# 球面調和関数表

{\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

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

### l = 1

${\displaystyle Y_{1}^{-1}(x)={\frac {1}{2}}{\sqrt {\frac {3}{2\pi }}}\cdot e^{-i\varphi }\cdot \sin \theta ={\frac {1}{2}}{\sqrt {\frac {3}{2\pi }}}\cdot {\frac {x-iy}{r}}}$
${\displaystyle Y_{1}^{0}(x)={\frac {1}{2}}{\sqrt {\frac {3}{\pi }}}\cdot \cos \theta ={\frac {1}{2}}{\sqrt {\frac {3}{\pi }}}\cdot {\frac {z}{r}}}$
${\displaystyle Y_{1}^{1}(x)=-{\frac {1}{2}}{\sqrt {\frac {3}{2\pi }}}\cdot e^{i\varphi }\cdot \sin \theta =-{\frac {1}{2}}{\sqrt {\frac {3}{2\pi }}}\cdot {\frac {x+iy}{r}}}$

### l = 2

${\displaystyle Y_{2}^{-2}(x)={\frac {1}{4}}{\sqrt {\frac {15}{2\pi }}}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta ={\frac {1}{4}}{\sqrt {\frac {15}{2\pi }}}\cdot {\frac {x^{2}-2ixy-y^{2}}{r^{2}}}}$
${\displaystyle Y_{2}^{-1}(x)={\frac {1}{2}}{\sqrt {\frac {15}{2\pi }}}\cdot e^{-i\varphi }\cdot \sin \theta \cdot \cos \theta ={\frac {1}{2}}{\sqrt {\frac {15}{2\pi }}}\cdot {\frac {xz-iyz}{r^{2}}}}$
${\displaystyle Y_{2}^{0}(x)={\frac {1}{4}}{\sqrt {\frac {5}{\pi }}}\cdot (3\cos ^{2}\theta -1)={\frac {1}{4}}{\sqrt {\frac {5}{\pi }}}\cdot {\frac {-x^{2}-y^{2}+2z^{2}}{r^{2}}}}$
${\displaystyle Y_{2}^{1}(x)=-{\frac {1}{2}}{\sqrt {\frac {15}{2\pi }}}\cdot e^{i\varphi }\cdot \sin \theta \cdot \cos \theta =-{\frac {1}{2}}{\sqrt {\frac {15}{2\pi }}}\cdot {\frac {xz+iyz}{r^{2}}}}$
${\displaystyle Y_{2}^{2}(x)={\frac {1}{4}}{\sqrt {\frac {15}{2\pi }}}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta ={\frac {1}{4}}{\sqrt {\frac {15}{2\pi }}}\cdot {\frac {x^{2}+2ixy-y^{2}}{r^{2}}}}$

