I am reading Koblitz's book Introduction to Elliptic Curves and Modular Forms. The author extends the usual topology on the half plane $\Bbb H$ to $\Bbb H \cup \{\infty\}$ in view of the map $z\mapsto q = e^{2\pi iz}$. This is done here.
The change of variables $(1.8)$ from $z$ to $q$ plays a basic role in the theory of modular functions. We use $(1.8)$ to define an analytic structure on $\Bbb H \cup \{\infty\}$. In other words, given a function on $\Bbb H$ of period $1$, we say that it is meromorphic at $\infty$ if it can be expressed as a power series in the variable $q$ having at most finitely many negative terms, i.e., it has a Fourier expansion of the form $$f(z) = \sum_{n\in \Bbb Z} a_n e^{2\pi inz}$$ in which $a_n = 0$ for $n \ll 0$.
The above paragraph makes sense to me, but it is not clear when a periodic function $f$ on $\Bbb H$ (with period $1$) admits a Fourier expansion of the above form in the first place. Are there any restrictions on $f$? I haven't taken a course in Fourier Analysis yet, and I believe that's where the answer may lie. Thanks!
The expression written is same as expression of an analytic function as power series evaluated at $z' = e^{i2\pi z}$ i.e., $f(z') = \sum_n a_n z'^n$. So the answer is when $f$ is analytic with Region of convergence (ROC) including $z'= e^{i2\pi z}$. If $z=x+iy$, $z'$ is in ROC when $\limsup_n |a_n|^{1/n} e^{-2 \pi y} < 1 $ => $\limsup_n |\frac{f^n(a)}{n!}|^{1/n} e^{-2 \pi y} < 1 $ where $a$ is the center of ROC we are targeting. This is a necessary condition. But not sure if this is sufficient.