I'll denote by $\mathbb{C}^*$ the punctured complex plane $\mathbb{C} \setminus \{0\}$. Say that I've got some open cover $\{V_j\}_{j \in J}$ of the closed unit interval $[0,1]$, and continuous functions $w_j: V_j \times \mathbb{C}^* \to \mathbb{C}^*$ such that, for any $t \in V_j$,
(a) $w_j(t, \cdot)$ is a holomorphic function $\mathbb{C}^* \to \mathbb{C}^*$ and
(b) $w_j(t, \cdot)$ has a holomorphic antiderivative, say $F: \mathbb{C}^* \to \mathbb{C}$.
I want to find a function $u: [0,1] \times \mathbb{C}^* \to \mathbb{C}^*$ that such that, for fixed $t \in [0,1]$, $u(t, \cdot)$ is still holomorphic and still has a holomorphic antiderivative.
My first thought was to use partitions of unity with respect to this cover, say $\{p_j\}_{j \in J}$ and to set $u(t, \cdot) = \sum_{j} p_j(t)w_j(t, \cdot)$. Of course, such a $u$ does still have an antiderivative for each $t$ but now we can't be sure whether the image of $u$ still lies in $\mathbb{C}^*$.
I get that this is a bit vague, but I'd appreciate any good advice and/or ideas...can someone think of another way of constructing $u$. Or perhaps I can pursue the partitions of unity idea, provided $w_j$ and/or the open cover have some additional features, and if so, what kinds of features would enable this? I'd appreciate any ideas.