Definition:Associated Legendre Function

Definition

The associated Legendre functions are the real functions defined and denoted as:

$\map { {P_n}^m} x = \paren {1 - x^2}^{m / 2} \dfrac {\d^m} {\d x^m} \map {P_n} x$

where $\map {P_n} x$ is the Legendre polynomial of order $n$.

The associated Legendre functions are the solutions to Legendre's associated differential equation.

Also see

• Definition:Legendre's Associated Differential Equation

• Results about the associated Legendre functions can be found here.

Source of Name

This entry was named for Adrien-Marie Legendre.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Legendre's differential equation
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Legendre's differential equation