### l = 3

${\displaystyle Y_{3}^{-3}(x)={\frac {1}{8}}{\sqrt {\frac {35}{\pi }}}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta ={\frac {1}{8}}{\sqrt {\frac {35}{\pi }}}\cdot {\frac {x^{3}-3ix^{2}y-3xy^{2}+iy^{3}}{r^{3}}}}$
${\displaystyle Y_{3}^{-2}(x)={\frac {1}{4}}{\sqrt {\frac {105}{2\pi }}}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot \cos \theta ={\frac {1}{4}}{\sqrt {\frac {105}{2\pi }}}\cdot {\frac {x^{2}z-2ixyz-y^{2}z}{r^{3}}}}$
${\displaystyle Y_{3}^{-1}(x)={\frac {1}{8}}{\sqrt {\frac {21}{\pi }}}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (5\cos ^{2}\theta -1)={\frac {1}{8}}{\sqrt {\frac {21}{\pi }}}\cdot {\frac {-x^{3}+ix^{2}y-xy^{2}+4xz^{2}+iy^{3}-4iyz^{2}}{r^{3}}}}$
${\displaystyle Y_{3}^{0}(x)={\frac {1}{4}}{\sqrt {\frac {7}{\pi }}}\cdot (5\cos ^{3}\theta -3\cos \theta )={\frac {1}{4}}{\sqrt {\frac {7}{\pi }}}\cdot {\frac {-3x^{2}z-3y^{2}z+2z^{3}}{r^{3}}}}$
${\displaystyle Y_{3}^{1}(x)=-{\frac {1}{8}}{\sqrt {\frac {21}{\pi }}}\cdot e^{i\varphi }\cdot \sin \theta \cdot (5\cos ^{2}\theta -1)=-{\frac {1}{8}}{\sqrt {\frac {21}{\pi }}}\cdot {\frac {-x^{3}-ix^{2}y-xy^{2}+4xz^{2}-iy^{3}+4iyz^{2}}{r^{3}}}}$
${\displaystyle Y_{3}^{2}(x)={\frac {1}{4}}{\sqrt {\frac {105}{2\pi }}}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot \cos \theta ={\frac {1}{4}}{\sqrt {\frac {105}{2\pi }}}\cdot {\frac {x^{2}z+2ixyz-y^{2}z}{r^{3}}}}$
${\displaystyle Y_{3}^{3}(x)=-{\frac {1}{8}}{\sqrt {\frac {35}{\pi }}}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta =-{\frac {1}{8}}{\sqrt {\frac {35}{\pi }}}\cdot {\frac {x^{3}+3ix^{2}y-3xy^{2}-iy^{3}}{r^{3}}}}$

### l = 4

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

### l = 5

${\displaystyle Y_{5}^{-5}(x)={\frac {3}{32}}{\sqrt {\frac {77}{\pi }}}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta }$
${\displaystyle Y_{5}^{-4}(x)={\frac {3}{16}}{\sqrt {\frac {385}{2\pi }}}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot \cos \theta }$
${\displaystyle Y_{5}^{-3}(x)={\frac {1}{32}}{\sqrt {\frac {385}{\pi }}}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (9\cos ^{2}\theta -1)}$
${\displaystyle Y_{5}^{-2}(x)={\frac {1}{8}}{\sqrt {\frac {1155}{2\pi }}}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (3\cos ^{3}\theta -1\cos \theta )}$
${\displaystyle Y_{5}^{-1}(x)={\frac {1}{16}}{\sqrt {\frac {165}{2\pi }}}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (21\cos ^{4}\theta -14\cos ^{2}\theta +1)}$
${\displaystyle Y_{5}^{0}(x)={\frac {1}{16}}{\sqrt {\frac {11}{\pi }}}\cdot (63\cos ^{5}\theta -70\cos ^{3}\theta +15\cos \theta )}$
${\displaystyle Y_{5}^{1}(x)=-{\frac {1}{16}}{\sqrt {\frac {165}{2\pi }}}\cdot e^{i\varphi }\cdot \sin \theta \cdot (21\cos ^{4}\theta -14\cos ^{2}\theta +1)}$
${\displaystyle Y_{5}^{2}(x)={\frac {1}{8}}{\sqrt {\frac {1155}{2\pi }}}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (3\cos ^{3}\theta -1\cos \theta )}$
${\displaystyle Y_{5}^{3}(x)=-{\frac {1}{32}}{\sqrt {\frac {385}{\pi }}}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (9\cos ^{2}\theta -1)}$
${\displaystyle Y_{5}^{4}(x)={\frac {3}{16}}{\sqrt {\frac {385}{2\pi }}}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot \cos \theta }$
${\displaystyle Y_{5}^{5}(x)=-{\frac {3}{32}}{\sqrt {\frac {77}{\pi }}}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta }$

