Associated Legendre function

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Revision as of 07:50, 22 August 2007 by imported>Paul Wormer (start of associated Legendres)
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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