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.

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.

$ \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.

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