# Associated Legendre function

*See Associated Legendre function/Catalogs for explicit equations through**ℓ*= 6.

In mathematics and physics, an **associated Legendre function** *P*_{ℓ}^{m} is related to a Legendre polynomial *P*_{ℓ} by the following equation

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.

## Differential equation

Define

where *P*_{ℓ}(*x*) is a Legendre polynomial.
Differentiating the Legendre differential equation:

*m* times gives an equation for Π^{m}_{l}

After substitution of

and after multiplying through with , we find the *associated Legendre differential equation*:

One often finds the equation written in the following equivalent way

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

## Extension to negative m

By the Rodrigues formula, one obtains

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 *P*_{ℓ}^{ ±m}, resulting from this expression, are proportional.^{[4]}

To obtain the proportionality constant we consider

and we bring the factor (1−*x*²)^{−m/2} to the other side.
Equate the coefficient of the highest power of *x* on the left and right hand side of

and it follows that the proportionality constant is

so that the associated Legendre functions of same |*m*| are related to each other by

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−*x*²)^{m}.

## Orthogonality relations

Important integral relations are:

and:

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

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]}

## 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 where 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 . The Rodrigues formula shows that 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)