### l = 6

${\displaystyle Y_{6}^{-6}(x)={\frac {1}{64}}{\sqrt {\frac {3003}{\pi }}}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta }$
${\displaystyle Y_{6}^{-5}(x)={\frac {3}{32}}{\sqrt {\frac {1001}{\pi }}}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot \cos \theta }$
${\displaystyle Y_{6}^{-4}(x)={\frac {3}{32}}{\sqrt {\frac {91}{2\pi }}}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (11\cos ^{2}\theta -1)}$
${\displaystyle Y_{6}^{-3}(x)={\frac {1}{32}}{\sqrt {\frac {1365}{\pi }}}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (11\cos ^{3}\theta -3\cos \theta )}$
${\displaystyle Y_{6}^{-2}(x)={\frac {1}{64}}{\sqrt {\frac {1365}{\pi }}}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (33\cos ^{4}\theta -18\cos ^{2}\theta +1)}$
${\displaystyle Y_{6}^{-1}(x)={\frac {1}{16}}{\sqrt {\frac {273}{2\pi }}}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (33\cos ^{5}\theta -30\cos ^{3}\theta +5\cos \theta )}$
${\displaystyle Y_{6}^{0}(x)={\frac {1}{32}}{\sqrt {\frac {13}{\pi }}}\cdot (231\cos ^{6}\theta -315\cos ^{4}\theta +105\cos ^{2}\theta -5)}$
${\displaystyle Y_{6}^{1}(x)=-{\frac {1}{16}}{\sqrt {\frac {273}{2\pi }}}\cdot e^{i\varphi }\cdot \sin \theta \cdot (33\cos ^{5}\theta -30\cos ^{3}\theta +5\cos \theta )}$
${\displaystyle Y_{6}^{2}(x)={\frac {1}{64}}{\sqrt {\frac {1365}{\pi }}}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (33\cos ^{4}\theta -18\cos ^{2}\theta +1)}$
${\displaystyle Y_{6}^{3}(x)=-{\frac {1}{32}}{\sqrt {\frac {1365}{\pi }}}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (11\cos ^{3}\theta -3\cos \theta )}$
${\displaystyle Y_{6}^{4}(x)={\frac {3}{32}}{\sqrt {\frac {91}{2\pi }}}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot (11\cos ^{2}\theta -1)}$
${\displaystyle Y_{6}^{5}(x)=-{\frac {3}{32}}{\sqrt {\frac {1001}{\pi }}}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot \cos \theta }$
${\displaystyle Y_{6}^{6}(x)={\frac {1}{64}}{\sqrt {\frac {3003}{\pi }}}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta }$

### l = 7

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

### l = 8

${\displaystyle Y_{8}^{-8}(x)={\frac {3}{256}}{\sqrt {\frac {12155}{2\pi }}}\cdot e^{-8i\varphi }\cdot \sin ^{8}\theta }$
${\displaystyle Y_{8}^{-7}(x)={\frac {3}{64}}{\sqrt {\frac {12155}{2\pi }}}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta \cdot \cos \theta }$
${\displaystyle Y_{8}^{-6}(x)={\frac {1}{128}}{\sqrt {\frac {7293}{\pi }}}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \cdot (15\cos ^{2}\theta -1)}$
