Definition:Legendre's Associated Differential Equation

Definition

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

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

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

Solutions of this equation are called associated Legendre functions.

Also see

• Results about Legendre's associated differential equation can be found here.

Source of Name

This entry was named for Adrien-Marie Legendre.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 26$: Associated Legendre Functions: $26.1$