Solution
= Solution
For $h=z^{-k}f$,
$$
\alpha=h^{-1}dh=\beta-kz^{-1}dz.
$$
The scalar-valued form $h^{-1}dh$ is closed, and
$$
\int_{S^1}\alpha=2\pi ik-k\int_{S^1}z^{-1}dz=0.
$$
By the period isomorphism from part (b), $[\alpha]=0$ in complexified <de Rham cohomology>. Hence there is a smooth complex-valued function $\varphi$ on $U_0\cap U_1$ with $\alpha=d\varphi$.