What is the residue of the following function: $$f(z)=\frac{\exp \left(1-\dfrac{1}{\frac{z-4i}{1-4i}-1}\right)}{\exp\left(\dfrac{-2\pi i}{\frac{z-4i}{1-4i}-1}\right)-1}$$ at $z=1$? I know it's equal to infinity at $z=1$, but I cannot compute it's residue there.
EDIT: I've been calculating it in sympy, to no avail. Also, I've plotted the function, and there is definitely a singularity at $z=1$. I have no idea how to go about it. Is there even a residue there (as in, does the limit as $z\to 1$ in the expression above exist?)