Revision as of 09:14, 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
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
Extension to negative m
By the Rodrigues formula, one obtains
This equation allows extension of the range of m to: -l ≤ m ≤ l.
Since the associated Legendre equation is invariant under the substitution m → -m, the equations for Pl( ±m), resulting from this expression, are proportional.
To obtain the proportionality constant we consider
and we bring the factor (1-x²)-m/2 to the other side.
Equate the coefficients of the same powers of x on the left and right hand side of
and it follows that the proportionality constant is
so that the associated Legendre functions of same |m| are related to each other by
Note that the phase factor (-1)m arising in this expression is not due to some arbitrary phase convention, but arises from expansion of (1-x²)m.