Does Associated Legendre Function of Second Kind Give Delta Function?

Question

Associated Legendre Function of Second Kind is singular at $x=\pm 1$. So I am wondering whether it satisfies the corresponding differential equation everywhere or there is a hidden functional of delta function on the right hand side of the equation. For example, consider the equation \begin{equation} \left((1-x^2)f'(x)\right)'+(2-\frac{4}{1-x^2})f(x)=0 \end{equation} which has a solution $f(x)=\frac{1}{1-x^2}$ singular at $x=\pm1$. The reason I ask this is an example of electric potential produced by a point particle and we have the equation $$\nabla^2\frac{1}{r}=\delta^3(r)$$ As one can see, the function $h=1/r$ satisfies $\nabla^2h=0$ everywhere except $r=0$. Actually, we have a delta function $\delta(r)$ as the source or inhomogeneous term of the differential equation. Therefore, I want to know if something similar happens to the associated Legendre function of second kind.

Answer 1

You are correct, and this applies to all Sturm-Liouville-type problems on finite intervals, with Dirichlet boundary conditions. Namely, as you observe, the classical formulation only refers to satisfaction of the differential equation on the interior of the interval... and since the solutions/eigenfunctions, extended by $0$ outside the interval, are not smooth at the endpoints, something "non-classical" must happen at the endpoints. Depending on the specifics of the situation of course, with Dirichlet condition, the multiples of Dirac $\delta$s at the endpoints are essentially the values of the one-sided derivative at those endpoints.