Bu bir **Küresel harmonikler ortonormalize tablosu**dur ve Bu Condon-Shortley fazı *l* = 10 dereceye kadar sağlanır.Bazen bu formüllerin "Kartezyen" yorumu verilir.Bu varsayım *x*, *y*, *z*, ve *r* Kartezyen-e-küresel koordinat dönüşümü yoluyla {\displaystyle \theta } ve {\displaystyle \varphi \,} ye ilişkindir:

{\displaystyle {\begin{aligned}x&=r\sin \theta \cos \varphi \\y&=r\sin \theta \sin \varphi \\z&=r\cos \theta \end{aligned}}}

Küresel harmonikler
-------------------

Burada dikkat çekmesi gereken nokta iki değişkenli fonksiyonların bir yüzeye karşılık geldiğidir.Yani bu harmonikler küre içinde farklı farklı yüzeylerin dalgalanmaları olacaktır

### l = 0

{\displaystyle Y\_{0}^{0}(\theta ,\varphi )={1 \over 2}{\sqrt {1 \over \pi }}}
### l = 1

{\displaystyle {\begin{aligned}Y\_{1}^{-1}(\theta ,\varphi )&={1 \over 2}{\sqrt {3 \over 2\pi }}\cdot e^{-i\varphi }\cdot \sin \theta \quad ={1 \over 2}{\sqrt {3 \over 2\pi }}\cdot {(x-iy) \over r}\\Y\_{1}^{0}(\theta ,\varphi )&={1 \over 2}{\sqrt {3 \over \pi }}\cdot \cos \theta \quad \quad ={1 \over 2}{\sqrt {3 \over \pi }}\cdot {z \over r}\\Y\_{1}^{1}(\theta ,\varphi )&={-1 \over 2}{\sqrt {3 \over 2\pi }}\cdot e^{i\varphi }\cdot \sin \theta \quad ={-1 \over 2}{\sqrt {3 \over 2\pi }}\cdot {(x+iy) \over r}\end{aligned}}}
### l = 2

{\displaystyle Y\_{2}^{-2}(\theta ,\varphi )={1 \over 4}{\sqrt {15 \over 2\pi }}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \quad ={1 \over 4}{\sqrt {15 \over 2\pi }}\cdot {(x-iy)^{2} \over r^{2}}}
{\displaystyle Y\_{2}^{-1}(\theta ,\varphi )={1 \over 2}{\sqrt {15 \over 2\pi }}\cdot e^{-i\varphi }\cdot \sin \theta \cdot \cos \theta \quad ={1 \over 2}{\sqrt {15 \over 2\pi }}\cdot {(x-iy)z \over r^{2}}}
{\displaystyle Y\_{2}^{0}(\theta ,\varphi )={1 \over 4}{\sqrt {5 \over \pi }}\cdot (3\cos ^{2}\theta -1)\quad ={1 \over 4}{\sqrt {5 \over \pi }}\cdot {(2z^{2}-x^{2}-y^{2}) \over r^{2}}}
{\displaystyle Y\_{2}^{1}(\theta ,\varphi )={-1 \over 2}{\sqrt {15 \over 2\pi }}\cdot e^{i\varphi }\cdot \sin \theta \cdot \cos \theta \quad ={-1 \over 2}{\sqrt {15 \over 2\pi }}\cdot {(x+iy)z \over r^{2}}}
{\displaystyle Y\_{2}^{2}(\theta ,\varphi )={1 \over 4}{\sqrt {15 \over 2\pi }}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \quad ={1 \over 4}{\sqrt {15 \over 2\pi }}\cdot {(x+iy)^{2} \over r^{2}}}
### l = 3

