I often see an integral such as this:
$$\int a^z dz$$ where $a \ne 0$, for $z$ complex.
And then in the solution I see justification like the following:
Since $a^z$ "is an entire function, its primitive is given by" $$\frac{a^{z}}{ Log (a)}$$.
This seems to imply the following:
If a function is entire, then its antiderivative exists.
Now, I know it is true that if a function is entire, then it is differentiable everywhere, but is it also true that its antiderivative exists if it is entire?
Yes. In any simply-connected domain, a holomorphic function has an anti-derivative. In particular, this is true on $\mathbb C$.
$F(z)=\int_{\gamma } f(\xi)d\xi$ where $\gamma$ is the line segment from $0$ to $z$ defines the anti-derivative when the domain is $\mathbb C$.