${\displaystyle Y_{8}^{-5}(x)={\frac {3}{64}}{\sqrt {\frac {17017}{2\pi }}}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot (5\cos ^{3}\theta -1\cos \theta )}$
${\displaystyle Y_{8}^{-4}(x)={\frac {3}{128}}{\sqrt {\frac {1309}{2\pi }}}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (65\cos ^{4}\theta -26\cos ^{2}\theta +1)}$
${\displaystyle Y_{8}^{-3}(x)={\frac {1}{64}}{\sqrt {\frac {19635}{2\pi }}}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (39\cos ^{5}\theta -26\cos ^{3}\theta +3\cos \theta )}$
${\displaystyle Y_{8}^{-2}(x)={\frac {3}{128}}{\sqrt {\frac {595}{\pi }}}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (143\cos ^{6}\theta -143\cos ^{4}\theta +33\cos ^{2}\theta -1)}$
${\displaystyle Y_{8}^{-1}(x)={\frac {3}{64}}{\sqrt {\frac {17}{2\pi }}}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (715\cos ^{7}\theta -1001\cos ^{5}\theta +385\cos ^{3}\theta -35\cos \theta )}$
${\displaystyle Y_{8}^{0}(x)={\frac {1}{256}}{\sqrt {\frac {17}{\pi }}}\cdot (6435\cos ^{8}\theta -12012\cos ^{6}\theta +6930\cos ^{4}\theta -1260\cos ^{2}\theta +35)}$
${\displaystyle Y_{8}^{1}(x)=-{\frac {3}{64}}{\sqrt {\frac {17}{2\pi }}}\cdot e^{i\varphi }\cdot \sin \theta \cdot (715\cos ^{7}\theta -1001\cos ^{5}\theta +385\cos ^{3}\theta -35\cos \theta )}$
${\displaystyle Y_{8}^{2}(x)={\frac {3}{128}}{\sqrt {\frac {595}{\pi }}}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (143\cos ^{6}\theta -143\cos ^{4}\theta +33\cos ^{2}\theta -1)}$
${\displaystyle Y_{8}^{3}(x)=-{\frac {1}{64}}{\sqrt {\frac {19635}{2\pi }}}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (39\cos ^{5}\theta -26\cos ^{3}\theta +3\cos \theta )}$
${\displaystyle Y_{8}^{4}(x)={\frac {3}{128}}{\sqrt {\frac {1309}{2\pi }}}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot (65\cos ^{4}\theta -26\cos ^{2}\theta +1)}$
${\displaystyle Y_{8}^{5}(x)=-{\frac {3}{64}}{\sqrt {\frac {17017}{2\pi }}}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot (5\cos ^{3}\theta -1\cos \theta )}$
${\displaystyle Y_{8}^{6}(x)={\frac {1}{128}}{\sqrt {\frac {7293}{\pi }}}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot (15\cos ^{2}\theta -1)}$
${\displaystyle Y_{8}^{7}(x)=-{\frac {3}{64}}{\sqrt {\frac {12155}{2\pi }}}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta \cdot \cos \theta }$
${\displaystyle Y_{8}^{8}(x)={\frac {3}{256}}{\sqrt {\frac {12155}{2\pi }}}\cdot e^{8i\varphi }\cdot \sin ^{8}\theta }$

