球面調和関数表
以下は球面調和関数の表である。ただし、x, y, z と r, θ, φ との関係としては
\begin{align}x&=r\sin\theta\cos\varphi\\y&=r\sin\theta\sin\varphi\\z&=r\cos\theta\end{align}
である。
l  0 から l  5 までは  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。
Y_{0}^{0}(x)=\frac{1}{2}\sqrt{\frac{1}{\pi}
Y_{1}^{-1}(x)=\frac{1}{2}\sqrt{\frac{3}{2\pi}\cdot e^{-i\varphi}\cdot\sin\theta=\frac{1}{2}\sqrt{\frac{3}{2\pi}\cdot\frac{x-iy}{r}
Y_{1}^{0}(x)=\frac{1}{2}\sqrt{\frac{3}{\pi}\cdot\cos\theta=\frac{1}{2}\sqrt{\frac{3}{\pi}\cdot\frac{z}{r}
Y_{1}^{1}(x)=-\frac{1}{2}\sqrt{\frac{3}{2\pi}\cdot e^{i\varphi}\cdot\sin\theta=-\frac{1}{2}\sqrt{\frac{3}{2\pi}\cdot\frac{x+iy}{r}
Y_{2}^{-2}(x)=\frac{1}{4}\sqrt{\frac{15}{2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta=\frac{1}{4}\sqrt{\frac{15}{2\pi}\cdot\frac{x^2-2ixy-y^2}{r^2}
Y_{2}^{-1}(x)=\frac{1}{2}\sqrt{\frac{15}{2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot\cos\theta=\frac{1}{2}\sqrt{\frac{15}{2\pi}\cdot\frac{xz-iyz}{r^2}
Y_{2}^{0}(x)=\frac{1}{4}\sqrt{\frac{5}{\pi}\cdot(3\cos^{2}\theta-1)=\frac{1}{4}\sqrt{\frac{5}{\pi}\cdot\frac{-x^2-y^2+2z^2}{r^2}
Y_{2}^{1}(x)=-\frac{1}{2}\sqrt{\frac{15}{2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot\cos\theta=-\frac{1}{2}\sqrt{\frac{15}{2\pi}\cdot\frac{xz+iyz}{r^2}
Y_{2}^{2}(x)=\frac{1}{4}\sqrt{\frac{15}{2\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta=\frac{1}{4}\sqrt{\frac{15}{2\pi}\cdot\frac{x^2+2ixy-y^2}{r^2}
Y_{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^2y-3xy^2+iy^3}{r^3}
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^2z-2ixyz-y^2z}{r^3}
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^2y-xy^2+4xz^2+iy^3-4iyz^2}{r^3}
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^2z-3y^2z+2z^3}{r^3}
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^2y-xy^2+4xz^2-iy^3+4iyz^2}{r^3}
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^2z+2ixyz-y^2z}{r^3}
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^2y-3xy^2-iy^3}{r^3}
Y_{4}^{-4}(x)=\frac{3}{16}\sqrt{\frac{35}{2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta
Y_{4}^{-3}(x)=\frac{3}{8}\sqrt{\frac{35}{\pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot\cos\theta
Y_{4}^{-2}(x)=\frac{3}{8}\sqrt{\frac{5}{2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(7\cos^{2}\theta-1)
Y_{4}^{-1}(x)=\frac{3}{8}\sqrt{\frac{5}{\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(7\cos^{3}\theta-3\cos\theta)
Y_{4}^{0}(x)=\frac{3}{16}\sqrt{\frac{1}{\pi}\cdot(35\cos^{4}\theta-30\cos^{2}\theta+3)
Y_{4}^{1}(x)=-\frac{3}{8}\sqrt{\frac{5}{\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(7\cos^{3}\theta-3\cos\theta)
Y_{4}^{2}(x)=\frac{3}{8}\sqrt{\frac{5}{2\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot(7\cos^{2}\theta-1)
Y_{4}^{3}(x)=-\frac{3}{8}\sqrt{\frac{35}{\pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\cdot\cos\theta
Y_{4}^{4}(x)=\frac{3}{16}\sqrt{\frac{35}{2\pi}\cdot e^{4i\varphi}\cdot\sin^{4}\theta
