EDIT: I can't seem to delete this question, so I've posted the solution below (I must have made an algebra error-someone checking all this would still be appreciated!). I've left the question as is.
The Bardeen-Press (i.e. Teukolsky equation for a Schwarzschild black hole) in Schwarzschild coordinates reads:
\begin{equation} \frac{r^4}{r^2-2Mr}\partial_t^2\psi -2s\left(\frac{Mr^2}{r^2-2Mr}-r\right)\partial_t\psi -\left(r^2-2Mr\right)^{-s}\partial_r\left(\left(r^2-2Mr\right)^{s+1}\partial_r\psi\right) +(l-s)(l+s+1)\psi = 0 \end{equation}
This equation is not regular at the horizon (r=2M). I want to transform this equation to horizon penetrating (more specifically, ingoing Eddington-Finkelstein) coordinates. I apply
\begin{equation} v=t+r+2M\mathrm{log}\left(r-2M\right) , \end{equation}
use the chain rule, and get an equation of the form
\begin{equation} -2r^2\partial_v\partial_r\psi +\frac{2r\left(\left(s-3\right)M+\left(2+s\right)r\right)}{2M-r}\partial_v\psi +\frac{q(r)}{(2M-r)r}\psi +\cdots = 0 , \end{equation}
where $q(r)$ is a long function and $\cdots$ contains terms regular in $r-2M$. I have read sources that seem to suggest that if I perform the field redefinition
\begin{equation} \psi\equiv r^{-1}\left(r^2-2Mr\right)^{-s}\Psi , \end{equation}
and rewrite everything in terms of $\Psi$ the equation becomes regular at the horizon (still at $r=2M$). I have tried this and for the life of me I cannot seem to get a regular equation at the horizon. When I plug in the above transformation I get
\begin{equation} -2r^2\partial_v\partial_r\Psi -\frac{2r\left(\left(1-3s\right)M+\left(s-1\right)r\right)}{2M-r}\partial_v\Psi +\cdots = 0 , \end{equation}
the coefficient for $\partial_v\Psi$ is still singular! I feel like I may be making a algebraic error but I cannot find where the problem is.
I am looking for a reference or an explicit calculation that the above manipulations lead to a wave equation regular on the black hole horizon.
