# Associated Legendre Function

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.

## Contents

## 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

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

## Sources

- 1861: E. Heine:
*Handbuch der Kugelfunctionen* - 1875: Isaac Todhunter:
*Functions of Laplace, Bessel and Legendre* - 1877: N.M. Ferrers:
*An Elementary Treatise on Spherical Harmonics and Subjects Connected With Them*

*This article incorporates material from Associated_Legendre_function on Citizendium, which is licensed under the Creative Commons Attribution/Share-Alike License.*