{\displaystyle Y\_{3}^{-3}(\theta ,\varphi )={1 \over 8}{\sqrt {35 \over \pi }}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \quad ={1 \over 8}{\sqrt {35 \over \pi }}\cdot {(x-iy)^{3} \over r^{3}}}
{\displaystyle Y\_{3}^{-2}(\theta ,\varphi )={1 \over 4}{\sqrt {105 \over 2\pi }}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot \cos \theta \quad ={1 \over 4}{\sqrt {105 \over 2\pi }}\cdot {(x-iy)^{2}z \over r^{3}}}
{\displaystyle Y\_{3}^{-1}(\theta ,\varphi )={1 \over 8}{\sqrt {21 \over \pi }}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (5\cos ^{2}\theta -1)\quad ={1 \over 8}{\sqrt {21 \over \pi }}\cdot {(x-iy)(4z^{2}-x^{2}-y^{2}) \over r^{3}}}
{\displaystyle Y\_{3}^{0}(\theta ,\varphi )={1 \over 4}{\sqrt {7 \over \pi }}\cdot (5\cos ^{3}\theta -3\cos \theta )\quad ={1 \over 4}{\sqrt {7 \over \pi }}\cdot {z(2z^{2}-3x^{2}-3y^{2}) \over r^{3}}}
{\displaystyle Y\_{3}^{1}(\theta ,\varphi )={-1 \over 8}{\sqrt {21 \over \pi }}\cdot e^{i\varphi }\cdot \sin \theta \cdot (5\cos ^{2}\theta -1)\quad ={-1 \over 8}{\sqrt {21 \over \pi }}\cdot {(x+iy)(4z^{2}-x^{2}-y^{2}) \over r^{3}}}
{\displaystyle Y\_{3}^{2}(\theta ,\varphi )={1 \over 4}{\sqrt {105 \over 2\pi }}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot \cos \theta \quad ={1 \over 4}{\sqrt {105 \over 2\pi }}\cdot {(x+iy)^{2}z \over r^{3}}}
{\displaystyle Y\_{3}^{3}(\theta ,\varphi )={-1 \over 8}{\sqrt {35 \over \pi }}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \quad ={-1 \over 8}{\sqrt {35 \over \pi }}\cdot {(x+iy)^{3} \over r^{3}}}
### l = 4

{\displaystyle Y\_{4}^{-4}(\theta ,\varphi )={3 \over 16}{\sqrt {35 \over 2\pi }}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta ={\frac {3}{16}}{\sqrt {\frac {35}{2\pi }}}\cdot {\frac {(x-iy)^{4}}{r^{4}}}}
{\displaystyle Y\_{4}^{-3}(\theta ,\varphi )={3 \over 8}{\sqrt {35 \over \pi }}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot \cos \theta ={\frac {3}{8}}{\sqrt {\frac {35}{\pi }}}\cdot {\frac {(x-iy)^{3}z}{r^{4}}}}
{\displaystyle Y\_{4}^{-2}(\theta ,\varphi )={3 \over 8}{\sqrt {5 \over 2\pi }}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (7\cos ^{2}\theta -1)={\frac {3}{8}}{\sqrt {\frac {5}{2\pi }}}\cdot {\frac {(x-iy)^{2}\cdot (7z^{2}-r^{2})}{r^{4}}}}
{\displaystyle Y\_{4}^{-1}(\theta ,\varphi )={3 \over 8}{\sqrt {5 \over \pi }}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (7\cos ^{3}\theta -3\cos \theta )={\frac {3}{8}}{\sqrt {\frac {5}{\pi }}}\cdot {\frac {(x-iy)\cdot z\cdot (7z^{2}-3r^{2})}{r^{4}}}}
{\displaystyle Y\_{4}^{0}(\theta ,\varphi )={3 \over 16}{\sqrt {1 \over \pi }}\cdot (35\cos ^{4}\theta -30\cos ^{2}\theta +3)={\frac {3}{16}}{\sqrt {\frac {1}{\pi }}}\cdot {\frac {(35z^{4}-30z^{2}r^{2}+3r^{4})}{r^{4}}}}
{\displaystyle Y\_{4}^{1}(\theta ,\varphi )={-3 \over 8}{\sqrt {5 \over \pi }}\cdot e^{i\varphi }\cdot \sin \theta \cdot (7\cos ^{3}\theta -3\cos \theta )={\frac {-3}{8}}{\sqrt {\frac {5}{\pi }}}\cdot {\frac {(x+iy)\cdot z\cdot (7z^{2}-3r^{2})}{r^{4}}}}
{\displaystyle Y\_{4}^{2}(\theta ,\varphi )={3 \over 8}{\sqrt {5 \over 2\pi }}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (7\cos ^{2}\theta -1)={\frac {3}{8}}{\sqrt {\frac {5}{2\pi }}}\cdot {\frac {(x+iy)^{2}\cdot (7z^{2}-r^{2})}{r^{4}}}}
{\displaystyle Y\_{4}^{3}(\theta ,\varphi )={-3 \over 8}{\sqrt {35 \over \pi }}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot \cos \theta ={\frac {-3}{8}}{\sqrt {\frac {35}{\pi }}}\cdot {\frac {(x+iy)^{3}z}{r^{4}}}}
{\displaystyle Y\_{4}^{4}(\theta ,\varphi )={3 \over 16}{\sqrt {35 \over 2\pi }}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta ={\frac {3}{16}}{\sqrt {\frac {35}{2\pi }}}\cdot {\frac {(x+iy)^{4}}{r^{4}}}}
### l = 5

