Definability of complex functions in o-minimal structures

We prove that some holomorphic continuations of functions in the classes ${\mathbf{a}\mathbf{n}}^{\ast }$ and $\mathcal{G}$ are definable in the o-minimal structures ${\mathbb{R}}_{{\mathrm{a}\mathrm{n}}^{\ast }}$ 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 ${\mathbb{R}}_{{\mathrm{a}\mathrm{n}}^{\ast },\exp }$ and on which the $\mathrm{\Gamma }$ function is definable in o-minimal expansions of ${\mathbb{R}}_{\mathcal{G},\exp }$.