Associated Legendre function

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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

P^{(m)}_\ell(x) = (1-x^2)^{m/2} \frac{d^m P_\ell(x)}{dx^m}, \qquad 0 \le m \le \ell.

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.

Contents

Differential equation

Define

\Pi^{(m)}_\ell(x) \equiv \frac{d^m P_\ell(x)}{dx^m},

where Pl(x) is a Legendre polynomial. Differentiating the Legendre differential equation:

(1-x^2) \frac{d^2 \Pi^{(0)}_\ell(x)}{dx^2} - 2 x \frac{d\Pi^{(0)}_\ell(x)}{dx} + \ell(\ell+1) \Pi^{(0)}_\ell(x) = 0,

m times gives an equation for Π(m)l

(1-x^2) \frac{d^2 \Pi^{(m)}_\ell(x)}{dx^2} - 2(m+1) x \frac{d\Pi^{(m)}_\ell(x)}{dx} + \left[\ell(\ell+1) -m(m+1) \right] \Pi^{(m)}_\ell(x) = 0 .

After substitution of

\Pi^{(m)}_\ell(x) = (1-x^2)^{-m/2} P^{(m)}_\ell(x),

and after multiplying through with (1 − x2)m / 2, we find the associated Legendre differential equation:

(1-x^2) \frac{d^2 P^{(m)}_\ell(x)}{dx^2} -2x\frac{d P^{(m)}_\ell(x)}{dx} + \left[ \ell(\ell+1) - \frac{m^2}{1-x^2}\right] P^{(m)}_\ell(x)= 0 .

In physical applications it is usually the case that x = cosθ, then the associated Legendre differential equation takes the form

\frac{1}{\sin \theta}\frac{d}{d\theta} \sin\theta \frac{d}{d\theta}P^{(m)}_\ell +\left[ \ell(\ell+1) - \frac{m^2}{\sin^2\theta}\right] P^{(m)}_\ell = 0.

Extension to negative m

By the Rodrigues formula, one obtains

P_\ell^{(m)}(x) = \frac{1}{2^\ell \ell!} (1-x^2)^{m/2}\ \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^\ell.

This equation allows extension of the range of m to: -m \le \ell \le m.

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

(1-x^2)^{-m/2} \frac{d^{\ell-m}}{dx^{\ell-m}} (x^2-1)^{\ell} = c_{lm} (1-x^2)^{m/2} \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^{\ell},\qquad 0 \le m \le \ell,

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

\frac{d^{\ell-m}}{dx^{\ell-m}} (x^2-1)^{\ell} = c_{lm} (1-x^2)^m \frac{d^{\ell+m}}{dx^{\ell+m}}(x^2-1)^{\ell},\qquad 0 \le m \le \ell,

and it follows that the proportionality constant is

c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} ,\qquad 0 \le m \le \ell,

so that the associated Legendre functions of same |m| are related to each other by

P^{(-|m|)}_\ell(x) = (-1)^m \frac{(\ell-|m|)!}{(\ell+|m|)!} P^{(|m|)}_\ell(x).

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

\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)!}
\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)!}

Recurrence relations

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

(\ell-m+1)P_{\ell+1}^{(m)}(x) - (2\ell+1)xP_{\ell}^{(m)}(x) + (\ell+m)P_{\ell-1}^{(m)}(x)=0
xP_{\ell}^{(m)}(x) -(\ell-m+1)(1-x^2)^{1/2} P_{\ell}^{(m-1)}(x) - P_{\ell-1}^{(m)}(x)=0
P_{\ell+1}^{(m)}(x) - x P_{\ell}^{(m)}(x)-(\ell+m)(1-x^2)^{1/2}P_{\ell}^{(m-1)}(x)=0
(\ell-m+1)P_{\ell+1}^{(m)}(x)+(1-x^2)^{1/2}P_{\ell}^{(m+1)}(x)- (\ell+m+1) xP_{\ell}^{(m)}(x)=0
(1-x^2)^{1/2}P_{\ell}^{(m+1)}(x)-2mxP_{\ell}^{(m)}(x)+ (\ell+m)(\ell-m+1)(1-x^2)^{1/2}P_{\ell}^{(m-1)}(x)=0
(1-x^2)\frac{dP_{\ell}^{(m)}}{dx}(x) =(\ell+1)xP_{\ell}^{(m)}(x) -(\ell-m+1)P_{\ell+1}^{(m)}(x)
=(\ell+m)P_{\ell-1}^{(m)}(x)-\ell x P_{\ell}^{(m)}(x)

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]

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