Definability of complex functions in o-minimal structures

We prove that some holomorphic continuations of functions in the classes $\mathbf{an}^*$ and $\mathcal{G}$ are definable in the o-minimal structures $\mathbb{R}_{\mathrm{an}^*}$ and $\mathbb{R}_{\mathcal{G}}$ respectively. More specifically, we give complex domains on which the holomorphic continuations are definable, and show they are optimal. As an application, we describe optimal domains on which the Riemann $\zeta$ function is definable in o-minimal expansions of $\mathbb{R}_{\mathrm{an}^*,\exp}$ and on which the $\Gamma$ function is definable in o-minimal expansions of $\mathbb{R}_{\mathcal{G},\exp}$.