球面調和関数表のソースを表示

以下は'''[[球面調和関数]]の表'''である。ただし、{{Mvar|x, y, z}} と {{Mvar|r, &theta;, &phi;}} との関係としては
:<math>\begin{align}x&=r\sin\theta\cos\varphi\\y&=r\sin\theta\sin\varphi\\z&=r\cos\theta\end{align}</math>
である。

== 球面調和関数 ==
{{Math|''l'' {{=}} 0}} から {{Math|''l'' {{=}} 5}} までは {{Harvtxt|Varshalovich|Moskalev|Khersonskii|1988}} を典拠としている。また、{{Math|''l'' {{=}} 0}} から {{Math|''l'' {{=}} 3}} までの {{Mvar|&theta;}} 形式での関数は [[#Reference-Mathworld-Spherical Harmonic|MathWorld]] でも確認できる。

=== ''l'' = 0 ===
:<math>Y_{0}^{0}(x)=\frac{1}{2}\sqrt{\frac{1}{\pi}}</math>

=== ''l'' = 1 ===
:<math>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}</math>
:<math>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}</math>
:<math>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}</math>

=== ''l'' = 2 ===
:<math>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}</math>
:<math>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}</math>
:<math>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}</math>
:<math>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}</math>
:<math>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}</math>

=== ''l'' = 3 ===
:<math>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}</math>
:<math>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}</math>
:<math>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}</math>
:<math>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}</math>
:<math>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}</math>
:<math>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}</math>
:<math>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}</math>

=== ''l'' = 4 ===
:<math>Y_{4}^{-4}(x)=\frac{3}{16}\sqrt{\frac{35}{2\pi}}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta</math>
:<math>Y_{4}^{-3}(x)=\frac{3}{8}\sqrt{\frac{35}{\pi}}\cdot e^{-3i\varphi}\cdot\sin^{3}\theta\cdot\cos\theta</math>
:<math>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)</math>
:<math>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)</math>
:<math>Y_{4}^{0}(x)=\frac{3}{16}\sqrt{\frac{1}{\pi}}\cdot(35\cos^{4}\theta-30\cos^{2}\theta+3)</math>
:<math>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)</math>
:<math>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)</math>
:<math>Y_{4}^{3}(x)=-\frac{3}{8}\sqrt{\frac{35}{\pi}}\cdot e^{3i\varphi}\cdot\sin^{3}\theta\cdot\cos\theta</math>
:<math>Y_{4}^{4}(x)=\frac{3}{16}\sqrt{\frac{35}{2\pi}}\cdot e^{4i\varphi}\cdot\sin^{4}\theta</math>

=== ''l'' = 5 ===
:<math>Y_{5}^{-5}(x)=\frac{3}{32}\sqrt{\frac{77}{\pi}}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta</math>
:<math>Y_{5}^{-4}(x)=\frac{3}{16}\sqrt{\frac{385}{2\pi}}\cdot e^{-4i\varphi}\cdot\sin^{4}\theta\cdot\cos\theta</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>Y_{5}^{0}(x)=\frac{1}{16}\sqrt{\frac{11}{\pi}}\cdot(63\cos^{5}\theta-70\cos^{3}\theta+15\cos\theta)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>Y_{5}^{4}(x)=\frac{3}{16}\sqrt{\frac{385}{2\pi}}\cdot e^{4i\varphi}\cdot\sin^{4}\theta\cdot\cos\theta</math>
:<math>Y_{5}^{5}(x)=-\frac{3}{32}\sqrt{\frac{77}{\pi}}\cdot e^{5i\varphi}\cdot\sin^{5}\theta</math>

=== ''l'' = 6 ===
:<math>Y_{6}^{-6}(x)=\frac{1}{64}\sqrt{\frac{3003}{\pi}}\cdot e^{-6i\varphi}\cdot\sin^{6}\theta</math>
:<math>Y_{6}^{-5}(x)=\frac{3}{32}\sqrt{\frac{1001}{\pi}}\cdot e^{-5i\varphi}\cdot\sin^{5}\theta\cdot\cos\theta</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>Y_{6}^{5}(x)=-\frac{3}{32}\sqrt{\frac{1001}{\pi}}\cdot e^{5i\varphi}\cdot\sin^{5}\theta\cdot\cos\theta</math>
:<math>Y_{6}^{6}(x)=\frac{1}{64}\sqrt{\frac{3003}{\pi}}\cdot e^{6i\varphi}\cdot\sin^{6}\theta</math>

