Differential Forms on the Riemann Sphere I am struggling with the following exercise of Rick Miranda's "Algebraic Curves and Riemann Surfaces" (page 111):
Let $X$ be the Riemann Sphere with local coordinate $z$ in one chart and $w=1/z$ in another chart. Let $\omega$ be a meromorphic $1$-form on $X$. Show that if $\omega=f(z)\,dz$ in the coordinate $z$, then $f$ must be a rational function of $z$.
I have unfortunately no idea how I should begin the proof (since I am new to this topic). Can someone give me a hint?
Edit: The transition map is $T:z\rightarrow 1/z$. We have $\omega=f(z)dz$ in the coordinate $z$. Then we know that $\omega$ transforms into $\omega_2=f_2(w)\,dw$ as follows: $f_2(w)=f(T(w))T'(w)=f(1/w)(-1/w^2)$, but how does this help?

EDIT: By multiplying by an appropriate polynomial, we may assume that $\omega$ has poles (at most) at $0$ and $\infty$.
On $\Bbb C-\{0\}$ you now have holomorphic functions $f$ and $g$ (your $f_2$) with $$z^2f(z)=-g(1/z).$$ Since $f$ and $g$ have at worst poles at $0$, this equation tells us that each of their Laurent series has only finitely many nonzero terms.
