Mathematical Functions - Legendre Polynomials

Definition of the Legendre polynomial P_n(x).

Even or odd Legendre polynomials.

Rodrigues formular for Legendre polynomials.

The Legendre polynomials are solutions of the Legendre differential equation.

Executing the derivative d/dx will result in the following differential equation:

Legendre Functions of the Second Kind

Recusive Definition:

Associated Legendre Polynomials

The associated Legendre polynomials are defined with the help of the derivatives of the Legendre polynomials as follows:

m = 0 .. l

The P_l(x) are the Legendre polynomials.

The P_l⁽⁰⁾(x) of the associated Legendre polynomials are the normal Legendre polynomials.

Inserting Rodrigues formular:

will result in the following expression for the associated Legendre polynomials:

The associated Legendre polynomials for negative m are then defined by

The value of m can now be the range -l..m..l.

Next Page:

Gamma Function