A First Course in Complex Analysis · Problem 4

Consequences of Cauchy's Theorem — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · complex-integration

Problem

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

\int_{C[w,r]} \frac{dz}{z - w} = 2\pi i.