### l = 9

${\displaystyle Y_{9}^{-9}(x)={\frac {1}{512}}{\sqrt {\frac {230945}{\pi }}}\cdot e^{-9i\varphi }\cdot \sin ^{9}\theta }$
${\displaystyle Y_{9}^{-8}(x)={\frac {3}{256}}{\sqrt {\frac {230945}{2\pi }}}\cdot e^{-8i\varphi }\cdot \sin ^{8}\theta \cdot \cos \theta }$
${\displaystyle Y_{9}^{-7}(x)={\frac {3}{512}}{\sqrt {\frac {13585}{\pi }}}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta \cdot (17\cos ^{2}\theta -1)}$
${\displaystyle Y_{9}^{-6}(x)={\frac {1}{128}}{\sqrt {\frac {40755}{\pi }}}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \cdot (17\cos ^{3}\theta -3\cos \theta )}$
${\displaystyle Y_{9}^{-5}(x)={\frac {3}{256}}{\sqrt {\frac {2717}{\pi }}}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot (85\cos ^{4}\theta -30\cos ^{2}\theta +1)}$
${\displaystyle Y_{9}^{-4}(x)={\frac {3}{128}}{\sqrt {\frac {95095}{2\pi }}}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (17\cos ^{5}\theta -10\cos ^{3}\theta +1\cos \theta )}$
${\displaystyle Y_{9}^{-3}(x)={\frac {1}{256}}{\sqrt {\frac {21945}{\pi }}}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (221\cos ^{6}\theta -195\cos ^{4}\theta +39\cos ^{2}\theta -1)}$
${\displaystyle Y_{9}^{-2}(x)={\frac {3}{128}}{\sqrt {\frac {1045}{\pi }}}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (221\cos ^{7}\theta -273\cos ^{5}\theta +91\cos ^{3}\theta -7\cos \theta )}$
${\displaystyle Y_{9}^{-1}(x)={\frac {3}{256}}{\sqrt {\frac {95}{2\pi }}}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (2431\cos ^{8}\theta -4004\cos ^{6}\theta +2002\cos ^{4}\theta -308\cos ^{2}\theta +7)}$
${\displaystyle Y_{9}^{0}(x)={\frac {1}{256}}{\sqrt {\frac {19}{\pi }}}\cdot (12155\cos ^{9}\theta -25740\cos ^{7}\theta +18018\cos ^{5}\theta -4620\cos ^{3}\theta +315\cos \theta )}$
${\displaystyle Y_{9}^{1}(x)=-{\frac {3}{256}}{\sqrt {\frac {95}{2\pi }}}\cdot e^{i\varphi }\cdot \sin \theta \cdot (2431\cos ^{8}\theta -4004\cos ^{6}\theta +2002\cos ^{4}\theta -308\cos ^{2}\theta +7)}$
${\displaystyle Y_{9}^{2}(x)={\frac {3}{128}}{\sqrt {\frac {1045}{\pi }}}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (221\cos ^{7}\theta -273\cos ^{5}\theta +91\cos ^{3}\theta -7\cos \theta )}$
${\displaystyle Y_{9}^{3}(x)=-{\frac {1}{256}}{\sqrt {\frac {21945}{\pi }}}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (221\cos ^{6}\theta -195\cos ^{4}\theta +39\cos ^{2}\theta -1)}$
${\displaystyle Y_{9}^{4}(x)={\frac {3}{128}}{\sqrt {\frac {95095}{2\pi }}}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot (17\cos ^{5}\theta -10\cos ^{3}\theta +1\cos \theta )}$
${\displaystyle Y_{9}^{5}(x)=-{\frac {3}{256}}{\sqrt {\frac {2717}{\pi }}}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot (85\cos ^{4}\theta -30\cos ^{2}\theta +1)}$
${\displaystyle Y_{9}^{6}(x)={\frac {1}{128}}{\sqrt {\frac {40755}{\pi }}}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot (17\cos ^{3}\theta -3\cos \theta )}$
${\displaystyle Y_{9}^{7}(x)=-{\frac {3}{512}}{\sqrt {\frac {13585}{\pi }}}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta \cdot (17\cos ^{2}\theta -1)}$
${\displaystyle Y_{9}^{8}(x)={\frac {3}{256}}{\sqrt {\frac {230945}{2\pi }}}\cdot e^{8i\varphi }\cdot \sin ^{8}\theta \cdot \cos \theta }$
${\displaystyle Y_{9}^{9}(x)=-{\frac {1}{512}}{\sqrt {\frac {230945}{\pi }}}\cdot e^{9i\varphi }\cdot \sin ^{9}\theta }$

