Associated Legendre Function

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: $-m \le l \le m$.
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.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.
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 and 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.A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 2nd edition (1960)
:$ \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 $
 
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 in 1861.
It was claimed by in his An Elementary Treatise on Spherical Harmonics that the Legendre polynomials were named associated Legendre function by in Functions of Laplace, Bessel and Legendre. In fact, referred to them as Legendre coefficients.
== References ==
== Sources ==
* 1861: : Handbuch der Kugelfunctionen
* 1875: : Functions of Laplace, Bessel and Legendre
* 1877: : An Elementary Treatise on Spherical Harmonics and Subjects Connected With Them