There is a proposition that every meromorphic differential on Riemann Sphere (or $\mathbb{P}^1 = \mathbb{C} \cup \{ \infty \}$) can be written as $f dz$ where $f$ is a meromorphic function on $\mathbb{P}^1$, and that $f dz$ is meromorphic if and only if $f$ is holomorphic on $\mathbb{C}$ and $z^2 f(z)$ tends to a finite limit as $z \rightarrow \infty$.

This is *Exercise 6.3.* on Page 178 of Frances Kirwan's *Complex algebraic curves*. I know how to prove the former half of the proposition (every meromorphic differential can be written as $f dz$), but I cannot find a proof of **the second half**. In fact, I'm not sure whether the propostion is true, because I seem to find a meromorphic differential which is not holomorphic on $\mathbb{C}$ (e.g. $1/z$ $dz$).

Does anyone can answer my question?

holomorphiciff those conditions are satisfied. $\endgroup$ – Sam Gunningham Apr 25 at 14:32