{\displaystyle Y\_{5}^{-5}(\theta ,\varphi )={3 \over 32}{\sqrt {77 \over \pi }}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta }
{\displaystyle Y\_{5}^{-4}(\theta ,\varphi )={3 \over 16}{\sqrt {385 \over 2\pi }}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot \cos \theta }
{\displaystyle Y\_{5}^{-3}(\theta ,\varphi )={1 \over 32}{\sqrt {385 \over \pi }}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (9\cos ^{2}\theta -1)}
{\displaystyle Y\_{5}^{-2}(\theta ,\varphi )={1 \over 8}{\sqrt {1155 \over 2\pi }}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (3\cos ^{3}\theta -\cos \theta )}
{\displaystyle Y\_{5}^{-1}(\theta ,\varphi )={1 \over 16}{\sqrt {165 \over 2\pi }}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (21\cos ^{4}\theta -14\cos ^{2}\theta +1)}
{\displaystyle Y\_{5}^{0}(\theta ,\varphi )={1 \over 16}{\sqrt {11 \over \pi }}\cdot (63\cos ^{5}\theta -70\cos ^{3}\theta +15\cos \theta )}
{\displaystyle Y\_{5}^{1}(\theta ,\varphi )={-1 \over 16}{\sqrt {165 \over 2\pi }}\cdot e^{i\varphi }\cdot \sin \theta \cdot (21\cos ^{4}\theta -14\cos ^{2}\theta +1)}
{\displaystyle Y\_{5}^{2}(\theta ,\varphi )={1 \over 8}{\sqrt {1155 \over 2\pi }}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (3\cos ^{3}\theta -\cos \theta )}
{\displaystyle Y\_{5}^{3}(\theta ,\varphi )={-1 \over 32}{\sqrt {385 \over \pi }}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (9\cos ^{2}\theta -1)}
{\displaystyle Y\_{5}^{4}(\theta ,\varphi )={3 \over 16}{\sqrt {385 \over 2\pi }}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot \cos \theta }
{\displaystyle Y\_{5}^{5}(\theta ,\varphi )={-3 \over 32}{\sqrt {77 \over \pi }}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta }
### l = 6

