Denote by $\mathbb{U}$ the complex unit disk. Let $\mathcal{O}$ an nonempty open subset of $\mathbb{R}^n$ $(n\geq 1)$, and $f\in\mathcal{C}^1(\mathcal{O}\times\mathbb{R},\mathbb{U})$ such that for all $x\in\mathcal{O}$, $f(x,0)=1$. Then, the following function $\Gamma$ defined over $\mathcal{O}\times\mathbb{R}$ by
$$\Gamma(x,t)=\int_0^t\frac{\partial_s f(x,s)}{i f(x,s)}\mathrm{d}s$$
satisfies $f(x,t)=e^{i\Gamma(x,t)}$ and for all $x\in\mathcal{O}$, $\Gamma(x,\cdot)\in\mathcal{C}^1(\mathbb{R},\mathbb{U})$.
Question: Is $\Gamma\in\mathcal{C}^1(\mathcal{O}\times\mathbb{R},\mathbb{U})$?
