Editing Associated Legendre polynomials (section)

==The first few associated Legendre functions==
[[File:Mplwp legendreP04a0.svg|thumb|300px|Associated Legendre functions for ''m'' = 0]]
[[File:Mplwp legendreP15a1.svg|thumb|300px|Associated Legendre functions for ''m'' = 1]]
[[File:Mplwp legendreP26a2.svg|thumb|300px|Associated Legendre functions for ''m'' = 2]]
The first few associated Legendre functions, including those for negative values of ''m'', are:

<math display="block">P_{0}^{0}(x)=1</math>

<math display="block">\begin{align}
P_{1}^{-1}(x)&=-\tfrac{1}{2}P_{1}^{1}(x) \\
P_{1}^{0}(x)&=x \\
P_{1}^{1}(x)&=-(1-x^2)^{1/2}
\end{align}</math>

<math display="block">\begin{align}
P_{2}^{-2}(x)&=\tfrac{1}{24}P_{2}^{2}(x) \\
P_{2}^{-1}(x)&=-\tfrac{1}{6}P_{2}^{1}(x) \\
P_{2}^{0}(x)&=\tfrac{1}{2}(3x^{2}-1) \\
P_{2}^{1}(x)&=-3x(1-x^2)^{1/2} \\
P_{2}^{2}(x)&=3(1-x^2)
\end{align}</math>

<math display="block">\begin{align}
P_{3}^{-3}(x)&=-\tfrac{1}{720}P_{3}^{3}(x) \\
P_{3}^{-2}(x)&=\tfrac{1}{120}P_{3}^{2}(x) \\
P_{3}^{-1}(x)&=-\tfrac{1}{12}P_{3}^{1}(x) \\
P_{3}^{0}(x)&=\tfrac{1}{2}(5x^3-3x) \\
P_{3}^{1}(x)&=\tfrac{3}{2}(1-5x^{2})(1-x^2)^{1/2} \\
P_{3}^{2}(x)&=15x(1-x^2) \\
P_{3}^{3}(x)&=-15(1-x^2)^{3/2}
\end{align}</math>

<math display="block">\begin{align}
P_{4}^{-4}(x)&=\tfrac{1}{40320}P_{4}^{4}(x) \\
P_{4}^{-3}(x)&=-\tfrac{1}{5040}P_{4}^{3}(x) \\
P_{4}^{-2}(x)&=\tfrac{1}{360}P_{4}^{2}(x) \\
P_{4}^{-1}(x)&=-\tfrac{1}{20}P_{4}^{1}(x) \\
P_{4}^{0}(x)&=\tfrac{1}{8}(35x^{4}-30x^{2}+3) \\
P_{4}^{1}(x)&=-\tfrac{5}{2}(7x^3-3x)(1-x^2)^{1/2} \\
P_{4}^{2}(x)&=\tfrac{15}{2}(7x^2-1)(1-x^2) \\
P_{4}^{3}(x)&= - 105x(1-x^2)^{3/2} \\
P_{4}^{4}(x)&=105(1-x^2)^{2}
\end{align}</math>
<!--
<math display="block">\begin{align}
P_{5}^{-5}(x)&={1\over 3840}\left(\sqrt{1-x^2}\right)^{5} \\
P_{5}^{-4}(x)&={1\over 384}\left(\sqrt{1-x^2}\right)^{4}x \\
P_{5}^{-3}(x)&={1\over 384}\left(\sqrt{1-x^2}\right)^{3}(9x^{2}-1) \\
P_{5}^{-2}(x)&={1\over 16}\left(\sqrt{1-x^2}\right)^{2}(3x^{3}-1x) \\
P_{5}^{-1}(x)&={1\over 16}\left(\sqrt{1-x^2}\right)(21x^{4}-14x^{2}+1) \\
P_{5}^{0}(x)&={1\over 8}(63x^{5}-70x^{3}+15x) \\
P_{5}^{1}(x)&={-15\over 8}\left(\sqrt{1-x^2}\right)(21x^{4}-14x^{2}+1) \\
P_{5}^{2}(x)&={105\over 2}\left(\sqrt{1-x^2}\right)^{2}(3x^{3}-1x) \\
P_{5}^{3}(x)&={-105\over 2}\left(\sqrt{1-x^2}\right)^{3}(9x^{2}-1) \\
P_{5}^{4}(x)&=945\left(\sqrt{1-x^2}\right)^{4}x \\
P_{5}^{5}(x)&=-945\left(\sqrt{1-x^2}\right)^{5}
\end{align}</math>