Y_{5}^{-5}(x)=\frac{3}{32}\sqrt{\frac{77}{\pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta
Y_{5}^{-4}(x)=\frac{3}{16}\sqrt{\frac{385}{2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot\cos\theta
Y_{5}^{-3}(x)=\frac{1}{32}\sqrt{\frac{385}{\pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(9\cos^{2}\theta-1)
Y_{5}^{-2}(x)=\frac{1}{8}\sqrt{\frac{1155}{2\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(3\cos^{3}\theta-1\cos\theta)
Y_{5}^{-1}(x)=\frac{1}{16}\sqrt{\frac{165}{2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(21\cos^{4}\theta-14\cos^{2}\theta+1)
Y_{5}^{0}(x)=\frac{1}{16}\sqrt{\frac{11}{\pi}\cdot(63\cos^{5}\theta-70\cos^{3}\theta+15\cos\theta)
Y_{5}^{1}(x)=-\frac{1}{16}\sqrt{\frac{165}{2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(21\cos^{4}\theta-14\cos^{2}\theta+1)
Y_{5}^{2}(x)=\frac{1}{8}\sqrt{\frac{1155}{2\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot(3\cos^{3}\theta-1\cos\theta)
Y_{5}^{3}(x)=-\frac{1}{32}\sqrt{\frac{385}{\pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\cdot(9\cos^{2}\theta-1)
Y_{5}^{4}(x)=\frac{3}{16}\sqrt{\frac{385}{2\pi}\cdot e^{4i\varphi}\cdot\sin^{4}\theta\cdot\cos\theta
Y_{5}^{5}(x)=-\frac{3}{32}\sqrt{\frac{77}{\pi}\cdot e^{5i\varphi}\cdot\sin^{5}\theta
Y_{6}^{-6}(x)=\frac{1}{64}\sqrt{\frac{3003}{\pi}\cdot e^{-6i\varphi}\cdot\sin^{6}\theta
Y_{6}^{-5}(x)=\frac{3}{32}\sqrt{\frac{1001}{\pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\cdot\cos\theta
Y_{6}^{-4}(x)=\frac{3}{32}\sqrt{\frac{91}{2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot(11\cos^{2}\theta-1)
Y_{6}^{-3}(x)=\frac{1}{32}\sqrt{\frac{1365}{\pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(11\cos^{3}\theta-3\cos\theta)
Y_{6}^{-2}(x)=\frac{1}{64}\sqrt{\frac{1365}{\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(33\cos^{4}\theta-18\cos^{2}\theta+1)
Y_{6}^{-1}(x)=\frac{1}{16}\sqrt{\frac{273}{2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(33\cos^{5}\theta-30\cos^{3}\theta+5\cos\theta)
Y_{6}^{0}(x)=\frac{1}{32}\sqrt{\frac{13}{\pi}\cdot(231\cos^{6}\theta-315\cos^{4}\theta+105\cos^{2}\theta-5)
Y_{6}^{1}(x)=-\frac{1}{16}\sqrt{\frac{273}{2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(33\cos^{5}\theta-30\cos^{3}\theta+5\cos\theta)
Y_{6}^{2}(x)=\frac{1}{64}\sqrt{\frac{1365}{\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot(33\cos^{4}\theta-18\cos^{2}\theta+1)
Y_{6}^{3}(x)=-\frac{1}{32}\sqrt{\frac{1365}{\pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\cdot(11\cos^{3}\theta-3\cos\theta)
Y_{6}^{4}(x)=\frac{3}{32}\sqrt{\frac{91}{2\pi}\cdot e^{4i\varphi}\cdot\sin^{4}\theta\cdot(11\cos^{2}\theta-1)
Y_{6}^{5}(x)=-\frac{3}{32}\sqrt{\frac{1001}{\pi}\cdot e^{5i\varphi}\cdot\sin^{5}\theta\cdot\cos\theta
Y_{6}^{6}(x)=\frac{1}{64}\sqrt{\frac{3003}{\pi}\cdot e^{6i\varphi}\cdot\sin^{6}\theta
Y_{7}^{-7}(x)=\frac{3}{64}\sqrt{\frac{715}{2\pi}\cdot e^{-7i\varphi}\cdot\sin^{7}\theta
