Associated Legendre function

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1 Differential equation
2 Extension to negative m
3 Orthogonality relations
4 Recurrence relations
5 Reference

See Associated Legendre function/Catalogs for explicit equations through $\ell=6.\,$

In mathematics and physics, an associated Legendre function P_l^m is related to a Legendre polynomial P_l by the following equation

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

Although extensions are possible, in this article $\ell\,$ and m are restricted to integer numbers. 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.

According to Ferrers^[1] the polynomials were named "Associated Legendre functions" by the British mathematician Isaac Todhunter in 1875,^[2] where "associated function" is Todhunter's translation of the German term zugeordnete Function, coined in 1861 by Heine,^[3] and "Legendre" is in honor of the French mathematician Adrien-Marie Legendre (1752–1833), who was the first to introduce and study the functions.

Differential equation

Define

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

where P_l(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 $(1-x^2)^{m/2}$ , we find the associated Legendre differential equation:

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

One often finds the equation written in the following equivalent way

$\left( (1-x^{2})\; y\,' \right)' +\left( \ell(\ell+1) -\frac{m^{2} }{1-x^{2} } \right) y=0,$

where the primes indicate differentiation with respect to x.

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 P_l^±m, resulting from this expression, are proportional.^[4]

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:

$\int\limits_{-1}^{1}P_{l}^{m} \left( x\right) P_{k}^{m} \left( x\right) dx =\frac{2}{2l+1} \frac{\left( l+m\right) !}{\left( l-m\right) !} \delta _{lk},$

and:

$\int_{-1}^{1} P^{m}_{\ell}(x) P^{n}_{\ell}(x) \frac{d x}{1-x^2} = \frac{\delta_{mn}(\ell+m)!}{m(\ell-m)!}, \qquad m \ne 0 .$

The latter integral for n = m = 0

$\int_{-1}^{1} P^{0}_{\ell}(x) P^{0}_{\ell}(x) \frac{d x}{1-x^2}$

is undetermined (infinite). (see the subpage Proofs for detailed proofs of these relations.)

Recurrence relations

The functions satisfy the following difference equations, which are taken from Edmonds.^[5]

$(\ell-m+1)P_{\ell+1}^{m}(x) - (2\ell+1)xP_{\ell}^{m}(x) + (\ell+m)P_{\ell-1}^{m}(x)=0$

$xP_{\ell}^{m}(x) -(\ell-m+1)(1-x^2)^{1/2} P_{\ell}^{m-1}(x) - P_{\ell-1}^{m}(x)=0$

$P_{\ell+1}^{m}(x) - x P_{\ell}^{m}(x)-(\ell+m)(1-x^2)^{1/2}P_{\ell}^{m-1}(x)=0$

$(\ell-m+1)P_{\ell+1}^{m}(x)+(1-x^2)^{1/2}P_{\ell}^{m+1}(x)- (\ell+m+1) xP_{\ell}^{m}(x)=0$

$(1-x^2)^{1/2}P_{\ell}^{m+1}(x)-2mxP_{\ell}^{m}(x)+ (\ell+m)(\ell-m+1)(1-x^2)^{1/2}P_{\ell}^{m-1}(x)=0$

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

$=(\ell+m)P_{\ell-1}^{m}(x)-\ell x P_{\ell}^{m}(x)$

Reference

↑ N. M. Ferrers, An Elementary Treatise on Spherical Harmonics, MacMillan, 1877 (London), p. 77. Online.
↑ I. Todhunter, An Elementary Treatise on Laplace's, Lamé's, and Bessel's Functions, MacMillan, 1875 (London). In fact, Todhunter called the Legendre polynomials "Legendre coefficients".
↑ E. Heine, Handbuch der Kugelfunctionen, G. Reimer, 1861 (Berlin).Google book online
↑ The associated Legendre differential equation being of second order, the general solution is of the form $AP_\ell^m + BQ_\ell^m$ where $Q_\ell^m$ is a Legendre polynomial of the second kind, which has a singularity at x = 0. Hence solutions that are regular at x = 0 have B = 0 and are proportional to $P_\ell^m$ . The Rodrigues formula shows that $P_\ell^{-m}$ is a regular (at x=0) solution and the proportionality follows.
↑ A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 2nd edition (1960)