 Solution

Changing the two local frames by the nowhere-zero functions ${\psi }_j$ replaces the transition function $f$ byWith the standard convention that the tautological bundle ${\mathcal{O}}_{{\mathbb{CP}}^1}(-1)$ has transition function $z$, its $k$th tensor power ${\mathcal{O}}_{{\mathbb{CP}}^1}(-k)$ has transition function $z^k$. ThusThis is the smooth classification of complex line bundles on the complex projective line by their winding number.