\begin{definition}[Definition:Legendre's Differential Equation]
'''Legendre's differential equation''' is a second order ODE of the form:
:$\paren {1 - x^2} \dfrac {\d^2 y} {\d x^2} - 2 x \dfrac {\d y} {\d x} + p \paren {p + 1} y = 0$
The parameter $p$ may be any arbitrary real or complex number.
Solutions of this equation are called '''Legendre functions of order $p$'''.
\end{definition}