Associated Legendre Function

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An associated Legendre function $P_l^m$ is related to a Legendre polynomial $P_l$ by the equation:

$ P^m_l \left({x}\right) = \left({1 - x^2}\right)^{m/2} \dfrac {\mathrm d^m P_l \left({x}\right)}{\mathrm d x^m}, \qquad 0 \le m \le l$

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 $\left({1 - x^2}\right)^ \frac 1 2$ and hence is not a polynomial.

The associated Legendre functions are important in quantum mechanics and potential theory.


Differential equation

Define:

$\displaystyle \Pi^m_l \left({x}\right) \equiv \frac{\mathrm d^m P_l \left({x}\right)}{\mathrm d x^m}$

where $P_l \left({x}\right)$ is a Legendre polynomial.

Differentiating Legendre's differential equation:

$\left({1 - x^2}\right) \dfrac {\mathrm d^2 \Pi^0_l \left({x}\right)} {\mathrm d x^2} - 2 x \dfrac {\mathrm d \Pi^0_l \left({x}\right)}{\mathrm d x} + l \left({l + 1}\right) \Pi^0_l \left({x}\right) = 0$

$m$ times gives an equation for $\Pi^m_l$:

$\left({1 - x^2}\right) \dfrac {\mathrm d^2 \Pi^m_l \left({x}\right)} {\mathrm d x^2} - 2 \left({m + 1}\right) x \dfrac {\mathrm d \Pi^m_l \left({x}\right)} {\mathrm d x} + \left({l \left({l + 1}\right) -m \left({m + 1}\right)} \right) \Pi^m_l \left({x}\right) = 0$

After substitution of:

$\Pi^m_l \left({x}\right) = \left({1 - x^2}\right)^{-m/2} P^m_l \left({x}\right)$

and after multiplying through with $\left({1 - x^2}\right)^{m/2}$, we find the associated Legendre differential equation:

$\displaystyle \left({1 - x^2}\right) \frac{\mathrm d^2 P^m_l\left({x}\right)}{\mathrm d x^2} - 2 x \frac {\mathrm d P^m_l \left({x}\right)} {\mathrm d x} + \left({l \left({l+1}\right) - \frac{m^2}{1 - x^2}}\right) P^m_l\left({x}\right)= 0$

One often finds the equation written in the following equivalent way:

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

where the primes indicate differentiation with respect to $x$.

In physical applications it is usually the case that $x = \cos \theta$, then the associated Legendre differential equation takes the form:

$ \displaystyle \frac 1 {\sin \theta} \frac{\mathrm d} {\mathrm d \theta} \sin \theta \frac{\mathrm d} {\mathrm d \theta} P^m_l + \left[{l \left({l+1}\right) - \frac{m^2}{\sin^2 \theta}}\right] P^m_l = 0$


Extension to negative m

By the Rodrigues formula, one obtains:

$\displaystyle P_l^m \left({x}\right) = \frac 1 {2^l l!} \left({1 - x^2}\right)^{m/2} \ \frac{\mathrm d^{l+m}}{\mathrm d x^{l+m}} \left({x^2-1}\right)^l$

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

Since the associated Legendre equation is invariant under the substitution $m \to -m$, the equations for $\displaystyle P_l^{\pm m}$, resulting from this expression, are proportional.[1]

To obtain the proportionality constant consider:

$\displaystyle \left({1 - x^2}\right)^{-m/2} \frac{\mathrm d^{l-m}}{\mathrm d x^{l-m}} \left({x^2-1}\right)^l = c_{lm} \left({1 - x^2}\right)^{m/2} \frac{\mathrm d^{l+m}}{\mathrm d x^{l+m}}\left({x^2-1}\right)^l: \qquad 0 \le m \le l $

and bring the factor $\left({1 - x^2}\right)^{-m/2}$ to the other side.

