A differential equation that occurs frequently in physics (as part of the solution of Laplace's equation, which occurs in such areas as electrodynamics and quantum mechanics, among others) is Legendre's equation. In this post, we'll have a look at the equations and some of the properties of its simplest solutions: the Legendre polynomials.
The equation occurs while solving Laplace's equation (which we'll consider in other posts) in spherical coordinates. Although the origins of the equation are important in the physical applications, for our purposes here we need concern ourselves only with the equation itself, which is usually first encountered in the follow for :
where I and m are constants and the angle theta is the spherical coordinate which can range over the interval [0, pi] and serves as the independent variable in the equation. The problem is to determine the function P(theta).
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The solution ...
The solution shows why the solutions to Legendre equation for m=0 retain the parity of l.