Let us consider the following function: $\phi(s)=\phi_1(s)+i\phi_2(s)$, given uniquely by the polar form $\phi(s)=\rho(s)\exp(i\theta(s))$, where $\rho(s)=\sqrt{\phi_1^2(s)+\phi_2^2(s)}\neq 0$ and $\theta(s)=\arg\phi(s)\in\mathbb{R}$, is the argument of $\phi(s)$.
My question is: What happen if the argument $\theta(s)=\arg\phi(s)\in\mathbb{R}$ is zero for $s$ in the set $D=\{s=\alpha+i\beta\in\mathbb{C}, 0<\alpha<1\}$. I know about the analytic continuation. But I am not sure if this principle would be applied here just like the case of modulus.