=== ''l'' = 7 ===
:<math>Y_{7}^{-7}(x)=\frac{3}{64}\sqrt{\frac{715}{2\pi}}\cdot e^{-7i\varphi}\cdot\sin^{7}\theta</math>
:<math>Y_{7}^{-6}(x)=\frac{3}{64}\sqrt{\frac{5005}{\pi}}\cdot e^{-6i\varphi}\cdot\sin^{6}\theta\cdot\cos\theta</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>Y_{7}^{6}(x)=\frac{3}{64}\sqrt{\frac{5005}{\pi}}\cdot e^{6i\varphi}\cdot\sin^{6}\theta\cdot\cos\theta</math>
:<math>Y_{7}^{7}(x)=-\frac{3}{64}\sqrt{\frac{715}{2\pi}}\cdot e^{7i\varphi}\cdot\sin^{7}\theta</math>

=== ''l'' = 8 ===
:<math>Y_{8}^{-8}(x)=\frac{3}{256}\sqrt{\frac{12155}{2\pi}}\cdot e^{-8i\varphi}\cdot\sin^{8}\theta</math>
:<math>Y_{8}^{-7}(x)=\frac{3}{64}\sqrt{\frac{12155}{2\pi}}\cdot e^{-7i\varphi}\cdot\sin^{7}\theta\cdot\cos\theta</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>Y_{8}^{7}(x)=-\frac{3}{64}\sqrt{\frac{12155}{2\pi}}\cdot e^{7i\varphi}\cdot\sin^{7}\theta\cdot\cos\theta</math>
:<math>Y_{8}^{8}(x)=\frac{3}{256}\sqrt{\frac{12155}{2\pi}}\cdot e^{8i\varphi}\cdot\sin^{8}\theta</math>

=== ''l'' = 9 ===
:<math>Y_{9}^{-9}(x)=\frac{1}{512}\sqrt{\frac{230945}{\pi}}\cdot e^{-9i\varphi}\cdot\sin^{9}\theta</math>
:<math>Y_{9}^{-8}(x)=\frac{3}{256}\sqrt{\frac{230945}{2\pi}}\cdot e^{-8i\varphi}\cdot\sin^{8}\theta\cdot\cos\theta</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>Y_{9}^{8}(x)=\frac{3}{256}\sqrt{\frac{230945}{2\pi}}\cdot e^{8i\varphi}\cdot\sin^{8}\theta\cdot\cos\theta</math>
:<math>Y_{9}^{9}(x)=-\frac{1}{512}\sqrt{\frac{230945}{\pi}}\cdot e^{9i\varphi}\cdot\sin^{9}\theta</math>

=== ''l'' = 10 ===
:<math>Y_{10}^{-10}(x)=\frac{1}{1024}\sqrt{\frac{969969}{\pi}}\cdot e^{-10i\varphi}\cdot\sin^{10}\theta</math>
:<math>Y_{10}^{-9}(x)=\frac{1}{512}\sqrt{\frac{4849845}{\pi}}\cdot e^{-9i\varphi}\cdot\sin^{9}\theta\cdot\cos\theta</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>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)</math>
:<math>Y_{10}^{9}(x)=-\frac{1}{512}\sqrt{\frac{4849845}{\pi}}\cdot e^{9i\varphi}\cdot\sin^{9}\theta\cdot\cos\theta</math>
:<math>Y_{10}^{10}(x)=\frac{1}{1024}\sqrt{\frac{969969}{\pi}}\cdot e^{10i\varphi}\cdot\sin^{10}\theta</math>

== 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。{{Math|''l'' {{=}} 0}} から {{Math|''l'' {{=}} 2}} までは {{Harvtxt|Chisholm|1976}} 及び {{Harvtxt|Blanco|Flórez|Bermejo|1996}} を、{{Math|''l'' {{=}} 3}} は {{Harvtxt|Chisholm|1976}} のみを典拠としている。

