Associated Legendre Functions

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Section 22.16 Associated Legendre Functions

We now return to (22.10.6) to consider the cases with \(m\ne0\text{.}\) We can solve these equations with (a slightly more sophisticated version of) the series techniques from the \(m=0\) case. We would again find solutions that are regular at \(z=\pm1\) whenever we choose \(A=-\ell(\ell+1)\) for \(\ell\in\{0,1,2,3,...\}\text{.}\) With this value for \(A\text{,}\) we obtain the standard form of Legendre’s associated equation, namely

\begin{equation} \left(\frac{\partial^2}{\partial z^2} -\frac{2z}{1-z^2}\frac{\partial}{\partial z} -\frac{m^2}{(1-z^2)^2}+\frac{\ell(\ell+1)}{1-z^2}\right) P(z) = 0\tag{22.16.1} \end{equation}

Recall that this equation was obtained by separating variables in spherical coordinates. Solutions of this equation which are regular at \(z=\pm1\) are called associated Legendre functions, and turn out to be given by

\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{22.16.2} \end{align}

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

¹

Some authors define \(P_\ell^{-m}(z)\) differently.

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{22.16.3} \end{equation}

so that \(P_\ell^m(z)\) 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 \(m\ne0\) (cf. factor of \((1-z^2)^{m/2}\))
\(P_\ell^m(-z) = (-1)^{\ell_m} P_\ell^m(z)\) (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 given value of \(m\text{,}\) the Associated Legendre functions form an orthonormal basis on the interval \(−1\le z\le1\text{.}\) Any function on this interval can be expanded in terms of anyone of these bases.