# The cotangent bundle of a non-compact Riemann surface

Suppose that $$M$$ is a non-compact Riemann surface obtained by removing several points from a compact one. It is known that the holomorphic cotangent bundle of $$M$$ is trivial. Therefore there exists a holomorphic 1-form $$\omega$$ that does not have any zeroes on $$M$$. I am interested to know whether we can find an exact 1-form $$\omega=df$$, where $$f$$ is some holomorphic function on $$M$$, such that $$\omega$$ does not have zeroes on $$M$$.

Such $$f$$ exists on every open Riemann surface: