Let w be a meromorphic differential on a Riemann surface C. Show that we can choose an appropriate coordinate chart so that w = (z^u)dz, u is an integer, in some neighborhood of a pole. Prove also that this integer u is independent of the coordinate chart selected.
Here is my take on the uniqueness part:
Let w be a meromorphic differential which have the local representation
w = p(z)dz = ((a_n) z^n + (a_(n+1)) z^(n+1) + ...)dz, where a_n is non-zero and n is a negative integer. Then w has a pole of order n at the point q. Let z = f(w) be a mapping such that f(0)=0 and f'(0) is nonzero.Then we get gbar(w) = g(z)dz = g(f(w))(df/dw). So,
lim (w->0) (w^(-n))*gbar(w) = lim (w/f(w))^(-n)*[f(w)^(-n) g(f(w))(df/dw)
= f'(0)* lim (z->0) (z^(-n)) p(z).
Here f'(0) is nonzero by construction, and thus the last expression is finite and non-zero when so is lim (z^(-n)) p(z) (this can be proved easily). So this number n is independent of the local chart.
Is this proof correct?