I'm reading about modular forms and am told that any meromorphic function $f:\mathbb{H}\rightarrow\mathbb{C}$ can be realised as a meromorphic function $f:\mathbb{D}\setminus\{0\}\rightarrow\mathbb{C}$ instead.
The image I have in my head is a transformation that takes lines parallel to the real axis to circles centred at the origin, so that the real axis corresponds to the boundary of the open unit disc and the point at infinity corresponds to the origin (i.e. the circle of zero radius). But I don't know how to explicitly construct such a function? It's been a while since I've done any complex analysis, forgive me if this is simple! Thanks
EDIT: I also have the fact that $f(z)=f(z+1)$ for all $z\in\mathbb{H}$, but I'm not sure if that's relevant.
EDIT 2: I realise that my first edit might be quite useful; it seems that the map $z\mapsto e^{2\pi i z}$ seems to work?
To answer the question in your title: $$ f(z) = e^{2\pi i z} $$ (where the $2\pi$ is so that the property in your edit also holds) This matches with the image that you had in your head, as long as you were also imagining the radii of the circles becoming smaller exponentially.
Edit: Ah, I see that you already discovered this on your own :)