Primitive function: set of definition

Question

I hae to determinate the set of definition of the premitive of this complex function: $$ f = \frac{e^z -1}{ ( z^2-z )^2 } $$

This is easy if I get the primitive but is there a faster way to determinate it?

Thank you.

Answer 1

The poles are at $z=0$ and at $z=1$, with nonzero residues at each ($1$ and $2-e$ respectively). Therefore a primitive would have a branch point around each of those poles. Of course you have a choice of where to put the branch cuts, and thus what domain to use for the primitive. The sum of the residues is nonzero, so the branch cut will have to go out to $\infty$. For example, you could define a primitive on the complex plane with the ray $(-\infty, 1]$ on the real line removed, or with the ray $[0,+\infty)$ removed, or with separate rays in some directions from $0$ and from $1$ removed.