### l = 10

${\displaystyle Y_{10}^{-10}(x)={\frac {1}{1024}}{\sqrt {\frac {969969}{\pi }}}\cdot e^{-10i\varphi }\cdot \sin ^{10}\theta }$
${\displaystyle Y_{10}^{-9}(x)={\frac {1}{512}}{\sqrt {\frac {4849845}{\pi }}}\cdot e^{-9i\varphi }\cdot \sin ^{9}\theta \cdot \cos \theta }$
${\displaystyle Y_{10}^{-8}(x)={\frac {1}{512}}{\sqrt {\frac {255255}{2\pi }}}\cdot e^{-8i\varphi }\cdot \sin ^{8}\theta \cdot (19\cos ^{2}\theta -1)}$
${\displaystyle Y_{10}^{-7}(x)={\frac {3}{512}}{\sqrt {\frac {85085}{\pi }}}\cdot e^{-7i\varphi }\cdot \sin ^{7}\theta \cdot (19\cos ^{3}\theta -3\cos \theta )}$
${\displaystyle Y_{10}^{-6}(x)={\frac {3}{1024}}{\sqrt {\frac {5005}{\pi }}}\cdot e^{-6i\varphi }\cdot \sin ^{6}\theta \cdot (323\cos ^{4}\theta -102\cos ^{2}\theta +3)}$
${\displaystyle Y_{10}^{-5}(x)={\frac {3}{256}}{\sqrt {\frac {1001}{\pi }}}\cdot e^{-5i\varphi }\cdot \sin ^{5}\theta \cdot (323\cos ^{5}\theta -170\cos ^{3}\theta +15\cos \theta )}$
${\displaystyle Y_{10}^{-4}(x)={\frac {3}{256}}{\sqrt {\frac {5005}{2\pi }}}\cdot e^{-4i\varphi }\cdot \sin ^{4}\theta \cdot (323\cos ^{6}\theta -255\cos ^{4}\theta +45\cos ^{2}\theta -1)}$
${\displaystyle Y_{10}^{-3}(x)={\frac {3}{256}}{\sqrt {\frac {5005}{\pi }}}\cdot e^{-3i\varphi }\cdot \sin ^{3}\theta \cdot (323\cos ^{7}\theta -357\cos ^{5}\theta +105\cos ^{3}\theta -7\cos \theta )}$
${\displaystyle Y_{10}^{-2}(x)={\frac {3}{512}}{\sqrt {\frac {385}{2\pi }}}\cdot e^{-2i\varphi }\cdot \sin ^{2}\theta \cdot (4199\cos ^{8}\theta -6188\cos ^{6}\theta +2730\cos ^{4}\theta -364\cos ^{2}\theta +7)}$
${\displaystyle Y_{10}^{-1}(x)={\frac {1}{256}}{\sqrt {\frac {1155}{2\pi }}}\cdot e^{-i\varphi }\cdot \sin \theta \cdot (4199\cos ^{9}\theta -7956\cos ^{7}\theta +4914\cos ^{5}\theta -1092\cos ^{3}\theta +63\cos \theta )}$
${\displaystyle Y_{10}^{0}(x)={\frac {1}{512}}{\sqrt {\frac {21}{\pi }}}\cdot (46189\cos ^{10}\theta -109395\cos ^{8}\theta +90090\cos ^{6}\theta -30030\cos ^{4}\theta +3465\cos ^{2}\theta -63)}$
${\displaystyle Y_{10}^{1}(x)=-{\frac {1}{256}}{\sqrt {\frac {1155}{2\pi }}}\cdot e^{i\varphi }\cdot \sin \theta \cdot (4199\cos ^{9}\theta -7956\cos ^{7}\theta +4914\cos ^{5}\theta -1092\cos ^{3}\theta +63\cos \theta )}$
${\displaystyle Y_{10}^{2}(x)={\frac {3}{512}}{\sqrt {\frac {385}{2\pi }}}\cdot e^{2i\varphi }\cdot \sin ^{2}\theta \cdot (4199\cos ^{8}\theta -6188\cos ^{6}\theta +2730\cos ^{4}\theta -364\cos ^{2}\theta +7)}$
${\displaystyle Y_{10}^{3}(x)=-{\frac {3}{256}}{\sqrt {\frac {5005}{\pi }}}\cdot e^{3i\varphi }\cdot \sin ^{3}\theta \cdot (323\cos ^{7}\theta -357\cos ^{5}\theta +105\cos ^{3}\theta -7\cos \theta )}$
${\displaystyle Y_{10}^{4}(x)={\frac {3}{256}}{\sqrt {\frac {5005}{2\pi }}}\cdot e^{4i\varphi }\cdot \sin ^{4}\theta \cdot (323\cos ^{6}\theta -255\cos ^{4}\theta +45\cos ^{2}\theta -1)}$
${\displaystyle Y_{10}^{5}(x)=-{\frac {3}{256}}{\sqrt {\frac {1001}{\pi }}}\cdot e^{5i\varphi }\cdot \sin ^{5}\theta \cdot (323\cos ^{5}\theta -170\cos ^{3}\theta +15\cos \theta )}$
${\displaystyle Y_{10}^{6}(x)={\frac {3}{1024}}{\sqrt {\frac {5005}{\pi }}}\cdot e^{6i\varphi }\cdot \sin ^{6}\theta \cdot (323\cos ^{4}\theta -102\cos ^{2}\theta +3)}$
${\displaystyle Y_{10}^{7}(x)=-{\frac {3}{512}}{\sqrt {\frac {85085}{\pi }}}\cdot e^{7i\varphi }\cdot \sin ^{7}\theta \cdot (19\cos ^{3}\theta -3\cos \theta )}$
${\displaystyle Y_{10}^{8}(x)={\frac {1}{512}}{\sqrt {\frac {255255}{2\pi }}}\cdot e^{8i\varphi }\cdot \sin ^{8}\theta \cdot (19\cos ^{2}\theta -1)}$
${\displaystyle Y_{10}^{9}(x)=-{\frac {1}{512}}{\sqrt {\frac {4849845}{\pi }}}\cdot e^{9i\varphi }\cdot \sin ^{9}\theta \cdot \cos \theta }$
${\displaystyle Y_{10}^{10}(x)={\frac {1}{1024}}{\sqrt {\frac {969969}{\pi }}}\cdot e^{10i\varphi }\cdot \sin ^{10}\theta }$