{\displaystyle Y\_{6}^{-6}(\theta ,\varphi )={1 \over 64}{\sqrt {3003 \over \pi }}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta }
{\displaystyle Y\_{6}^{-5}(\theta ,\varphi )={3 \over 32}{\sqrt {1001 \over \pi }}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot \cos \theta }
{\displaystyle Y\_{6}^{-4}(\theta ,\varphi )={3 \over 32}{\sqrt {91 \over 2\pi }}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (11\cos ^{2}\theta -1)}
{\displaystyle Y\_{6}^{-3}(\theta ,\varphi )={1 \over 32}{\sqrt {1365 \over \pi }}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (11\cos ^{3}\theta -3\cos \theta )}
{\displaystyle Y\_{6}^{-2}(\theta ,\varphi )={1 \over 64}{\sqrt {1365 \over \pi }}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (33\cos ^{4}\theta -18\cos ^{2}\theta +1)}
{\displaystyle Y\_{6}^{-1}(\theta ,\varphi )={1 \over 16}{\sqrt {273 \over 2\pi }}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (33\cos ^{5}\theta -30\cos ^{3}\theta +5\cos \theta )}
{\displaystyle Y\_{6}^{0}(\theta ,\varphi )={1 \over 32}{\sqrt {13 \over \pi }}\cdot (231\cos ^{6}\theta -315\cos ^{4}\theta +105\cos ^{2}\theta -5)}
{\displaystyle Y\_{6}^{1}(\theta ,\varphi )={-1 \over 16}{\sqrt {273 \over 2\pi }}\cdot e^{i\varphi }\cdot \sin \theta \cdot (33\cos ^{5}\theta -30\cos ^{3}\theta +5\cos \theta )}
{\displaystyle Y\_{6}^{2}(\theta ,\varphi )={1 \over 64}{\sqrt {1365 \over \pi }}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (33\cos ^{4}\theta -18\cos ^{2}\theta +1)}
{\displaystyle Y\_{6}^{3}(\theta ,\varphi )={-1 \over 32}{\sqrt {1365 \over \pi }}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (11\cos ^{3}\theta -3\cos \theta )}
{\displaystyle Y\_{6}^{4}(\theta ,\varphi )={3 \over 32}{\sqrt {91 \over 2\pi }}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot (11\cos ^{2}\theta -1)}
{\displaystyle Y\_{6}^{5}(\theta ,\varphi )={-3 \over 32}{\sqrt {1001 \over \pi }}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot \cos \theta }
{\displaystyle Y\_{6}^{6}(\theta ,\varphi )={1 \over 64}{\sqrt {3003 \over \pi }}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta }
### l = 7

{\displaystyle Y\_{7}^{-7}(\theta ,\varphi )={3 \over 64}{\sqrt {715 \over 2\pi }}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta }
{\displaystyle Y\_{7}^{-6}(\theta ,\varphi )={3 \over 64}{\sqrt {5005 \over \pi }}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \cdot \cos \theta }
{\displaystyle Y\_{7}^{-5}(\theta ,\varphi )={3 \over 64}{\sqrt {385 \over 2\pi }}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot (13\cos ^{2}\theta -1)}
{\displaystyle Y\_{7}^{-4}(\theta ,\varphi )={3 \over 32}{\sqrt {385 \over 2\pi }}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (13\cos ^{3}\theta -3\cos \theta )}
{\displaystyle Y\_{7}^{-3}(\theta ,\varphi )={3 \over 64}{\sqrt {35 \over 2\pi }}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (143\cos ^{4}\theta -66\cos ^{2}\theta +3)}
{\displaystyle Y\_{7}^{-2}(\theta ,\varphi )={3 \over 64}{\sqrt {35 \over \pi }}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (143\cos ^{5}\theta -110\cos ^{3}\theta +15\cos \theta )}
{\displaystyle Y\_{7}^{-1}(\theta ,\varphi )={1 \over 64}{\sqrt {105 \over 2\pi }}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (429\cos ^{6}\theta -495\cos ^{4}\theta +135\cos ^{2}\theta -5)}
{\displaystyle Y\_{7}^{0}(\theta ,\varphi )={1 \over 32}{\sqrt {15 \over \pi }}\cdot (429\cos ^{7}\theta -693\cos ^{5}\theta +315\cos ^{3}\theta -35\cos \theta )}
{\displaystyle Y\_{7}^{1}(\theta ,\varphi )={-1 \over 64}{\sqrt {105 \over 2\pi }}\cdot e^{i\varphi }\cdot \sin \theta \cdot (429\cos ^{6}\theta -495\cos ^{4}\theta +135\cos ^{2}\theta -5)}
{\displaystyle Y\_{7}^{2}(\theta ,\varphi )={3 \over 64}{\sqrt {35 \over \pi }}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (143\cos ^{5}\theta -110\cos ^{3}\theta +15\cos \theta )}
{\displaystyle Y\_{7}^{3}(\theta ,\varphi )={-3 \over 64}{\sqrt {35 \over 2\pi }}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (143\cos ^{4}\theta -66\cos ^{2}\theta +3)}
{\displaystyle Y\_{7}^{4}(\theta ,\varphi )={3 \over 32}{\sqrt {385 \over 2\pi }}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot (13\cos ^{3}\theta -3\cos \theta )}
{\displaystyle Y\_{7}^{5}(\theta ,\varphi )={-3 \over 64}{\sqrt {385 \over 2\pi }}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot (13\cos ^{2}\theta -1)}
{\displaystyle Y\_{7}^{6}(\theta ,\varphi )={3 \over 64}{\sqrt {5005 \over \pi }}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot \cos \theta }
{\displaystyle Y\_{7}^{7}(\theta ,\varphi )={-3 \over 64}{\sqrt {715 \over 2\pi }}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta }
### l = 8

