Questions about Legendre Equation

[AlonsoMcLaren]

1. I understand that the x in Legendre Equation (1-x^2)y''-2xy'+l(l+1)y=0 is often related to θ in spherical coordinates. We want the latter equation to have a solution at θ=0 and θ=pi. Therefore, we require that Legendre Equation has a solution at x=±1

And it is claimed that "we require the equation to have a polynomial solution, and so l must be an integer. Furthermore, we also require the coefficient c2 of the function Ql(x) (Legendre's function of the second kind) to be zero" But this assumes that at x=±1 the solution of Legendre Equation MUST be found by a power series. Having a solution at x=±1 is not the same as having a power series solution at x=±1.

2. How to solve Legendre Equation with |x|>1?

 

[HallsofIvy]

The "Legendre polynomials" are eigenfunctions for the Legendre operator. That means that any solution can be written as a (possibly infinite) sum of Legendre polynomials.