Let $f\colon \mathbb{C} \rightarrow \mathbb{C}$ be a periodic (i.e. $f(z+1)=f(z)$ ), everywhere holomorphic over $\mathbb{C}$ function. We can write its Fourier series $f(z)=\sum_{n\in\mathbb{Z}} a_n q^n $ where $q=\exp(2n\pi i z)$. Is it true that this series is convergent for every value of $z\in\mathbb{C}$?
My idea is to use the fact that a $1$-periodic function is a function $g\colon \mathbb{C}^*\rightarrow \mathbb{C}$ of $q$. We can expand this in its Laurent series centered in $0$ and this should gives us the Fourier expansion.