Y_{7}^{-6}(x)=\frac{3}{64}\sqrt{\frac{5005}{\pi}\cdot e^{-6i\varphi}\cdot\sin^{6}\theta\cdot\cos\theta
Y_{7}^{-5}(x)=\frac{3}{64}\sqrt{\frac{385}{2\pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\cdot(13\cos^{2}\theta-1)
Y_{7}^{-4}(x)=\frac{3}{32}\sqrt{\frac{385}{2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot(13\cos^{3}\theta-3\cos\theta)
Y_{7}^{-3}(x)=\frac{3}{64}\sqrt{\frac{35}{2\pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(143\cos^{4}\theta-66\cos^{2}\theta+3)
Y_{7}^{-2}(x)=\frac{3}{64}\sqrt{\frac{35}{\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(143\cos^{5}\theta-110\cos^{3}\theta+15\cos\theta)
Y_{7}^{-1}(x)=\frac{1}{64}\sqrt{\frac{105}{2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(429\cos^{6}\theta-495\cos^{4}\theta+135\cos^{2}\theta-5)
Y_{7}^{0}(x)=\frac{1}{32}\sqrt{\frac{15}{\pi}\cdot(429\cos^{7}\theta-693\cos^{5}\theta+315\cos^{3}\theta-35\cos\theta)
Y_{7}^{1}(x)=-\frac{1}{64}\sqrt{\frac{105}{2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(429\cos^{6}\theta-495\cos^{4}\theta+135\cos^{2}\theta-5)
Y_{7}^{2}(x)=\frac{3}{64}\sqrt{\frac{35}{\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot(143\cos^{5}\theta-110\cos^{3}\theta+15\cos\theta)
Y_{7}^{3}(x)=-\frac{3}{64}\sqrt{\frac{35}{2\pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\cdot(143\cos^{4}\theta-66\cos^{2}\theta+3)
Y_{7}^{4}(x)=\frac{3}{32}\sqrt{\frac{385}{2\pi}\cdot e^{4i\varphi}\cdot\sin^{4}\theta\cdot(13\cos^{3}\theta-3\cos\theta)
Y_{7}^{5}(x)=-\frac{3}{64}\sqrt{\frac{385}{2\pi}\cdot e^{5i\varphi}\cdot\sin^{5}\theta\cdot(13\cos^{2}\theta-1)
Y_{7}^{6}(x)=\frac{3}{64}\sqrt{\frac{5005}{\pi}\cdot e^{6i\varphi}\cdot\sin^{6}\theta\cdot\cos\theta
Y_{7}^{7}(x)=-\frac{3}{64}\sqrt{\frac{715}{2\pi}\cdot e^{7i\varphi}\cdot\sin^{7}\theta
Y_{8}^{-8}(x)=\frac{3}{256}\sqrt{\frac{12155}{2\pi}\cdot e^{-8i\varphi}\cdot\sin^{8}\theta
Y_{8}^{-7}(x)=\frac{3}{64}\sqrt{\frac{12155}{2\pi}\cdot e^{-7i\varphi}\cdot\sin^{7}\theta\cdot\cos\theta
Y_{8}^{-6}(x)=\frac{1}{128}\sqrt{\frac{7293}{\pi}\cdot e^{-6i\varphi}\cdot\sin^{6}\theta\cdot(15\cos^{2}\theta-1)
Y_{8}^{-5}(x)=\frac{3}{64}\sqrt{\frac{17017}{2\pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\cdot(5\cos^{3}\theta-1\cos\theta)
Y_{8}^{-4}(x)=\frac{3}{128}\sqrt{\frac{1309}{2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot(65\cos^{4}\theta-26\cos^{2}\theta+1)
Y_{8}^{-3}(x)=\frac{1}{64}\sqrt{\frac{19635}{2\pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(39\cos^{5}\theta-26\cos^{3}\theta+3\cos\theta)
Y_{8}^{-2}(x)=\frac{3}{128}\sqrt{\frac{595}{\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(143\cos^{6}\theta-143\cos^{4}\theta+33\cos^{2}\theta-1)
Y_{8}^{-1}(x)=\frac{3}{64}\sqrt{\frac{17}{2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(715\cos^{7}\theta-1001\cos^{5}\theta+385\cos^{3}\theta-35\cos\theta)
Y_{8}^{0}(x)=\frac{1}{256}\sqrt{\frac{17}{\pi}\cdot(6435\cos^{8}\theta-12012\cos^{6}\theta+6930\cos^{4}\theta-1260\cos^{2}\theta+35)