{\displaystyle Y\_{8}^{-8}(\theta ,\varphi )={3 \over 256}{\sqrt {12155 \over 2\pi }}\cdot e^{-8i\varphi }\cdot \sin ^{8}\theta }
{\displaystyle Y\_{8}^{-7}(\theta ,\varphi )={3 \over 64}{\sqrt {12155 \over 2\pi }}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta \cdot \cos \theta }
{\displaystyle Y\_{8}^{-6}(\theta ,\varphi )={1 \over 128}{\sqrt {7293 \over \pi }}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \cdot (15\cos ^{2}\theta -1)}
{\displaystyle Y\_{8}^{-5}(\theta ,\varphi )={3 \over 64}{\sqrt {17017 \over 2\pi }}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot (5\cos ^{3}\theta -\cos \theta )}
{\displaystyle Y\_{8}^{-4}(\theta ,\varphi )={3 \over 128}{\sqrt {1309 \over 2\pi }}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (65\cos ^{4}\theta -26\cos ^{2}\theta +1)}
{\displaystyle Y\_{8}^{-3}(\theta ,\varphi )={1 \over 64}{\sqrt {19635 \over 2\pi }}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (39\cos ^{5}\theta -26\cos ^{3}\theta +3\cos \theta )}
{\displaystyle Y\_{8}^{-2}(\theta ,\varphi )={3 \over 128}{\sqrt {595 \over \pi }}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (143\cos ^{6}\theta -143\cos ^{4}\theta +33\cos ^{2}\theta -1)}
{\displaystyle Y\_{8}^{-1}(\theta ,\varphi )={3 \over 64}{\sqrt {17 \over 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 )}
{\displaystyle Y\_{8}^{0}(\theta ,\varphi )={1 \over 256}{\sqrt {17 \over \pi }}\cdot (6435\cos ^{8}\theta -12012\cos ^{6}\theta +6930\cos ^{4}\theta -1260\cos ^{2}\theta +35)}
{\displaystyle Y\_{8}^{1}(\theta ,\varphi )={-3 \over 64}{\sqrt {17 \over 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 )}
{\displaystyle Y\_{8}^{2}(\theta ,\varphi )={3 \over 128}{\sqrt {595 \over \pi }}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (143\cos ^{6}\theta -143\cos ^{4}\theta +33\cos ^{2}\theta -1)}
{\displaystyle Y\_{8}^{3}(\theta ,\varphi )={-1 \over 64}{\sqrt {19635 \over 2\pi }}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (39\cos ^{5}\theta -26\cos ^{3}\theta +3\cos \theta )}
{\displaystyle Y\_{8}^{4}(\theta ,\varphi )={3 \over 128}{\sqrt {1309 \over 2\pi }}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot (65\cos ^{4}\theta -26\cos ^{2}\theta +1)}
{\displaystyle Y\_{8}^{5}(\theta ,\varphi )={-3 \over 64}{\sqrt {17017 \over 2\pi }}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot (5\cos ^{3}\theta -\cos \theta )}
{\displaystyle Y\_{8}^{6}(\theta ,\varphi )={1 \over 128}{\sqrt {7293 \over \pi }}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot (15\cos ^{2}\theta -1)}
{\displaystyle Y\_{8}^{7}(\theta ,\varphi )={-3 \over 64}{\sqrt {12155 \over 2\pi }}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta \cdot \cos \theta }
{\displaystyle Y\_{8}^{8}(\theta ,\varphi )={3 \over 256}{\sqrt {12155 \over 2\pi }}\cdot e^{8i\varphi }\cdot \sin ^{8}\theta }
### l = 9

