Characterization of simply connected domain Let $D$ be a domain (open and connected) in $\mathbb{C}$. Then show that the following are equivalent:
(1) $D$ is simply connected (in homotopy sense);
(2) $\left( \mathbb{C} \cup \{\infty\}\right)\setminus D$ is connected;
(3) For each holomorphic function $f$ on $D$ such that $f(z) \neq 0$ for all $z \in D$, there exists a holomorphic function $g$ on $D$ such that $f = e^g$;
(4) For each holomorphic function $f$ on $D$ such that $f(z) \neq 0$ for all $z \in D$ and for each $n \in \mathbb{N}$, there exists a holomorphic function $h$ on $D$ such that $f = h^n.$
How unique are functions $g, h$ ?
I am not sure how to do $(1) \rightarrow (2), (2) \rightarrow (3)$ and $(4) \rightarrow (1)$. Actually in $(2)$, I think that $g := \log (f)$ which is not unique since $e^z$ is a periodic function. I am not sure why $(2)$ is needed to define $g := \log (f)$ in $(3)$. 
 A: We can show that $(1) \iff (2)$. This part is purely topological.
If $D$ is simply connected, we can carry a homotopy on 2-sphere such that this region is compressed to a point on sphere. $\mathbb{S}^2 \backslash \{pt\}$ is connected, so is $(\mathbb{C}\backslash D)\cup \{ \infty \} $. 
Ont the other hand, If $D$ is not simply connected, we can take a non-trivial loop inside $D$ so that the region bounded by this loop doesn't lie inside $D$ completely. That is, some point in the region is disconnected from $\infty$. So $(\mathbb{C}\backslash D)\cup \{ \infty \} $ is not connected.
(1) $\implies$ (3) or (4) is a direct use of complex logarithm. The function $g$ in
$$ f(z) = e^{g(z)} $$
is not unique unless you fix $g(z_0) \in \exp^{-1}(f(z_0))$ for a point $z_0 \in D$.
For (3) or (4) $\implies$ (1), it is again by topological argument. Assume $0$ is not in $D$, we can get $f: D \to \mathbb{C}^*$ as the inclusion map. If $\gamma$ is a loop in $D$, for example in the case of (3), 
$$ [\gamma] = f_*[\gamma] = exp_* \circ g_*[\gamma] = 0 $$
So, $\gamma$ is trivial in $\pi_1(D)$. Exactly, $\pi_1(D) = 0$.
