The Legendre Differential Equation

We study the Legendre differential equation which arises in the study of spherical harmonics and electrostatics. This second order linear differential equation corresponds to Sturm-Liouville eigenvalue problem. We will find a solution to this differential equation around the ordinary point of zero (it has three regular singular points at 1, -1 and infinity). We show that there are polynomial solutions to this differential equation for certain values of the parameter in the problem.

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