Parametrize the unit circle by a loop $\gamma:[0,2\pi]\to\mathbb C^*$. Part (d) and $f(\gamma(0))=f(\gamma(2\pi))$ imply
$$
\exp\left(-\int_{S^1}\beta\right)=1.
$$
The kernel of the <complex exponential function> is $2\pi i\mathbb Z$, so
$$
\int_{S^1}\beta=2\pi i k
$$
for some $k\in\mathbb Z$. This integer is the <winding number> of $f|_{S^1}$.

Solution

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

{title2=$e^z$}

The complex exponential is the entire function
$$
e^z=\sum_{n=0}^{\infty}\frac{z^n}{n!}.
$$
It maps $\mathbb C$ onto $\mathbb C^*$ and has kernel $2\pi i\mathbb Z$.


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