Deciding whether a complex function has a primitive in any open subset of $\mathbb{C}$ I need to decide whether $g(z)=\bar{z}$ has a primitive in any open subset of $\mathbb{C}$. I'm struggling to decide where to start. I was thinking of using Cauchy's integral theorem i.e. if $g(z)$ is holomorphic then $$\int_C g(z) \: \text{d}z=0,$$ where $C$ is a closed path. But I'm really not sure. Some tips would be great!

$\overline{z}$ is not even holomorphic. Let $\gamma$ be the unit circle in the positive orientation, then
$$ \int_\gamma \overline{z}\,dz = i \int_0^{2\pi} \overline{e^{i\phi}} e^{i\phi}\,d\phi = i\int_0^{2\pi} \,d\phi = 2\pi i
$$
By Goursat's theorem, $\overline{z}$ cannot be holomorphic in the unit disk. You can expand $\gamma$ to show that $\overline{z}$ is not holomorphic in any disk.