<math display="block">\begin{align}
P_{6}^{-6}(x)={1\over 46080}\left(\sqrt{1-x^2}\right)^{6} \\
P_{6}^{-5}(x)={1\over 3840}\left(\sqrt{1-x^2}\right)^{5}x \\
P_{6}^{-4}(x)={1\over 3840}\left(\sqrt{1-x^2}\right)^{4}(11x^{2}-1) \\
P_{6}^{-3}(x)={1\over 384}\left(\sqrt{1-x^2}\right)^{3}(11x^{3}-3x) \\
P_{6}^{-2}(x)={1\over 128}\left(\sqrt{1-x^2}\right)^{2}(33x^{4}-18x^{2}+1) \\
P_{6}^{-1}(x)={1\over 16}\left(\sqrt{1-x^2}\right)(33x^{5}-30x^{3}+5x) \\
P_{6}^{0}(x)={1\over 16}(231x^{6}-315x^{4}+105x^{2}-5) \\
P_{6}^{1}(x)={-21\over 8}\left(\sqrt{1-x^2}\right)(33x^{5}-30x^{3}+5x) \\
P_{6}^{2}(x)={105\over 8}\left(\sqrt{1-x^2}\right)^{2}(33x^{4}-18x^{2}+1) \\
P_{6}^{3}(x)={-315\over 2}\left(\sqrt{1-x^2}\right)^{3}(11x^{3}-3x) \\
P_{6}^{4}(x)={945\over 2}\left(\sqrt{1-x^2}\right)^{4}(11x^{2}-1) \\
P_{6}^{5}(x)=-10395\left(\sqrt{1-x^2}\right)^{5}x \\
P_{6}^{6}(x)=10395\left(\sqrt{1-x^2}\right)^{6}
\end{align}</math>

<math display="block">\begin{align}
P_{7}^{-7}(x)&={1\over 645120}\left(\sqrt{1-x^2}\right)^{7} \\
P_{7}^{-6}(x)&={1\over 46080}\left(\sqrt{1-x^2}\right)^{6}x \\
P_{7}^{-5}(x)&={1\over 46080}\left(\sqrt{1-x^2}\right)^{5}(13x^{2}-1) \\
P_{7}^{-4}(x)&={1\over 3840}\left(\sqrt{1-x^2}\right)^{4}(13x^{3}-3x) \\
P_{7}^{-3}(x)&={1\over 3840}\left(\sqrt{1-x^2}\right)^{3}(143x^{4}-66x^{2}+3) \\
P_{7}^{-2}(x)&={1\over 384}\left(\sqrt{1-x^2}\right)^{2}(143x^{5}-110x^{3}+15x) \\
P_{7}^{-1}(x)&={1\over 128}\left(\sqrt{1-x^2}\right)(429x^{6}-495x^{4}+135x^{2}-5) \\
P_{7}^{0}(x)&={1\over 16}(429x^{7}-693x^{5}+315x^{3}-35x) \\
P_{7}^{1}(x)&={-7\over 16}\left(\sqrt{1-x^2}\right)(429x^{6}-495x^{4}+135x^{2}-5) \\
P_{7}^{2}(x)&={63\over 8}\left(\sqrt{1-x^2}\right)^{2}(143x^{5}-110x^{3}+15x) \\
P_{7}^{3}(x)&={-315\over 8}\left(\sqrt{1-x^2}\right)^{3}(143x^{4}-66x^{2}+3) \\
P_{7}^{4}(x)&={3465\over 2}\left(\sqrt{1-x^2}\right)^{4}(13x^{3}-3x) \\
P_{7}^{5}(x)&={-10395\over 2}\left(\sqrt{1-x^2}\right)^{5}(13x^{2}-1) \\
P_{7}^{6}(x)&=135135\left(\sqrt{1-x^2}\right)^{6}x \\
P_{7}^{7}(x)&=-135135\left(\sqrt{1-x^2}\right)^{7}
\end{align}</math>