## 線型結合された球面調和関数

### l = 0

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

### l = 1

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

### l = 2

${\displaystyle Y_{2,-2}=d_{xy}=i{\sqrt {\frac {1}{2}}}\left(Y_{2}^{-2}-Y_{2}^{2}\right)={\frac {1}{2}}{\sqrt {\frac {15}{\pi }}}\cdot {\frac {xy}{r^{2}}}}$
${\displaystyle Y_{2,-1}=d_{yz}=i{\sqrt {\frac {1}{2}}}\left(Y_{2}^{-1}+Y_{2}^{1}\right)={\frac {1}{2}}{\sqrt {\frac {15}{\pi }}}\cdot {\frac {yz}{r^{2}}}}$
${\displaystyle Y_{20}=d_{z^{2}}=Y_{2}^{0}={\frac {1}{4}}{\sqrt {\frac {5}{\pi }}}\cdot {\frac {-x^{2}-y^{2}+2z^{2}}{r^{2}}}}$
${\displaystyle Y_{21}=d_{xz}={\sqrt {\frac {1}{2}}}\left(Y_{2}^{-1}-Y_{2}^{1}\right)={\frac {1}{2}}{\sqrt {\frac {15}{\pi }}}\cdot {\frac {zx}{r^{2}}}}$
${\displaystyle Y_{22}=d_{x^{2}-y^{2}}={\sqrt {\frac {1}{2}}}\left(Y_{2}^{-2}+Y_{2}^{2}\right)={\frac {1}{4}}{\sqrt {\frac {15}{\pi }}}\cdot {\frac {x^{2}-y^{2}}{r^{2}}}}$

### l = 3

${\displaystyle Y_{3,-3}=f_{y(3x^{2}-y^{2})}=i{\sqrt {\frac {1}{2}}}\left(Y_{3}^{-3}+Y_{3}^{3}\right)={\frac {1}{4}}{\sqrt {\frac {35}{2\pi }}}\cdot {\frac {\left(3x^{2}-y^{2}\right)y}{r^{3}}}}$
${\displaystyle Y_{3,-2}=f_{xyz}=i{\sqrt {\frac {1}{2}}}\left(Y_{3}^{-2}-Y_{3}^{2}\right)={\frac {1}{2}}{\sqrt {\frac {105}{\pi }}}\cdot {\frac {xyz}{r^{3}}}}$
${\displaystyle Y_{3,-1}=f_{yz^{2}}=i{\sqrt {\frac {1}{2}}}\left(Y_{3}^{-1}+Y_{3}^{1}\right)={\frac {1}{4}}{\sqrt {\frac {21}{2\pi }}}\cdot {\frac {y(4z^{2}-x^{2}-y^{2})}{r^{3}}}}$
${\displaystyle Y_{30}=f_{z^{3}}=Y_{3}^{0}={\frac {1}{4}}{\sqrt {\frac {7}{\pi }}}\cdot {\frac {z(2z^{2}-3x^{2}-3y^{2})}{r^{3}}}}$
${\displaystyle Y_{31}=f_{xz^{2}}={\sqrt {\frac {1}{2}}}\left(Y_{3}^{-1}-Y_{3}^{1}\right)={\frac {1}{4}}{\sqrt {\frac {21}{2\pi }}}\cdot {\frac {x(4z^{2}-x^{2}-y^{2})}{r^{3}}}}$
${\displaystyle Y_{32}=f_{z(x^{2}-y^{2})}={\sqrt {\frac {1}{2}}}\left(Y_{3}^{-2}+Y_{3}^{2}\right)={\frac {1}{4}}{\sqrt {\frac {105}{\pi }}}\cdot {\frac {\left(x^{2}-y^{2}\right)z}{r^{3}}}}$
${\displaystyle Y_{33}=f_{x(x^{2}-3y^{2})}={\sqrt {\frac {1}{2}}}\left(Y_{3}^{-3}-Y_{3}^{3}\right)={\frac {1}{4}}{\sqrt {\frac {35}{2\pi }}}\cdot {\frac {\left(x^{2}-3y^{2}\right)x}{r^{3}}}}$

