## Franklin's Notes

### Winding number

Given a closed curve $\gamma\subset\mathbb C$ and a point $z\in\mathbb C$ not on $\gamma$, we define the winding number, or the index of the curve $\gamma$ about the point $z$, as follows: Intuitively, this quantity measures how many times the curve $\gamma$ "wraps around" the point $z$ clockwise. In particular, if $\gamma$ is the boundary of a circle oriented counterclockwise, then $\mathrm{ind}(\gamma,z)=1$ for points $z$ inside the circle, and it equals $0$ for points $z$ outside the circle.

It is not entirely trivial that this quantity is necessarily even an integer, but we can prove this as follows. Let $y(t)$ be a parametrization of the curve $\gamma$, so that Let us define a function $E(\alpha)$ for $\alpha\in [0,1]$ as follows: so that we have Note that this is defined for all $\alpha\in [0,1]$ because $z$ does not lie on $\gamma$ by assumption, meaning that we always have $y(\alpha)\ne z$ and the denominator in this expression is nonzero. Now, consider the function and notice that we have meaning that $f$ is a constant function, since it is differentiable $\mathbb R\to\mathbb C$. This means that there is a constant $K$ such that and since $\gamma$ is closed, we have $y(0)=y(1)$ and therefore $E(0)=E(1)$, so that $E(1)/E(0)=1$. But this means that and therefore the value of the integral must be a multiple of $2\pi i$, so that the quantity is an integer as claimed!