= Associated Legendre function
{title2=$P_\nu^\mu(x),Q_\nu^\mu(x)$}
Associated Legendre functions solve $(1-x^2)y''-2xy'+[\nu(\nu+1)-\mu^2/(1-x^2)]y=0$. For integers $\ell\geq0$ and $0\leq m\leq\ell$, the regular solution is $P_\ell^m(x)=(-1)^m(1-x^2)^{m/2}(d/dx)^mP_\ell(x)$, using a <Legendre polynomial>. It enters normalized <spherical harmonics>. Despite the traditional term associated Legendre polynomial, the factor $(1-x^2)^{m/2}$ is not a polynomial when $m$ is odd. Negative integer orders satisfy $P_\ell^{-m}=(-1)^m(\ell-m)!P_\ell^m/(\ell+m)!$.
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