how can we find domain of analyticity of $f(z)$

Question

If I have a complex logarithm $Log(z+4)$, to find its domain of analyticity do I thought using the theorem;

The domain of analyticity of any function $f(z)=Log[g(z)]$, where $g(z)$ is analytic, will be the set of points $z$ such that $g(z)$ is defined and $g(z)$ does not belong to the set ${z=x+iy | −∞<x≤0,y=0}$.

So, is this set analytic or not?

Also, ı need to compute $f'(z)$, $f(i)$, and $f'(i)$ .

Answer 1

Just shift the domain of analyticity of $\log z$. And recall $\log'z=1/z$ and use the chain rule.

Also, sets aren't analytic, functions are.