imported>Paul Wormer |
imported>Paul Wormer |
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| -m(m+1) \right] \Pi^{(m)}_\ell(x) = 0 . | | -m(m+1) \right] \Pi^{(m)}_\ell(x) = 0 . |
| </math> | | </math> |
| SAfter substitutition of
| | After substitution of |
| :<math> | | :<math> |
| \Pi^{(m)}_\ell(x) = (1-x^2)^{-m/2} P^{(m)}_\ell(x), | | \Pi^{(m)}_\ell(x) = (1-x^2)^{-m/2} P^{(m)}_\ell(x), |
Revision as of 07:59, 22 August 2007
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
After substitution 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