{\displaystyle Y\_{9}^{-9}(\theta ,\varphi )={1 \over 512}{\sqrt {230945 \over \pi }}\cdot e^{-9i\varphi }\cdot \sin ^{9}\theta }
{\displaystyle Y\_{9}^{-8}(\theta ,\varphi )={3 \over 256}{\sqrt {230945 \over 2\pi }}\cdot e^{-8i\varphi }\cdot \sin ^{8}\theta \cdot \cos \theta }
{\displaystyle Y\_{9}^{-7}(\theta ,\varphi )={3 \over 512}{\sqrt {13585 \over \pi }}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta \cdot (17\cos ^{2}\theta -1)}
{\displaystyle Y\_{9}^{-6}(\theta ,\varphi )={1 \over 128}{\sqrt {40755 \over \pi }}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \cdot (17\cos ^{3}\theta -3\cos \theta )}
{\displaystyle Y\_{9}^{-5}(\theta ,\varphi )={3 \over 256}{\sqrt {2717 \over \pi }}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot (85\cos ^{4}\theta -30\cos ^{2}\theta +1)}
{\displaystyle Y\_{9}^{-4}(\theta ,\varphi )={3 \over 128}{\sqrt {95095 \over 2\pi }}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (17\cos ^{5}\theta -10\cos ^{3}\theta +\cos \theta )}
{\displaystyle Y\_{9}^{-3}(\theta ,\varphi )={1 \over 256}{\sqrt {21945 \over \pi }}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (221\cos ^{6}\theta -195\cos ^{4}\theta +39\cos ^{2}\theta -1)}
{\displaystyle Y\_{9}^{-2}(\theta ,\varphi )={3 \over 128}{\sqrt {1045 \over \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 )}
{\displaystyle Y\_{9}^{-1}(\theta ,\varphi )={3 \over 256}{\sqrt {95 \over 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)}
{\displaystyle Y\_{9}^{0}(\theta ,\varphi )={1 \over 256}{\sqrt {19 \over \pi }}\cdot (12155\cos ^{9}\theta -25740\cos ^{7}\theta +18018\cos ^{5}\theta -4620\cos ^{3}\theta +315\cos \theta )}
{\displaystyle Y\_{9}^{1}(\theta ,\varphi )={-3 \over 256}{\sqrt {95 \over 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)}
{\displaystyle Y\_{9}^{2}(\theta ,\varphi )={3 \over 128}{\sqrt {1045 \over \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 )}
{\displaystyle Y\_{9}^{3}(\theta ,\varphi )={-1 \over 256}{\sqrt {21945 \over \pi }}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (221\cos ^{6}\theta -195\cos ^{4}\theta +39\cos ^{2}\theta -1)}
{\displaystyle Y\_{9}^{4}(\theta ,\varphi )={3 \over 128}{\sqrt {95095 \over 2\pi }}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot (17\cos ^{5}\theta -10\cos ^{3}\theta +\cos \theta )}
{\displaystyle Y\_{9}^{5}(\theta ,\varphi )={-3 \over 256}{\sqrt {2717 \over \pi }}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot (85\cos ^{4}\theta -30\cos ^{2}\theta +1)}
{\displaystyle Y\_{9}^{6}(\theta ,\varphi )={1 \over 128}{\sqrt {40755 \over \pi }}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot (17\cos ^{3}\theta -3\cos \theta )}
{\displaystyle Y\_{9}^{7}(\theta ,\varphi )={-3 \over 512}{\sqrt {13585 \over \pi }}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta \cdot (17\cos ^{2}\theta -1)}
{\displaystyle Y\_{9}^{8}(\theta ,\varphi )={3 \over 256}{\sqrt {230945 \over 2\pi }}\cdot e^{8i\varphi }\cdot \sin ^{8}\theta \cdot \cos \theta }
{\displaystyle Y\_{9}^{9}(\theta ,\varphi )={-1 \over 512}{\sqrt {230945 \over \pi }}\cdot e^{9i\varphi }\cdot \sin ^{9}\theta }
### l = 10