### l = 4

${\displaystyle Y_{4,-4}=g_{xy(x^{2}-y^{2})}=i{\sqrt {\frac {1}{2}}}\left(Y_{4}^{-4}-Y_{4}^{4}\right)={\frac {3}{4}}{\sqrt {\frac {35}{\pi }}}\cdot {\frac {xy\left(x^{2}-y^{2}\right)}{r^{4}}}}$
${\displaystyle Y_{4,-3}=g_{zy^{3}}=i{\sqrt {\frac {1}{2}}}\left(Y_{4}^{-3}+Y_{4}^{3}\right)={\frac {3}{4}}{\sqrt {\frac {35}{2\pi }}}\cdot {\frac {(3x^{2}-y^{2})yz}{r^{4}}}}$
${\displaystyle Y_{4,-2}=g_{z^{2}xy}=i{\sqrt {\frac {1}{2}}}\left(Y_{4}^{-2}-Y_{4}^{2}\right)={\frac {3}{4}}{\sqrt {\frac {5}{\pi }}}\cdot {\frac {xy\cdot (7z^{2}-r^{2})}{r^{4}}}}$
${\displaystyle Y_{4,-1}=g_{z^{3}y}=i{\sqrt {\frac {1}{2}}}\left(Y_{4}^{-1}+Y_{4}^{1}\right)={\frac {3}{4}}{\sqrt {\frac {5}{2\pi }}}\cdot {\frac {yz\cdot (7z^{2}-3r^{2})}{r^{4}}}}$
${\displaystyle Y_{40}=g_{z^{4}}=Y_{4}^{0}={\frac {3}{16}}{\sqrt {\frac {1}{\pi }}}\cdot {\frac {(35z^{4}-30z^{2}r^{2}+3r^{4})}{r^{4}}}}$
${\displaystyle Y_{41}=g_{z^{3}x}={\sqrt {\frac {1}{2}}}\left(Y_{4}^{-1}-Y_{4}^{1}\right)={\frac {3}{4}}{\sqrt {\frac {5}{2\pi }}}\cdot {\frac {xz\cdot (7z^{2}-3r^{2})}{r^{4}}}}$
${\displaystyle Y_{42}=g_{z^{2}xy}={\sqrt {\frac {1}{2}}}\left(Y_{4}^{-2}+Y_{4}^{2}\right)={\frac {3}{8}}{\sqrt {\frac {5}{\pi }}}\cdot {\frac {(x^{2}-y^{2})\cdot (7z^{2}-r^{2})}{r^{4}}}}$
${\displaystyle Y_{43}=g_{zx^{3}}={\sqrt {\frac {1}{2}}}\left(Y_{4}^{-3}-Y_{4}^{3}\right)={\frac {3}{4}}{\sqrt {\frac {35}{2\pi }}}\cdot {\frac {(x^{2}-3y^{2})xz}{r^{4}}}}$
${\displaystyle Y_{44}=g_{x^{4}+y^{4}}={\sqrt {\frac {1}{2}}}\left(Y_{4}^{-4}+Y_{4}^{4}\right)={\frac {3}{16}}{\sqrt {\frac {35}{\pi }}}\cdot {\frac {x^{2}\left(x^{2}-3y^{2}\right)-y^{2}\left(3x^{2}-y^{2}\right)}{r^{4}}}}$

## 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.