# 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:

- $ \map {P^m_l} x = \paren {1 - x^2}^{m / 2} \dfrac {\d^m \map {P_l} x} {\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 $\paren {1 - x^2}^\frac 1 2$ and hence is not a polynomial.

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

## Differential equation

Define:

- $\ds \map {\Pi^m_l} x \equiv \frac {\d^m \map {P_l} x} {\d x^m}$

where $\map {P_l} x$ is a Legendre polynomial.

Differentiating Legendre's differential equation:

- $\paren {1 - x^2} \dfrac {\d^2 \map {\Pi^0_l} x} {\d x^2} - 2 x \dfrac {\d \map {\Pi^0_l} x} {\d x} + l \paren {l + 1} \map {\Pi^0_l} x = 0$

$m$ times and using the Leibniz rule gives an equation for $\Pi^m_l$:

- $\paren {1 - x^2} \dfrac {\d^2 \map {\Pi^m_l} x} {\d x^2} - 2 \paren {m + 1} x \dfrac {\d \map {\Pi^m_l} x} {\d x} + \paren {l \paren {l + 1} - m \paren {m + 1} } \map {\Pi^m_l} x = 0$

since many derivatives, such as $\dfrac {\d^2} {\d x^2} (x) $, cancel.

After substitution of:

- $\map {\Pi^m_l} x = \paren {1 - x^2}^{-m / 2} \map {P^m_l} x$

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

- $\ds \paren {1 - x^2} \frac {\d^2 \map {P^m_l} x}{\d x^2} - 2 x \frac {\d \map {P^m_l} x} {\d x} + \paren {l \paren {l + 1} - \frac {m^2} {1 - x^2} } \map {P^m_l} x = 0$

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

- $\ds \paren {\paren {1 - x^2} \ y\,' }' + \paren {l \paren {l + 1} - \frac {m^2} {1 - x^2} } y = 0$

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:

- $ \ds \frac 1 {\sin \theta} \frac \d {\d \theta} \sin \theta \frac \d {\d \theta} P^m_l + \left[{l \paren {l + 1} - \frac {m^2} {\sin^2 \theta}}\right] P^m_l = 0$

## Extension to negative $m$

By the Rodrigues formula, one obtains:

- $\ds \map {P_l^m} x = \frac 1 {2^l l!} \paren {1 - x^2}^{m / 2} \ \frac {\d^{l + m} } {\d x^{l + m} } \paren {x^2 - 1}^l$

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

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

To obtain the proportionality constant consider:

- $\ds \paren {1 - x^2}^{-m / 2} \frac{\d^{l - m} }{\d x^{l - m} } \paren {x^2 - 1}^l = c_{l m} \paren {1 - x^2}^{m / 2} \frac {\d^{l + m} } {\d x^{l + m} } \paren {x^2 - 1}^l: \qquad 0 \le m \le l$

and bring the factor $\paren {1 - x^2}^{-m / 2}$ to the other side.

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

- $\ds \frac {\d^{l - m} } {\d x^{l - m} } \paren {x^2 - 1}^l = c_{l m} \paren {1 - x^2}^m \frac {\d^{l + m} } {\d x^{l + m} } \paren {x^2 - 1}^l, \qquad 0 \le m \le l$

and it follows that the proportionality constant is:

- $\ds c_{l m} = \paren {-1}^m \frac {\paren {l - m}!} {\paren {l + m}!}, \qquad 0 \le m \le l$

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

- $\ds \map {P^{- \size m}_l} x = \paren {-1}^m \frac {\paren {l - \size m}!} {\paren {l + \size m}!} \map {P^{\size m}_l} x$

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

## Orthogonality relations

Important integral relations are:

- $\ds \int_{-1}^1 \map {P_l^m} x \map {P_k^m} x \rd x = \frac 2 {2 l + 1} \frac {\paren {l + m}!} {\paren {l - m}!} \delta_{l k}$

and:

- $\ds \int_{-1}^1 \map {P^m_l} x \map {P^n_l} x \frac {\rd x} {1 - x^2} = \frac {\map {\delta_{m n} } {l + m}!} {m \paren {l - m}!}, \qquad m \ne 0$

The latter integral for *n* = *m* = 0

- $\ds \int_{-1}^1 \map {P^0_l} x \map {P^0_l} x \frac {\rd x} {1 - x^2}$

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]}

- $ \ds \paren {l - m + 1} \map {P_{l + 1}^m} x - \paren {2 l + 1} x \map {P_l^m} x + \paren {l + m} \map {P_{l - 1}^m} x = 0 $

- $ \ds x \map {P_l^m} x - \paren {l - m + 1} \paren {1 - x^2}^{1 / 2} \map {P_l^{m - 1} } x - \map {P_{l - 1}^m} x = 0 $

- $ \ds \map {P_{l + 1}^m} x - x \map {P_l^m} x - \paren {l + m} \paren {1 - x^2}^{1 / 2} \map {P_l^{m - 1} } x = 0$

- $ \ds \paren {l - m + 1} \map {P_{l + 1}^m} x + \paren {1 - x^2}^{1 / 2} \map {P_l^{m + 1} } x - \paren {l + m + 1} x \map {P_l^m} x = 0 $

- $ \ds \paren {1 - x^2}^{1 / 2} \map {P_l^{m + 1} } x - 2 m x \map {P_l^m} x + \paren {l + m} \paren {l - m + 1} \paren {1 - x^2}^{1 / 2} \map {P_l^{m - 1} } x = 0 $

\(\ds \paren {1 - x^2} \map {\frac {\d P_l^m} {\d x} } x\) | \(=\) | \(\ds \paren {l + 1} x \map {P_l^m} x - \paren {l - m + 1} \map {P_{l + 1}^m} x\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \paren {l + m} \map {P_{l - 1}^m} x - l x \map {P_l^m} x\) |

## 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

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- 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*