Is the principle part of a meromorphic differential independent of local parameter?

Question

Suppose we have a Riemann surface $S$, a local parameter $z$ around point $p$ and a principle part $f(z)=\sum_{i\ge -n} a_iz^i$. Then $\eta=f(z)dz$ is a meromorphic differential. We use another parameter $w$ and $\eta=f(z)dz$ becomes $\sum_{i\ge -n} b_iw^i dw$. Do we have $a_i=b_i$ for $i<0$ ? I know $a_{-1}=b_{-1}=1/2\pi i \oint \eta$

Answer 1

No. For example, consider $$\alpha = \left( \frac{1}{z^2} + \frac{1}{z}\right) dz$$ on $\mathbb C$. With another local coordinates $z=2\omega$,

$$ \alpha =\left( \frac{1}{(2\omega)^2} + \frac{1}{2\omega}\right) d(2\omega) = \left( \frac{1}{2\omega^2} + \frac{1}{\omega}\right) d\omega.$$