Associated Legendre function: Difference between revisions

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imported>Paul Wormer
(start of associated Legendres)
 
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