d8a25ece798b55ede7ed4)
l = 6
![{\displaystyle Y_{6}^{-6}(x)={\frac {1}{64}}{\sqrt {\frac {3003}{\pi }}}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a7995e48ec8e097090d03c1f32ba33159f148b5)
![{\displaystyle Y_{6}^{-5}(x)={\frac {3}{32}}{\sqrt {\frac {1001}{\pi }}}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot \cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3003e7db60eed8cffbf05891cd0a9832a04184b)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6007f904a365a92c696ffd326623bdba5725b23)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/548725e5d8492cd4de5844d539f9a426c6ab83f9)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da761d2e354ca08b2d3ad684a687337a555ca81e)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4480784434aa1be48105bfcc7bd9ab7c688bd79)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8f100f36498a6174322afcda92b5eee287e20a2)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/267cd44086a2caff438971000fa226a0a56b01b2)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc3e2c91828bfc4878e2bf953c27816847574cb8)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b70ee1d4de2fdec35159f3baeb1f0d2d045c606a)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e6b7c913aca987afc68ec2120f333df845d91b)
![{\displaystyle Y_{6}^{5}(x)=-{\frac {3}{32}}{\sqrt {\frac {1001}{\pi }}}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot \cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/26c23ba8d6dfcfe5a942f5cd7bdc168cd53f2a57)
![{\displaystyle Y_{6}^{6}(x)={\frac {1}{64}}{\sqrt {\frac {3003}{\pi }}}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/974be9f5e3381c640b35e7c6831442f8fc7a4871)
l = 7
![{\displaystyle Y_{7}^{-7}(x)={\frac {3}{64}}{\sqrt {\frac {715}{2\pi }}}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/470a9442efea91b1de06d59d9f11bafab5c98ea7)
![{\displaystyle Y_{7}^{-6}(x)={\frac {3}{64}}{\sqrt {\frac {5005}{\pi }}}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \cdot \cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/88347d006ce50147aadb11453e0ad161d5283107)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be1adb9d14c24b529f1dfc10f316d12d66fe660b)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6112cb10cab0388afb645700cdf9029624fe96e1)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6270099401c52c79e590b5935e50e4a5556eccf)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f51a35cab5689799ad603926202a3fbdf237640a)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4dcdbaec9fc2411009549229b71fbd8488cf5ff)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5a0cd92dc463a766fed1b2a439359b335243271)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4680d6c7ca96596f02970bd2f891873af1960c71)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcbc33defa2853dc22e76be7ff5fee03f09ccfc0)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7dc5d704b0dc59067ebf98a547927d2e0689a4b1)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70b744d0dbfd9a32a149b0474403c32517008f4a)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d6a07671d65ab10234d3d7809fbdd6c0bb4599)
![{\displaystyle Y_{7}^{6}(x)={\frac {3}{64}}{\sqrt {\frac {5005}{\pi }}}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot \cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d090fac042c536b29327923b82d3a2ca7c75be87)
![{\displaystyle Y_{7}^{7}(x)=-{\frac {3}{64}}{\sqrt {\frac {715}{2\pi }}}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/625a2e431c5807f08d7821b738e4bc66daea5023)
l = 8
![{\displaystyle Y_{8}^{-8}(x)={\frac {3}{256}}{\sqrt {\frac {12155}{2\pi }}}\cdot e^{-8i\varphi }\cdot \sin ^{8}\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/307f569433502a75ae3d9ff0eda8a3767bb03394)
![{\displaystyle Y_{8}^{-7}(x)={\frac {3}{64}}{\sqrt {\frac {12155}{2\pi }}}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta \cdot \cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c51b8aacc0422147056f2f407072f727e228eb1f)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fc1fe843d2914a9849c9c1c37ce08519c30e425)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07e759a05a0ee4597a273210bf9686d00a2fcee3)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fc1dbf02aa8ccd953fc56d5d31a32fcca01153c)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eff211d433300343f60a1ec698f47e4028ddce86)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05a14a1fd3a2cbcd527ac93ccc50ebbc05ee06e1)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ac584842ad3ee14f10e5e2e36f98b2d7314d271)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c3cd9006223c3020a10a9cb0e56e8bb16723669)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9155bc820d4ad5160090dd30bad940e6d99cd6e6)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2e56114d340ecfa1240463a0ec1e4e6d9cd8775)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e033a87711238b6646ea50e363cc3887562ac36e)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b81f41a83b01077de5582646b3b5648f8f18f49)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a37c2469b54a4ca0ba1839584ffdbd6b04bfdff)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01dea065f605064bc1869cfa4c11292bdcadc362)
![{\displaystyle Y_{8}^{7}(x)=-{\frac {3}{64}}{\sqrt {\frac {12155}{2\pi }}}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta \cdot \cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/09dc88b410fafe497e1d0319fc2f836bda441e36)
![{\displaystyle Y_{8}^{8}(x)={\frac {3}{256}}{\sqrt {\frac {12155}{2\pi }}}\cdot e^{8i\varphi }\cdot \sin ^{8}\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c07e0b8b80fdbdafee27e5c42a449dff3d8da0d3)
l = 9
![{\displaystyle Y_{9}^{-9}(x)={\frac {1}{512}}{\sqrt {\frac {230945}{\pi }}}\cdot e^{-9i\varphi }\cdot \sin ^{9}\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b06efd6e0703e3f9eae0f2f41968be9d40767cd)
![{\displaystyle Y_{9}^{-8}(x)={\frac {3}{256}}{\sqrt {\frac {230945}{2\pi }}}\cdot e^{-8i\varphi }\cdot \sin ^{8}\theta \cdot \cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d94616fa98207799edaa84f53e60c7abac7445c)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d116ade2936917984c6d9d0bd9c71d1c72fc5094)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc2673e9869e32e9689896d43b6b8c5de1ae50e8)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8c0e2828ae5954ab040d391d6a70f337ed453d4)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c53015d6ee44495ce03e625cbc4f5faa9df3eba5)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d20b619d74ef7e860201c1892e3ff975a01d202)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8091bb08b97e93cb138c84e07c2e50dcbe266a6)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fe7278b116784d782e5a5e3a0a5f2e8e9734b15)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f6988ec3dcbe02c0b1cc6ca3e0b260d4ed767c3)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb257255ccb312ac87b8a285abc69c0eb0c7d68b)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4d3c246a7ef7984b529b10cac8de75220e918f0)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/125537114c9ba4530c36cbcf5510cd9beea77dd4)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2537be80999283d4b15b0dfee78861b9a79d6da6)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1f42617dad56415296adc9fa8372def71e57e16)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25fa0d347527ed24d2dee3aa79d33f918ea919a8)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08d625461e25593509b364bbfab071e466d0f1ac)
![{\displaystyle Y_{9}^{8}(x)={\frac {3}{256}}{\sqrt {\frac {230945}{2\pi }}}\cdot e^{8i\varphi }\cdot \sin ^{8}\theta \cdot \cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b7a7594bf7e8d3098377aa158042ef1cde10ca2)
![{\displaystyle Y_{9}^{9}(x)=-{\frac {1}{512}}{\sqrt {\frac {230945}{\pi }}}\cdot e^{9i\varphi }\cdot \sin ^{9}\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e293b3208ab55de1b86df42ec5971d025fd716ce)
l = 10
![{\displaystyle Y_{10}^{-10}(x)={\frac {1}{1024}}{\sqrt {\frac {969969}{\pi }}}\cdot e^{-10i\varphi }\cdot \sin ^{10}\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/de7aba6b990acc6a5fc15c7464a6d4d04c45300b)
![{\displaystyle Y_{10}^{-9}(x)={\frac {1}{512}}{\sqrt {\frac {4849845}{\pi }}}\cdot e^{-9i\varphi }\cdot \sin ^{9}\theta \cdot \cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/07468554b093b943da8e35a964a986601d75140b)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/593e11bbaa9302fa99f5322ad5ae65fb4bc73ada)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b82d40e77b2ab29dc99d4fb5175bb0bcdaf2136)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6f66ab0e0b859af17e9bc00a8193aa902bf504f)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06d802fd4dadb6ea7e7fcd6b7f7ee92e35252497)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a26e338ae00f465c896b2e056ef338931d2e513)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42d6ff3a21a035d713139ab26f879ce012625fae)