Let $ϕ(s)$ be an analytic function that has zeros outside a simply connected domain $D$. The function $ϕ(s)$ can be written as $ϕ(s)=ϕ₁(s)+iϕ₂(s)$ and therefore it is given uniquely by the polar form $ϕ(s)=ρ(s)exp(iθ(s))$, where, $ρ(s)=√(ϕ₁²(s)+ϕ₂²(s))≠0$ is the magnitude and $θ(s)=argϕ(s)∈ℝ$, is the argument of $ϕ(s)$.
We know that:
(P): If $ϕ(s)≠0$ and $ϕ$ is entire, then $θ(s)$ and $ρ(s)$ can be well-defined without branch cuts (the main reason here is that poles and zeros require branch cuts).
My question is: Does the property (P) holds true (at least locally) if we replace $ℂ$ by the simply connected domain $D$.
Yes. Starting with some $z_0\in D$, pick a value for $\log \phi(z_0)$ and proceed to define $\log \phi$ in $D$ by analytic continuation. The monodromy theorem tells you that $\log \phi$ is single-valued, no branch cuts. The real and imaginary parts of $\log \phi$ give what you want.