Definition:Legendre's Associated Differential Equation

Definition
Legendre's associated differential equation is a second order ODE of the form:


 * $\displaystyle \paren {1 - x^2} \frac {\d^2 y} {\d x^2} - 2 x \frac {\d y} {\d x} + \paren {n \paren {n + 1} - \frac {m^2} {1 - x^2} } y = 0$

where $m$ and $n$ are complex numbers.

Solutions of this equation are called associated Legendre functions.