Associated Legendre function

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In mathematics and physics, an associated Legendre function Pℓm 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.

Although extensions are possible, in this article ℓ 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.

Contents

[edit] 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 Πml

(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 .

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.

[edit] 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 ≤ ℓ ≤ m.

Since the associated Legendre equation is invariant under the substitution m → −m, the equations for Pℓ ±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.

[edit] 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 Associated_Legendre_function/Proofs for detailed proofs of these relations.)

[edit] 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)

[edit] Reference

  1. ↑ N. M. Ferrers, An Elementary Treatise on Spherical Harmonics, MacMillan, 1877 (London), p. 77. Online.
  2. ↑ 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".
  3. ↑ E. Heine, Handbuch der Kugelfunctionen, G. Reimer, 1861 (Berlin).Google book online
  4. ↑ 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.
  5. ↑ A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 2nd edition (1960)
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