{\displaystyle Y\_{10}^{-10}(\theta ,\varphi )={1 \over 1024}{\sqrt {969969 \over \pi }}\cdot e^{-10i\varphi }\cdot \sin ^{10}\theta }
{\displaystyle Y\_{10}^{-9}(\theta ,\varphi )={1 \over 512}{\sqrt {4849845 \over \pi }}\cdot e^{-9i\varphi }\cdot \sin ^{9}\theta \cdot \cos \theta }
{\displaystyle Y\_{10}^{-8}(\theta ,\varphi )={1 \over 512}{\sqrt {255255 \over 2\pi }}\cdot e^{-8i\varphi }\cdot \sin ^{8}\theta \cdot (19\cos ^{2}\theta -1)}
{\displaystyle Y\_{10}^{-7}(\theta ,\varphi )={3 \over 512}{\sqrt {85085 \over \pi }}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta \cdot (19\cos ^{3}\theta -3\cos \theta )}
{\displaystyle Y\_{10}^{-6}(\theta ,\varphi )={3 \over 1024}{\sqrt {5005 \over \pi }}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \cdot (323\cos ^{4}\theta -102\cos ^{2}\theta +3)}
{\displaystyle Y\_{10}^{-5}(\theta ,\varphi )={3 \over 256}{\sqrt {1001 \over \pi }}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot (323\cos ^{5}\theta -170\cos ^{3}\theta +15\cos \theta )}
{\displaystyle Y\_{10}^{-4}(\theta ,\varphi )={3 \over 256}{\sqrt {5005 \over 2\pi }}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (323\cos ^{6}\theta -255\cos ^{4}\theta +45\cos ^{2}\theta -1)}
{\displaystyle Y\_{10}^{-3}(\theta ,\varphi )={3 \over 256}{\sqrt {5005 \over \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 )}
{\displaystyle Y\_{10}^{-2}(\theta ,\varphi )={3 \over 512}{\sqrt {385 \over 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)}
{\displaystyle Y\_{10}^{-1}(\theta ,\varphi )={1 \over 256}{\sqrt {1155 \over 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 )}
{\displaystyle Y\_{10}^{0}(\theta ,\varphi )={1 \over 512}{\sqrt {21 \over \pi }}\cdot (46189\cos ^{10}\theta -109395\cos ^{8}\theta +90090\cos ^{6}\theta -30030\cos ^{4}\theta +3465\cos ^{2}\theta -63)}
{\displaystyle Y\_{10}^{1}(\theta ,\varphi )={-1 \over 256}{\sqrt {1155 \over 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 )}
{\displaystyle Y\_{10}^{2}(\theta ,\varphi )={3 \over 512}{\sqrt {385 \over 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)}
{\displaystyle Y\_{10}^{3}(\theta ,\varphi )={-3 \over 256}{\sqrt {5005 \over \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 )}
{\displaystyle Y\_{10}^{4}(\theta ,\varphi )={3 \over 256}{\sqrt {5005 \over 2\pi }}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot (323\cos ^{6}\theta -255\cos ^{4}\theta +45\cos ^{2}\theta -1)}
{\displaystyle Y\_{10}^{5}(\theta ,\varphi )={-3 \over 256}{\sqrt {1001 \over \pi }}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot (323\cos ^{5}\theta -170\cos ^{3}\theta +15\cos \theta )}
{\displaystyle Y\_{10}^{6}(\theta ,\varphi )={3 \over 1024}{\sqrt {5005 \over \pi }}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot (323\cos ^{4}\theta -102\cos ^{2}\theta +3)}
{\displaystyle Y\_{10}^{7}(\theta ,\varphi )={-3 \over 512}{\sqrt {85085 \over \pi }}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta \cdot (19\cos ^{3}\theta -3\cos \theta )}
{\displaystyle Y\_{10}^{8}(\theta ,\varphi )={1 \over 512}{\sqrt {255255 \over 2\pi }}\cdot e^{8i\varphi }\cdot \sin ^{8}\theta \cdot (19\cos ^{2}\theta -1)}
{\displaystyle Y\_{10}^{9}(\theta ,\varphi )={-1 \over 512}{\sqrt {4849845 \over \pi }}\cdot e^{9i\varphi }\cdot \sin ^{9}\theta \cdot \cos \theta }
{\displaystyle Y\_{10}^{10}(\theta ,\varphi )={1 \over 1024}{\sqrt {969969 \over \pi }}\cdot e^{10i\varphi }\cdot \sin ^{10}\theta }

