Proposition

compute $\int_{\gamma} \frac{dz}{z}$ where $\gamma$ is the unit circle, oriented counterclockwise. More generally, show that for any $w \in \mathbb{C}$ and $r > 0$,

Solution

It suffices to prove the second statement. Let $w \in \mathbb{C}$ and $r > 0$ be given. Let $f(z) = 1$. Then $f(z)$ is holomorphic on $\mathbb{C}$, which is an open set containing $\overline{D}[w, r]$. By Theorem 4.24, $f(w) = \frac{1}{2\pi i}\int_{C[w, r]}\frac{f(z)}{z - w}dz$. This implies $\int_{C[w, r]} \frac{dz}{z - w} = 2\pi i$.