What is Winding number?

For a closed path $\gamma$ and $z \notin \operatorname{im}\gamma$, the index is

It is an integer: setting $\varphi(t) = \int_a^t\frac{\gamma'(s)}{\gamma(s) - z}\dd s$, the function $(\gamma(t) - z)\eu^{-\varphi(t)}$ has zero derivative (piecewise), hence is constant; at $t = b$, $\eu^{\varphi(b)} = \frac{\gamma(b) - z}{\gamma(a) - z} = 1$, so $\varphi(b) \in 2\iu\pi\Z$. As a function of $z$, the index is continuous on $\C\setminus\operatorname{im}\gamma$ (dominated convergence), hence constant on each connected component, and $0$ on the unbounded component (ML: the integral tends to $0$ as $z \to \infty$). For the circle $\gamma(t) = a + r\eu^{\iu t}$, $t \in \intcc0{2\pi}$: $\operatorname{Ind}_\gamma(z) = 1$ for $z \in D(a,r)$ (compute at $z = a$: $\frac1{2\iu\pi} \int_0^{2\pi}\frac{r\iu\eu^{\iu t}}{r\eu^{\iu t}}\dd t = 1$; constancy does the rest).

Let $\gamma\colon\intcc01\to\R^2\setminus\{a\}$ be a smooth closed curve. Its winding number around $a$ is