Associated Legendre Functions

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Section 19.7 Associated Legendre Functions

The standard form of Legendre’s associated equation, is given by

\begin{equation} \left(\frac{d^2}{d z^2} -\frac{2z}{1-z^2}\frac{d}{d z} -\frac{m^2}{(1-z^2)^2}+\frac{\ell(\ell+1)}{1-z^2}\right) P_{\ell}^m(z) = 0\tag{19.7.1} \end{equation}

where \(\ell\in\{0,1,2,3,...\}\) and \(m\in\{0,\pm 1,\pm 2,\pm 3,...\pm \ell\}\text{.}\)

This equation arises during separation of variables for the Laplacian in spherical coordinates, Section 15.9. The restrictions on the values of the constants \(\ell\) and \(m\) arise from imposing the boundary condition that the solutions \(P_{\ell}^m\) are regular (i.e. don’t blow up) at \(z=\pm1\text{.}\) These restrictions can be found from a straightforward but messy calculation using power series solutions, analogous to Section 16.2.

The solutions \(P_{\ell}^m\) are called associated Legendre functions, and can be generated quickly from the Legendre polynomials, Section 19.2, by a generalization of Rodrigues’ formula, Section 19.3:

\begin{align} P_\ell^{-m}(z) = P_\ell^m(z) \amp= (1-z^2)^{m/2} \frac{d^m}{dz^m} \left(P_\ell(z)\right)\notag\\ \amp= (1-z^2)^{m/2} \frac{d^{m+\ell}}{dz^{m+\ell}} \left((z^2-1)^\ell\right)\tag{19.7.2} \end{align}

where \(m\ge0\text{.}\)

Note that if \(z=\cos\theta\text{,}\) then \(P_\ell(z)\) is a polynomial in \(\cos\theta\text{,}\) while

\begin{equation} (1-z^2)^{m/2} = (\sin^2\!\theta)^{m/2} = \sin^m\!\theta\tag{19.7.3} \end{equation}

so that \(P_\ell^m(\cos\theta)\) is a polynomial in \(\cos\theta\) times a factor of \(\sin^m\!\theta\text{.}\)

Some other properties of the associated Legendre functions are

\(P_\ell^m(z) = 0\) if \(m>\ell\)
\(\displaystyle P_\ell^{-m}(z) = P_\ell^m(z)\)
\(P_\ell^m(\pm1) = 0\) for \(m\ne0\text{,}\) (from the factor \((1-z^2)^{m/2}\))
\(P_\ell^m(-z) = (-1)^{\ell m} P_\ell^m(z)\text{,}\) (behavior under parity)
\(\displaystyle \displaystyle\int\limits_{-1}^{1} P_\ell^m(z) P_q^m(z) \, dz = \frac{2}{(2\ell+1)} \> \frac{(\ell+m)!}{(\ell-m)!} \> \delta_{\ell q}\)

The last property shows that for each value of \(m\text{,}\) the Associated Legendre functions form an orthogonal basis on the interval \(-1\le z\le 1\text{.}\) Any function on this interval can be expanded in terms of any one of these eigenbases.