I have a generating function that has a closed form like $1/(\log(z-a)+b)$ and I would like to get asymptotics for the size of the coefficients of it.

I was going to use the methods in Chapter 5 of Wilf's Generatingfunctionology using the singularities. However, there is a branch point of log(z-a) at z=a. In Section 5.3 of Wilf's book, he discusses how to handle algebraic singularities (e.g., the branch point at $z=1$ of $\sqrt{z-1}$, but I have not been able to find anything about how to handle branch points when the function is transcendental (like the log function that I am considering).

Are there any known techniques for handling the branch points of such functions? If so, would anyone have references they could provide on this, please?