I am currently reading Forster. Let $X$ be a Riemann suface, and $Y$ an open subset of $X$. Assume we have a meromorphic 1-form $\omega \in \Omega(Y \setminus \{a\})$ with a pole at $a \in Y$. One defines the order of $\omega$ at $a$ as the order of $f$ at $a$, where $\left. \omega \right|_{Y \cap U} = f \, \mathrm{d}z$ in some chart $(U,z)$ around $a$.
How exactly can I prove that this definition is independent on the choice of chart? My strategy was to try to show that different charts give rise to functions that differ by a non-zero constant, so that the respective power series have the same order. However, having done very little differential geometry, I'm not very confident working with the technicalities involved.
Thanks.
Not exactly a nonzero constant, rather a non-vanishing holomorphic function.
Let $z : U_1 \subset X \to V_1 \subset \mathbb{C}$, $z': U_2 \subset X \to V_2 \subset \mathbb{C}$ be two coordinate systems involved, and $\phi :z(U_1 \cap U_2) \to z'(U_2 \cap U_1)$ sends the coordinate system of $z'$ to that of $z$. Here $\phi$ is a biholomorphic map, i.e. $\frac{d\phi}{dz'}$ is nonvanishing. Then you can check that $$fdz = (f\circ \phi) \frac{d\phi}{dz'}dz'$$ So the coefficient is off by $\frac{d\phi}{dz'}$, a nonvanishing holomorphic function.