=== ''l'' = 0 ===
:<math>Y_{00}=s=Y_0^0=\frac{1}{2}\sqrt{\frac{1}{\pi}}</math>

=== ''l'' = 1 ===
:<math>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}</math>
:<math>Y_{10}=p_z=Y_1^0=\sqrt{\frac{3}{4\pi}}\cdot\frac{z}{r}</math>
:<math>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}</math>

=== ''l'' = 2 ===
:<math>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}</math>
:<math>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}</math>
:<math>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}</math>
:<math>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}</math>
:<math>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}</math>

=== ''l'' = 3 ===
:<math>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}</math>
:<math>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}</math>
:<math>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}</math>
:<math>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}</math>
:<math>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}</math>
:<math>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}</math>
:<math>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}</math>

===''l'' = 4===
:<math>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}</math>
:<math>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}</math>
:<math>Y_{4,-2}=g_{z^2xy}=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}</math>
:<math>Y_{4,-1}=g_{z^3y}=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}</math>
:<math>Y_{40}=g_{z^4}=Y_4^0=\frac{3}{16}\sqrt{\frac{1}{\pi}}\cdot\frac{(35z^4-30z^2r^2+3r^4)}{r^4}</math>
:<math>Y_{41}=g_{z^3x}=\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}</math>
:<math>Y_{42}=g_{z^2xy}=\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}</math>
:<math>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}</math>
:<math>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}</math>

== 参考文献 ==
=== 原論文 ===
* {{cite journal|last=Blanco|first=Miguel A.|last2=Flórez|first2=M.|last3=Bermejo|first3=M.|url=http://ac.els-cdn.com/S0166128097001851/1-s2.0-S0166128097001851-main.pdf?_tid=9b5146ea-a25c-11e6-9353-00000aab0f02&acdnat=1478243110_840f06f35087d21e1c1809eb2de68a1e|title=Evaluation of the rotation matrices in the basis of real spherical harmonics|format=[[Portable Document Format|PDF]]|journal={{enlink|Journal of Molecular Structure|Journal of Molecular Structure: THEOCHEM|p=off|s=off}}|publisher=[[エルゼビア|Elsevier]] [[ScienceDirect]]|location=[[アムステルダム|Amsterdam]]|date=1 November 1996|volume=419|issue=1–3|pages=19–27|issn=0022-2860|oclc=224506237|doi=10.1016/S0166-1280(97)00185-1|ref=harv}}

=== 書籍 ===
* {{cite book|title=Group theoretical techniques in quantum chemistry|series=Theoretical chemistry: a series of monographs|volume=5|edition=1st|date=March 8, 1976|publisher={{enlink|Academic Press|p=off|s=off}}|location=[[ニューヨーク|New York]]|lccn=75027326|ncid=BA03187896|asin=0121729508|oclc=3104116|isbn=0-12-172950-8|first=C.D.H.|last=Chisholm|ref=harv}}
* {{cite book|title=Quantum theory of angular momentum : irreducible tensors, spherical harmonics, vector coupling coefficients, 3nj symbols|date=October 1, 1988|publisher={{enlink|World Scientific|World Scientific Pub.|p=off|s=off}}|location=[[シンガポール|Singapore]]|lccn=86009279|ncid=BA04808445|asin=9971501074|oclc=13525826|isbn=9971-50-107-4|first1=D. A.|last1=Varshalovich|first2=A. N.|last2=Moskalev|first3=V. K.|last3=Khersonskii|edition=1. repr.|pages=155–156|ref=harv}}

==関連項目==
* [[球面調和関数]]
* {{enlink|Category:Special hypergeometric functions|p=off|s=off}}

== 外部リンク ==
* {{MathWorld|title=Spherical Harmonic|urlname=SphericalHarmonic}}

{{DEFAULTSORT:きゆうめんちようわかんすうひよう}}
[[Category:特殊関数]]
[[Category:量子力学]]
[[Category:数学に関する記事]]