Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 115 / 2 / e / Solution

Parametrize the unit circle by a loop $\gamma :[0,2\pi ]\to {\mathbb{C}}^{\ast }$. Part (d) and $f(\gamma (0))=f(\gamma (2\pi ))$ implyThe kernel of the complex exponential function is $2\pi i\mathbb{Z}$, sofor some $k\in \mathbb{Z}$. This integer is the winding number of $f{|}_{S^1}$.