Associated Legendre function: Difference between revisions

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imported>Dan Nessett
(Added link to proof of orthogonality and derivation of normalization constant)
imported>Dan Nessett
(→‎Orthogonality relations: Redirected proof link to new proofs sub-page)
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\int_{-1}^{1} P^{(m)}_{\ell}(x) P^{(m)}_{\ell'}(x) d x =
\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)!}
\frac{2\delta_{\ell\ell'}(\ell+m)!}{(2\ell+1)(\ell-m)!}
</math> &nbsp;&nbsp;&nbsp;[[Associated_Legendre_function/Addendum | &#91;Proof&#93;]]
</math> &nbsp;&nbsp;&nbsp;[[Associated_Legendre_function/Proofs | &#91;Proof&#93;]]


:<math>
:<math>
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\frac{\delta_{mn}(\ell+m)!}{m(\ell-m)!}
\frac{\delta_{mn}(\ell+m)!}{m(\ell-m)!}
</math>
</math>
==Recurrence relations==
==Recurrence relations==



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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 functions 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

and after multiplying through with , we find the associated Legendre differential equation:

In physical applications it is usually the case that x = cosθ, then the 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: .

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

    [Proof]

Recurrence relations

The functions satisfy the following difference equations, which are taken from Edmonds[1]

Reference

  1. A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 2nd edition (1960)

External link

Weisstein, Eric W. "Legendre Polynomial." From MathWorld--A Wolfram Web Resource. [1]