The Spherical Harmonics Mathematical Interlude: The Legendre Mathematical Interlude: The Legendre

The Legendre Polynomials

The Legendre polynomials arise when we solve Laplace's equation in spherical polar coordinates for a problem possessing azimuthal symmetry. They are real functions (polynomials of order l) defined in the interval [-1,+1] and have the following properties:

Its generating functions is

where , and l is a non-negative integer. The arise from a power-series solution of the order differential equation, the Legendre Equation:

and satisfy the following recurrence relations

They are normalised as follows:

The first few polynomials are

Note that