I was reading this, and wasn't able to solve equation (2.34). The equation is:
$$\Big[\nu^2 + \frac{\rho^2 -1}{\rho^2} \partial_{\rho}(\rho^2 (\rho^2 -1)\partial_{\rho}) \Big]f(\rho) = 0,$$
where $\rho$'s range is $(1,\infty)$.
I tried solutions of the form $f(\rho) = \frac{g(\rho)}{\rho}$, and further $\rho = \cosh[x]$. Then in the asymptotic limit $x \to 0$, the solution goes like $$g(\cosh x) = \left(\coth {\frac{x}{2}}\right)^{i\nu} g_1(\cosh x) $$
The differential equation for $g_1$ becomes then $$\frac{d^2g_1}{dx^2} + [\coth x -2i\nu\, \text{cosech}\, x]\frac{dg_1}{dx}-2g_1=0$$
I don't know how to proceed from here. I tried out the solutions using Mathematica also, but that didn't help. How do I solve the same? Thanks.