Why the function $g(q)=f(\log(q)/(2\pi i))$ is well defined?

Question

Let $D'=\{q\in \mathbb{C}: |q|<1\} - \{0\}$. Let $g: D' \to \mathbb{C}$ be defined by $g(q)=f(\log(q)/(2\pi i))$. Here $f: \mathcal{H} \to \mathbb{C}$ is a weakly modular function such that $f(\tau + 1)=f(\tau)$ and $\mathcal{H}$ is the upper half plane. Why the function $g(q)=f(\log(q)/(2\pi i))$ is well defined? We know that $\log$ is multivalued. Is it because $f(\tau+1)=f(\tau)$ and hence $f(\tau + k) = f(\tau)$ for all $k \in \mathbb{Z}$? Thank you very much.

Answer 1

Yes, you are correct. Since $\log q$ is well-defined modulo $2\pi i \mathbb{Z}$, and $f(z) = f(z + k)$ for all $k \in \mathbb{Z}$, the function $g$ is well-defined.