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 $\mathbb{ R}_{ \mathcal{ G},\exp}$.
Categories: math.LO