While studying Complex Analysis from my professor's notes I came across the following theorem.
A demain $D$ in the complex plane is simply connected if and only if any analytic function $f(z)$ on $D$ that does not vanish at any point of $D$ has an analytic logarithm on $D$. For the proof, consider the function
$$ G(z) = \int_{z_0}^{z} \frac{f'(w)}{f(w)} \, dw \;\; \text{with} \ f(z) \neq 0. $$
It is my understanding that the proof of this theorem is very useful for problems/exercises involving the existence of analytic functions on simply connected domains.
I am looking for a complete proof of this theorem (with no "shortcuts", if possible) which will allow me to understand the above theorem in detail.