Domain for a complex function

Question

For which $z$ values is the complex function below undefined?

$$\frac{1}{e^z-1}$$

Am I under thinking this problem, or shouldn't the domain be all complex values except for $z=0$?

Answer 1

It's undefined when $e^z=1,$ and notice that $e^{\pm2\pi i} = 1$ and $e^{\pm4\pi i} = 1,$ etc.

The slightly more involved question is how do we know there are no other points where it's undefined? And there you need to observe that $e^z$ is defined for all values of $z.$ And also ask yourself how you know that there are no other values of $z$ for which $z=1$ than the ones included in the "etc." of the first paragraph above.

Answer 2

HINT: The complex exponential is periodic. Consider the trigonometric definition of $\exp(z)$ or more clearly $\exp(ix)$.