I am working with an ODE whose general solution is of the form, $$f(\theta) = \sin(\theta)^{-k}\left[c_1 P_\nu^\mu (\cos(\theta)) + c_2 Q_\nu^\mu(\cos(\theta))\right]$$ where $\mu,\nu\in \mathbb{R}$ (see here for more details) and $k>0.$ I have to obtain conditions on the constants under the boundary conditions $f(0)=f(\pi)=0.$
However, I am not sure what is the behavior of the functions $P_\nu^\mu(x)$ and $Q_\nu^\mu(x)$ when $x\to 1$ or $x\to -1.$ Thus to put it more clearly I would like to know whether for instance, $$\lim_{\theta \to 0} \sin(\theta)^{-k}\left[c_1 P_\nu^\mu (\cos(\theta)) + c_2 Q_\nu^\mu(\cos(\theta))\right]$$ is finite or not. Any comments regarding this will be much appreciated.