Y_{8}^{1}(x)=-\frac{3}{64}\sqrt{\frac{17}{2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(715\cos^{7}\theta-1001\cos^{5}\theta+385\cos^{3}\theta-35\cos\theta)
Y_{8}^{2}(x)=\frac{3}{128}\sqrt{\frac{595}{\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot(143\cos^{6}\theta-143\cos^{4}\theta+33\cos^{2}\theta-1)
Y_{8}^{3}(x)=-\frac{1}{64}\sqrt{\frac{19635}{2\pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\cdot(39\cos^{5}\theta-26\cos^{3}\theta+3\cos\theta)
Y_{8}^{4}(x)=\frac{3}{128}\sqrt{\frac{1309}{2\pi}\cdot e^{4i\varphi}\cdot\sin^{4}\theta\cdot(65\cos^{4}\theta-26\cos^{2}\theta+1)
Y_{8}^{5}(x)=-\frac{3}{64}\sqrt{\frac{17017}{2\pi}\cdot e^{5i\varphi}\cdot\sin^{5}\theta\cdot(5\cos^{3}\theta-1\cos\theta)
Y_{8}^{6}(x)=\frac{1}{128}\sqrt{\frac{7293}{\pi}\cdot e^{6i\varphi}\cdot\sin^{6}\theta\cdot(15\cos^{2}\theta-1)
Y_{8}^{7}(x)=-\frac{3}{64}\sqrt{\frac{12155}{2\pi}\cdot e^{7i\varphi}\cdot\sin^{7}\theta\cdot\cos\theta
Y_{8}^{8}(x)=\frac{3}{256}\sqrt{\frac{12155}{2\pi}\cdot e^{8i\varphi}\cdot\sin^{8}\theta
Y_{9}^{-9}(x)=\frac{1}{512}\sqrt{\frac{230945}{\pi}\cdot e^{-9i\varphi}\cdot\sin^{9}\theta
Y_{9}^{-8}(x)=\frac{3}{256}\sqrt{\frac{230945}{2\pi}\cdot e^{-8i\varphi}\cdot\sin^{8}\theta\cdot\cos\theta
Y_{9}^{-7}(x)=\frac{3}{512}\sqrt{\frac{13585}{\pi}\cdot e^{-7i\varphi}\cdot\sin^{7}\theta\cdot(17\cos^{2}\theta-1)
Y_{9}^{-6}(x)=\frac{1}{128}\sqrt{\frac{40755}{\pi}\cdot e^{-6i\varphi}\cdot\sin^{6}\theta\cdot(17\cos^{3}\theta-3\cos\theta)
Y_{9}^{-5}(x)=\frac{3}{256}\sqrt{\frac{2717}{\pi}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\cdot(85\cos^{4}\theta-30\cos^{2}\theta+1)
Y_{9}^{-4}(x)=\frac{3}{128}\sqrt{\frac{95095}{2\pi}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot(17\cos^{5}\theta-10\cos^{3}\theta+1\cos\theta)
Y_{9}^{-3}(x)=\frac{1}{256}\sqrt{\frac{21945}{\pi}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot(221\cos^{6}\theta-195\cos^{4}\theta+39\cos^{2}\theta-1)
Y_{9}^{-2}(x)=\frac{3}{128}\sqrt{\frac{1045}{\pi}\cdot e^{-2i\varphi}\cdot\sin^{2}\theta\cdot(221\cos^{7}\theta-273\cos^{5}\theta+91\cos^{3}\theta-7\cos\theta)
Y_{9}^{-1}(x)=\frac{3}{256}\sqrt{\frac{95}{2\pi}\cdot e^{-i\varphi}\cdot\sin\theta\cdot(2431\cos^{8}\theta-4004\cos^{6}\theta+2002\cos^{4}\theta-308\cos^{2}\theta+7)
Y_{9}^{0}(x)=\frac{1}{256}\sqrt{\frac{19}{\pi}\cdot(12155\cos^{9}\theta-25740\cos^{7}\theta+18018\cos^{5}\theta-4620\cos^{3}\theta+315\cos\theta)
Y_{9}^{1}(x)=-\frac{3}{256}\sqrt{\frac{95}{2\pi}\cdot e^{i\varphi}\cdot\sin\theta\cdot(2431\cos^{8}\theta-4004\cos^{6}\theta+2002\cos^{4}\theta-308\cos^{2}\theta+7)
Y_{9}^{2}(x)=\frac{3}{128}\sqrt{\frac{1045}{\pi}\cdot e^{2i\varphi}\cdot\sin^{2}\theta\cdot(221\cos^{7}\theta-273\cos^{5}\theta+91\cos^{3}\theta-7\cos\theta)
Y_{9}^{3}(x)=-\frac{1}{256}\sqrt{\frac{21945}{\pi}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\cdot(221\cos^{6}\theta-195\cos^{4}\theta+39\cos^{2}\theta-1)
Y_{9}^{4}(x)=\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)