How to calculate Associated Legendre Functions as a function of theta?

Question

I’m trying to solve Problem 4.4 in Griffiths Quantum Mechanics, and I need to calculate the associated legendre functions $P^{m}_{\ell}(x)$ to do this. $P^{m}_{\ell}(x)$ are functions of $x$, yet in Griffiths, it is said that $P^{1}_{1} = -\sin(\theta)$ (clearly not a function of $x$). I then thought that maybe I should be plugging in $\cos(\theta)$ for $x$, but when I do that I don’t get the answer for $P^{1}_{1}$. Are we doing a change of variables technique here? Why isn’t $P^{1}_{1}$ a function of $x$?

Answer 1

The associated Legendre polynomial $P^{1}_1$ is $-(1-x^2)^{1/2}$. Substituting $\cos\,x$ and using the trigonometric identity $1-\mathrm{cos}^2x=\mathrm{sin}^2x$ should give you the result.