= Associated Legendre differential equation
{title2=$[(1-x^2)y']'+[\ell(\ell+1)-m^2/(1-x^2)]y=0$}
= Associated Legendre equation
{synonym}
This second-order linear differential equation extends the <Legendre differential equation> by its order parameter $m$. Its solutions are <associated Legendre functions>. In a degree-one potential problem, with $x=\tanh\eta$, two explicit solutions are $e^{m\eta}(m-\tanh\eta)$ and $e^{-m\eta}(m+\tanh\eta)$. Their <Wronskian> with respect to $\eta$ is $2m(1-m^2)$, so the two formulas are not a fundamental pair at $m=0$ or $m=\pm1$.
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