<math display="block">\begin{align}
P_{8}^{-8}(x)={1\over 10321920}\left(\sqrt{1-x^2}\right)^{8} \\
P_{8}^{-7}(x)&={1\over 645120}\left(\sqrt{1-x^2}\right)^{7}x \\
P_{8}^{-6}(x)&={1\over 645120}\left(\sqrt{1-x^2}\right)^{6}(15x^{2}-1) \\
P_{8}^{-5}(x)&={1\over 15360}\left(\sqrt{1-x^2}\right)^{5}(5x^{3}-1x) \\
P_{8}^{-4}(x)&={1\over 15360}\left(\sqrt{1-x^2}\right)^{4}(65x^{4}-26x^{2}+1) \\
P_{8}^{-3}(x)&={1\over 768}\left(\sqrt{1-x^2}\right)^{3}(39x^{5}-26x^{3}+3x) \\
P_{8}^{-2}(x)&={1\over 256}\left(\sqrt{1-x^2}\right)^{2}(143x^{6}-143x^{4}+33x^{2}-1) \\
P_{8}^{-1}(x)&={1\over 128}\left(\sqrt{1-x^2}\right)(715x^{7}-1001x^{5}+385x^{3}-35x) \\
P_{8}^{0}(x)&={1\over 128}(6435x^{8}-12012x^{6}+6930x^{4}-1260x^{2}+35) \\
P_{8}^{1}(x)&={-9\over 16}\left(\sqrt{1-x^2}\right)(715x^{7}-1001x^{5}+385x^{3}-35x) \\
P_{8}^{2}(x)&={315\over 16}\left(\sqrt{1-x^2}\right)^{2}(143x^{6}-143x^{4}+33x^{2}-1) \\
P_{8}^{3}(x)&={-3465\over 8}\left(\sqrt{1-x^2}\right)^{3}(39x^{5}-26x^{3}+3x) \\
P_{8}^{4}(x)&={10395\over 8}\left(\sqrt{1-x^2}\right)^{4}(65x^{4}-26x^{2}+1) \\
P_{8}^{5}(x)&={-135135\over 2}\left(\sqrt{1-x^2}\right)^{5}(5x^{3}-1x) \\
P_{8}^{6}(x)&={135135\over 2}\left(\sqrt{1-x^2}\right)^{6}(15x^{2}-1) \\
P_{8}^{7}(x)&=-2027025\left(\sqrt{1-x^2}\right)^{7}x \\
P_{8}^{8}(x)&=2027025\left(\sqrt{1-x^2}\right)^{8}
\end{align}</math>

<math display="block">\begin{align}
P_{9}^{-9}(x)&={1\over 185794560}\left(\sqrt{1-x^2}\right)^{9} \\
P_{9}^{-8}(x)&={1\over 10321920}\left(\sqrt{1-x^2}\right)^{8}x \\
P_{9}^{-7}(x)&={1\over 10321920}\left(\sqrt{1-x^2}\right)^{7}(17x^{2}-1) \\
P_{9}^{-6}(x)&={1\over 645120}\left(\sqrt{1-x^2}\right)^{6}(17x^{3}-3x) \\
P_{9}^{-5}(x)&={1\over 215040}\left(\sqrt{1-x^2}\right)^{5}(85x^{4}-30x^{2}+1) \\
P_{9}^{-4}(x)&={1\over 3072}\left(\sqrt{1-x^2}\right)^{4}(17x^{5}-10x^{3}+1x) \\
P_{9}^{-3}(x)&={1\over 3072}\left(\sqrt{1-x^2}\right)^{3}(221x^{6}-195x^{4}+39x^{2}-1) \\
P_{9}^{-2}(x)&={1\over 256}\left(\sqrt{1-x^2}\right)^{2}(221x^{7}-273x^{5}+91x^{3}-7x) \\
P_{9}^{-1}(x)&={1\over 256}\left(\sqrt{1-x^2}\right)(2431x^{8}-4004x^{6}+2002x^{4}-308x^{2}+7) \\
P_{9}^{0}(x)&={1\over 128}(12155x^{9}-25740x^{7}+18018x^{5}-4620x^{3}+315x) \\
P_{9}^{1}(x)&={-45\over 128}\left(\sqrt{1-x^2}\right)(2431x^{8}-4004x^{6}+2002x^{4}-308x^{2}+7) \\
P_{9}^{2}(x)&={495\over 16}\left(\sqrt{1-x^2}\right)^{2}(221x^{7}-273x^{5}+91x^{3}-7x) \\
P_{9}^{3}(x)&={-3465\over 16}\left(\sqrt{1-x^2}\right)^{3}(221x^{6}-195x^{4}+39x^{2}-1) \\
P_{9}^{4}(x)&={135135\over 8}\left(\sqrt{1-x^2}\right)^{4}(17x^{5}-10x^{3}+1x) \\
P_{9}^{5}(x)&={-135135\over 8}\left(\sqrt{1-x^2}\right)^{5}(85x^{4}-30x^{2}+1) \\
P_{9}^{6}(x)&={675675\over 2}\left(\sqrt{1-x^2}\right)^{6}(17x^{3}-3x) \\
P_{9}^{7}(x)&={-2027025\over 2}\left(\sqrt{1-x^2}\right)^{7}(17x^{2}-1) \\
P_{9}^{8}(x)&=34459425\left(\sqrt{1-x^2}\right)^{8}x \\
P_{9}^{9}(x)&=-34459425\left(\sqrt{1-x^2}\right)^{9}
\end{align}</math>