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30379772909540d780dada1f1cd26717f4c108dd)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e367567d1445102f241502f1265eb0a35d7c44b)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a67bd154203457113a757d12d05a483f8a0c85c1)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ee31df6888e4ee91b50ebf51a22453781467ea2)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f0abed22c7a24d625d2714ee531f1256636bd89)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c361a45c72d5b1c2ce02de0a163f2af55aa96f0)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aaecb0d8cfa1e4595bdc274f37363c95f6e1b875)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aacc93df19a15800f48a2f86e6afe7f02cc6fbc6)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bea23bd834c7e330838cac1ef9775675069df2b)
![{\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 )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e091c5f92d1d633462220dcd13dad9fac6057858)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a750a8c401222dd34a7ac2fded2c678b83cb166c)
![{\displaystyle Y_{10}^{9}(x)=-{\frac {1}{512}}{\sqrt {\frac {4849845}{\pi }}}\cdot e^{9i\varphi }\cdot \sin ^{9}\theta \cdot \cos \theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d780730368651954b72946db18e2c444542d7c96)
![{\displaystyle Y_{10}^{10}(x)={\frac {1}{1024}}{\sqrt {\frac {969969}{\pi }}}\cdot e^{10i\varphi }\cdot \sin ^{10}\theta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1407299d6fff16323ba718436dede00733c6e0e)
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
![{\displaystyle Y_{00}=s=Y_{0}^{0}={\frac {1}{2}}{\sqrt {\frac {1}{\pi }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/887f4ab1191d9761b407624baef3975ceb8c0472)
l = 1
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5fe5d442cafd24af4bbc503397d1a3591f7305e)
![{\displaystyle Y_{10}=p_{z}=Y_{1}^{0}={\sqrt {\frac {3}{4\pi }}}\cdot {\frac {z}{r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0bc99fe181acbcc5f4294ced9f5e5b1331b5e0)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70f5868ee2965195fb49b5bc701a9ab4f55309c6)
l = 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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8fcdf1b4c7726ef19f00b1cf8c7a04b7b7297b3)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/358a54e5fc693cdf7f606159579f9d72b77136b7)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/512d0839e17b4bf6c7511cc66c2d6ebe55a78a38)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6bc55e8f336395cded1ee63a25ca32133303522)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a99525c5794b78ecbd72281fe1e6da2443455820)
l = 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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ba65f9bbdc6c11c07b1ce1e4a18de8f6b99d917)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2138b4da59c7af08f2c156ab0d410931857bea51)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7715a62c9f2e87c3cc38fb374bd125a3f679e787)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7d8689d36ee87a970b0ccbdaee31bd1696d31d7)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/141ccb6499a8c25a109e247cde06fb072dcdf682)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/820f114c1b3815f701618416fb8baa1cc8109985)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f038d694ab18b106d17e66ed600daa8c9e743f10)
l = 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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79efecf6f06959fb41c33efb8a1759831f72ffe6)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1858b4c85a890d092eb102bbcaf1a21a9bde4c20)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8447d9e68bb1570f9654a24b4bbe759ed5a9072)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fd77f699d47a12fe684cb0d47f6654270e722fc)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01cd6a8c5eda645dca8d62bc46a7190faf8ea632)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f16aed2835238b6135a4c18682a38ad96a1efd9)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94fc3ed4b24f528b22102fc490e84e0d00f796f2)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69d7feb381524d83fe19426b9f5073befdf9416b)
![{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d47007c71e2da1d6810fbbad2ee0e95e3c501d1)
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
- 球面調和関数
- Category:Special hypergeometric functions
外部リンク
- Weisstein, Eric W. "Spherical Harmonic". mathworld.wolfram.com (英語).