Associated Legendre function: Difference between revisions

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imported>Paul Wormer
imported>Paul Wormer
Line 64: Line 64:
</math>
</math>
Note that the phase factor (-1)<sup>''m''</sup> arising in this expression is ''not'' due to some arbitrary phase convention, but arises from expansion of (1-''x''&sup2;)<sup>m</sup>.
Note that the phase factor (-1)<sup>''m''</sup> arising in this expression is ''not'' due to some arbitrary phase convention, but arises from expansion of (1-''x''&sup2;)<sup>m</sup>.
==Orthogonality relations==
Important integral relations are
:<math>
\int_{-1}^{1} P^{(m)}_{\ell}(x) P^{(m)}_{\ell'}(x) d x =
\frac{2\delta_{\ell\ell'}(\ell+m)!}{(2\ell+1)(\ell-m)!}
</math>
:<math>
\int_{-1}^{1} P^{(m)}_{\ell}(x) P^{(n)}_{\ell}(x) \frac{d x}{1-x^2} =
\frac{\delta_{mn}(\ell+m)!}{m(\ell-m)!}
</math>

Revision as of 09:28, 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

Extension to negative m

By the Rodrigues formula, one obtains

This equation allows extension of the range of m to: -lml.

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 coefficient of the highest power 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.

Orthogonality relations

Important integral relations are