Gerçek küresel harmonikler
--------------------------

Her Gerçek küresel harmonik için,karşılık gelen (*s*, *p*, *d*, *f*, *g*) atomik orbital sembolü de bildirilmektedir.

### l = 0

{\displaystyle {\begin{aligned}Y\_{00}&=s=Y\_{0}^{0}={\frac {1}{2}}{\sqrt {\frac {1}{\pi }}}\end{aligned}}}
### l = 1

{\displaystyle {\begin{aligned}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}={\sqrt {\frac {3}{4\pi }}}\cdot {\frac {z}{r}}\\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}}\end{aligned}}}
### l = 2

**Ayrıştırılamadı (Mümkünse MathML (daha deneme aşamasında): Geçersiz yanıt ("Math extension cannot connect to Restbase.") sunucu "http://localhost:6011/tr.wikipedia.org/v1/":): {\displaystyle \begin{align} 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{x y}{r^2} \\ 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{y z}{r^2} \\ Y\_{20} & = d\_{z^2} = Y\_2^0 = \frac{1}{4} \sqrt{\frac{5}{\pi}} \cdot \frac{- x^2 - y^2 + 2 z^2}{r^2} \\ 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{z x}{r^2} \\ 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} \end{align} }**

### l = 3

{\displaystyle {\begin{aligned}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\_{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\_{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}={\frac {1}{4}}{\sqrt {\frac {7}{\pi }}}\cdot {\frac {z(2z^{2}-3x^{2}-3y^{2})}{r^{3}}}\\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})}={\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}-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}}}\end{aligned}}}
### l = 4

{\displaystyle {\begin{aligned}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\_{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}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{\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}={\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}={\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}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\_{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}}={\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}}}\end{aligned}}}

Ayrıca bakınız
--------------

* Küresel harmonikler

Dış bağlantılar
---------------

* Spherical Harmonic 25 Mart 2014 tarihinde Wayback Machine sitesinde arşivlendi. at MathWorld

Kaynakça
--------

Cite kaynakça

1. 1 2 3 4 5 6  D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii (1988). *Quantum theory of angular momentum : irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols* (1. repr. bas.). Singapore: World Scientific Pub. s. 155-156. ISBN 9971-50-107-4. KB1 bakım: Birden fazla ad: yazar listesi (link)
2. 1 2 3 4  C.D.H. Chisholm (1976). *Group theoretical techniques in quantum chemistry*. New York: Academic Press. ISBN 0-12-172950-8.
3. 1 2 3  Blanco, Miguel A.; Flórez, M.; Bermejo, M. (1 Aralık 1997). "Evaluation of the rotation matrices in the basis of real spherical harmonics". *Journal of Molecular Structure: THEOCHEM*. **419** (1–3). ss. 19-27. doi:10.1016/S0166-1280(97)00185-1.

Genel kaynakça
* See section 3 in Mathar, R. J. (2009). "Zernike basis to cartesian transformations". *Serbian Astronomical Journal*. **179** (179). ss. 107-120. arXiv:0809.2368 $2. Bibcode:2009SerAj.179..107M. doi:10.2298/SAJ0979107M.  (see section 3.3)
* For complex spherical harmonics, see also 27 Nisan 2014 tarihinde Wayback Machine sitesinde arşivlendi. SphericalHarmonicY[l,m,theta,phi] at Wolfram Alpha, especially for specific values of l and m.