Equate the coefficient of the highest power of $x$ on the left and right hand side of:

$\displaystyle \frac{\mathrm d^{l-m}}{\mathrm d x^{l-m}} \left({x^2-1}\right)^l = c_{lm} \left({1 - x^2}\right)^m \frac{\mathrm d^{l+m}}{\mathrm d x^{l+m}}\left({x^2-1}\right)^l,\qquad 0 \le m \le l$

and it follows that the proportionality constant is:

$\displaystyle c_{lm} = \left({-1}\right)^m \frac{\left({l-m}\right)!}{\left({l+m}\right)!} ,\qquad 0 \le m \le l$

so that the associated Legendre functions of same $\vert m \vert$ are related to each other by:

$\displaystyle P^{- \vert m \vert}_l \left({x}\right) = \left({-1}\right)^m \frac{\left({l - \vert m \vert}\right)!}{\left({l + \vert m \vert}\right)!} P^{ \vert m \vert}_l\left({x}\right)$

Note that the phase factor $\left({-1}\right)^m$ arising in this expression is not due to some arbitrary phase convention, but arises from expansion of $\left({1 - x^2}\right)^m$.


Orthogonality relations

Important integral relations are:

$\displaystyle \int\limits_{-1}^1 P_l^m \left({ x}\right) P_k^m \left({ x}\right) \ \mathrm d x = \frac 2 {2 l+1} \frac{\left({ l+m}\right) !} {\left({l-m}\right)!} \delta_{l k}$

and:

$\displaystyle \int_{-1}^1 P^m_l\left({x}\right) P^n_l\left({x}\right) \frac{\mathrm d x}{1 - x^2} = \frac {\delta_{m n}\left({l+m}\right)!} {m \left({l-m}\right)!}, \qquad m \ne 0$

The latter integral for n = m = 0

$\displaystyle \int_{-1}^1 P^0_l \left({x}\right) P^0_l \left({x}\right) \frac{\mathrm d x}{1 - x^2}$

is undetermined (infinite). (see Orthogonality of Associated Legendre Functions for details.)


Recurrence relations

The functions satisfy the following difference equations, which are taken from Edmonds.[2]

$ \displaystyle \left({l-m+1}\right)P_{l+1}^m\left({x}\right) - \left({2l+1}\right)xP_l^m\left({x}\right) + \left({l+m}\right)P_{l-1}^m\left({x}\right)=0 $
$ \displaystyle xP_l^m\left({x}\right) -\left({l-m+1}\right)\left({1 - x^2}\right)^{1/2} P_l^{m-1}\left({x}\right) - P_{l-1}^m\left({x}\right)=0 $
$ \displaystyle P_{l+1}^m\left({x}\right) - x P_l^m\left({x}\right)-\left({l+m}\right)\left({1 - x^2}\right)^{1/2}P_l^{m-1}\left({x}\right)=0$
$ \displaystyle \left({l-m+1}\right)P_{l+1}^m\left({x}\right) + \left({1 - x^2}\right)^{1/2}P_l^{m+1}\left({x}\right)- \left({l+m+1}\right) xP_l^m\left({x}\right)=0 $
$ \displaystyle \left({1 - x^2}\right)^{1/2}P_l^{m+1}\left({x}\right)-2mxP_l^m\left({x}\right)+ \left({l+m}\right)\left({l-m+1}\right) \left({1 - x^2}\right)^{1/2}P_l^{m-1}\left({x}\right)=0 $
\(\displaystyle \left({1 - x^2}\right) \frac{dP_l^m }{dx}\left({x}\right)\) \(=\) \(\displaystyle \left({l+1}\right)xP_l^m \left({x}\right) -\left({l-m+1}\right)P_{l+1}^m \left({x}\right)\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({l+m}\right)P_{l-1}^m \left({x}\right)-l x P_l^m \left({x}\right)\) $\quad$ $\quad$


Source of Name

This entry was named for Adrien-Marie Legendre.

He was the first to introduce and study these functions.

The term associated Legendre function is a translation of the German term zugeordnete Function, coined by Heinrich Eduard Heine in 1861.

It was claimed by N.M. Ferrers in his An Elementary Treatise on Spherical Harmonics that the Legendre polynomials were named associated Legendre function by Isaac Todhunter in Functions of Laplace, Bessel and Legendre. In fact, Todhunter referred to them as Legendre coefficients.


References

  1. The associated Legendre differential equation being of second order, the general solution is of the form $A P_l^m + B Q_l^m$ where $Q_l^m$ 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 $P_l^m$. The Rodrigues formula shows that $P_l^{-m}$ is a regular (at $x = 0$) solution and the proportionality follows.
  2. A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 2nd edition (1960)

Sources