Analytic properties of $\log (1-e^{-\lambda(z-a)})$.

Question

I am trying to figure out the resolvent corresponding to a particular Riemann-Hilbert problem with a singular kernel. In the process of calculation I require the branch cuts and poles (if any) of the function $\log (1-e^{-\lambda(z-a)})$. What I am seriously confused about is
1) Is $z=\infty$ a branch point?
2) Are there any other branch points other than $z=a+ \frac{2\pi i n}{\lambda}$ for $n \in \mathbb{Z}$?
3) If I have indeed got the branch points correctly, can one give one ``nice" realization of the branch cuts (I understand this choice depends on the kind of problem I am trying to address, but what would be the simplest thing one would do?)
Assumptions: It is fine to assume $\lambda>0$ and $a>0$. These are input that comes from physics (these functions come up when you are looking at Young diagrams growing under a $q$-deformed Plancherel Growth process. As a mathematical curiosity, I am sure it will be fun to study this for generic $\lambda$ and $a$, but for the time being getting to know the behaviour of this function under the above mentioned constraints is also fine.)

Answer 1

By translation and scaling, we may assume wlog $\lambda = 1$ and $a=0$, so your function is $\log(1-e^{-z})$.

Technically, $\infty$ is not a branch point because it is the limit of points $2\pi i n$ where the global analytic function does not exist.

You could take the branch cuts to be $\{iy: y \ge 2\pi \ \text{or}\ y \le 0\}$.