What is simply connected domain?

An open connected $\Omega \subseteq \C$ is simply connected (in the homological sense, sufficient for all our purposes) if $\operatorname{Ind}_\gamma(w) = 0$ for every cycle $\gamma$ in $\Omega$ and every $w \notin \Omega$ — “no cycle of $\Omega$ surrounds a hole”. By the global Cauchy theorem (Theorem 17.1) and Proposition 16.5, on such $\Omega$ every holomorphic function has a primitive; hence every zero-free $f \in \mathcal H(\Omega)$ has a holomorphic logarithm ($\exp\circ$(primitive of $f'/f$), adjusted by a constant, as $(f\eu^{-L})' = 0$) and holomorphic $n$-th roots $\eu^{L/n}$.