In general, in that kind of question, there isn't a domain defined (I think). If I ask you to tell me the answer of the contour integral $\int_{\gamma}\frac{1}{z-2} dz$ where $\gamma$ is simply the unit circle $C(0,1)$, you will tell me : it's easy by the Cauchy theorem, $\int_{\gamma}\frac{1}{z-2} dz = 0$ because $2 > 1$. But according to the definition of the Cauchy theorem, how is defined the domain? Or simply, what is the domain here? (To know that $\mathbb{C}$ is not the domain.)
Cauchy theorem : Let $D \subset \mathbb{C}$ a domain, $f : D \to \mathbb{C}$ a holomorphic function and $C$ a closed path contained with his interior in $D$. Then $\int_C f(z)dz=0$
A domain, or a region, is a connected open set.
The connectedness ensures that the antiderivative is unique up to a constant. The openness ensures that the derivative can be calculated for every point.