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Associated Legendre polynomials
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==Reparameterization in terms of angles== These functions are most useful when the argument is reparameterized in terms of angles, letting <math>x = \cos\theta</math>: <math display="block">P_\ell^{m}(\cos\theta) = (-1)^m (\sin \theta)^m\ \frac{d^m}{d(\cos\theta)^m}\left(P_\ell(\cos\theta)\right)</math> Using the relation <math>(1 - x^2)^{1 / 2} = \sin\theta</math>, [[#The first few associated Legendre functions|the list given above]] yields the first few polynomials, parameterized this way, as: <math display="block">\begin{align} P_0^0(\cos\theta) & = 1 \\[8pt] P_1^0(\cos\theta) & = \cos\theta \\[8pt] P_1^1(\cos\theta) & = -\sin\theta \\[8pt] P_2^0(\cos\theta) & = \tfrac{1}{2} (3\cos^2\theta-1) \\[8pt] P_2^1(\cos\theta) & = -3\cos\theta\sin\theta \\[8pt] P_2^2(\cos\theta) & = 3\sin^2\theta \\[8pt] P_3^0(\cos\theta) & = \tfrac{1}{2} (5\cos^3\theta-3\cos\theta) \\[8pt] P_3^1(\cos\theta) & = -\tfrac{3}{2} (5\cos^2\theta-1)\sin\theta \\[8pt] P_3^2(\cos\theta) & = 15\cos\theta\sin^2\theta \\[8pt] P_3^3(\cos\theta) & = -15\sin^3\theta \\[8pt] P_4^0(\cos\theta) & = \tfrac{1}{8} (35\cos^4\theta-30\cos^2\theta+3) \\[8pt] P_4^1(\cos\theta) & = - \tfrac{5}{2} (7\cos^3\theta-3\cos\theta)\sin\theta \\[8pt] P_4^2(\cos\theta) & = \tfrac{15}{2} (7\cos^2\theta-1)\sin^2\theta \\[8pt] P_4^3(\cos\theta) & = -105\cos\theta\sin^3\theta \\[8pt] P_4^4(\cos\theta) & = 105\sin^4\theta \end{align}</math> The orthogonality relations given above become in this formulation: for fixed ''m'', <math>P_\ell^m(\cos\theta)</math> are orthogonal, parameterized by ΞΈ over <math>[0, \pi]</math>, with weight <math>\sin \theta</math>: <math display="block">\int_0^\pi P_k^{m}(\cos\theta) P_\ell^{m}(\cos\theta)\,\sin\theta\,d\theta = \frac{2 (\ell+m)!}{(2\ell+1)(\ell-m)!}\ \delta _{k,\ell}</math> Also, for fixed ''β'': <math display="block">\int_0^\pi P_\ell^{m}(\cos\theta) P_\ell^{n}(\cos\theta) \csc\theta\,d\theta = \begin{cases} 0 & \text{if } m\neq n \\ \frac{(\ell+m)!}{m(\ell-m)!} & \text{if } m=n\neq0 \\ \infty & \text{if } m=n=0\end{cases}</math> In terms of ΞΈ, <math>P_\ell^{m}(\cos \theta)</math> are solutions of <math display="block">\frac{d^{2}y}{d\theta^2} + \cot \theta \frac{dy}{d\theta} + \left[\lambda - \frac{m^2}{\sin^2\theta}\right]\,y = 0\,</math> More precisely, given an integer ''m''<math>\ge</math>0, the above equation has nonsingular solutions only when <math>\lambda = \ell(\ell+1)\,</math> for ''β'' an integer β₯ ''m'', and those solutions are proportional to <math>P_\ell^{m}(\cos \theta)</math>.
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