<math display="block">P_{10}^{-10}(x)={1\over 3715891200}\left(\sqrt{1-x^2}\right)^{10}</math>
<math display="block">P_{10}^{-9}(x)={1\over 185794560}\left(\sqrt{1-x^2}\right)^{9}x</math>
<math display="block">P_{10}^{-8}(x)={1\over 185794560}\left(\sqrt{1-x^2}\right)^{8}(19x^{2}-1)</math>
<math display="block">P_{10}^{-7}(x)={1\over 10321920}\left(\sqrt{1-x^2}\right)^{7}(19x^{3}-3x)</math>
<math display="block">P_{10}^{-6}(x)={1\over 10321920}\left(\sqrt{1-x^2}\right)^{6}(323x^{4}-102x^{2}+3)</math>
<math display="block">P_{10}^{-5}(x)={1\over 645120}\left(\sqrt{1-x^2}\right)^{5}(323x^{5}-170x^{3}+15x)</math>
<math display="block">P_{10}^{-4}(x)={1\over 43008}\left(\sqrt{1-x^2}\right)^{4}(323x^{6}-255x^{4}+45x^{2}-1)</math>
<math display="block">P_{10}^{-3}(x)={1\over 3072}\left(\sqrt{1-x^2}\right)^{3}(323x^{7}-357x^{5}+105x^{3}-7x)</math>
<math display="block">P_{10}^{-2}(x)={1\over 3072}\left(\sqrt{1-x^2}\right)^{2}(4199x^{8}-6188x^{6}+2730x^{4}-364x^{2}+7)</math>
<math display="block">P_{10}^{-1}(x)={1\over 256}\left(\sqrt{1-x^2}\right)(4199x^{9}-7956x^{7}+4914x^{5}-1092x^{3}+63x)</math>
<math display="block">P_{10}^{0}(x)={1\over 256}(46189x^{10}-109395x^{8}+90090x^{6}-30030x^{4}+3465x^{2}-63)</math>
<math display="block">P_{10}^{1}(x)={-55\over 128}\left(\sqrt{1-x^2}\right)(4199x^{9}-7956x^{7}+4914x^{5}-1092x^{3}+63x)</math>
<math display="block">P_{10}^{2}(x)={495\over 128}\left(\sqrt{1-x^2}\right)^{2}(4199x^{8}-6188x^{6}+2730x^{4}-364x^{2}+7)</math>
<math display="block">P_{10}^{3}(x)={-6435\over 16}\left(\sqrt{1-x^2}\right)^{3}(323x^{7}-357x^{5}+105x^{3}-7x)</math>
<math display="block">P_{10}^{4}(x)={45045\over 16}\left(\sqrt{1-x^2}\right)^{4}(323x^{6}-255x^{4}+45x^{2}-1)</math>
<math display="block">P_{10}^{5}(x)={-135135\over 8}\left(\sqrt{1-x^2}\right)^{5}(323x^{5}-170x^{3}+15x)</math>
<math display="block">P_{10}^{6}(x)={675675\over 8}\left(\sqrt{1-x^2}\right)^{6}(323x^{4}-102x^{2}+3)</math>
<math display="block">P_{10}^{7}(x)={-11486475\over 2}\left(\sqrt{1-x^2}\right)^{7}(19x^{3}-3x)</math>
<math display="block">P_{10}^{8}(x)={34459425\over 2}\left(\sqrt{1-x^2}\right)^{8}(19x^{2}-1)</math>
<math display="block">P_{10}^{9}(x)=-654729075\left(\sqrt{1-x^2}\right)^{9}x</math>
<math display="block">P_{10}^{10}(x)=654729075\left(\sqrt{1-x^2}\right)^{10}</math>
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