Consider the function $f:(0,\infty)\to\mathbf{R}$ given by
$$ f(\alpha) = 1-\prod_{k=0}^{n} (1-Ae^{-\alpha k})$$
where $0<A<1$ is a fixed constant, and $n\in\mathbf{N}$ is fixed. For $A<1$, this function is invertible (for $A=1$, $f(\alpha)=1\ \forall\ \alpha$; for $A>1$ it is not injective).
My question is, is there an identity or something that might help me get an analytic expression for the inverse of this function? Or, does someone have an explanation of why it wouldn't exist?
I have tried expanding the product by hand but it got very messy and intractable-looking.