Revision as of 07:50, 22 August 2007 by imported>Paul Wormer
In mathematics and physics, an associated Legendre function Pl(m) is related to a Legendre polynomial Pl by the following equation
For even m the associated Legendre function is a polynomial, for odd m the function contains the factor (1-x ² )½ and hence is not a polynomial.
The associated Legendre polynomials are important in quantum mechanics and potential theory.
Differential equation
Define
where Pl(x) is a Legendre polynomial.
Differentiating the Legendre differential equation:
m times gives an equation for Π(m)l
SAfter substitutition of
we find, after multiplying through with , that the associated Legendre differential equation holds for the associated Legendre functions
In physical applications usually x = cosθ, then then associated Legendre differential equation takes the form