Past exam of the mathematics course of the University of Cambridge / 2005 / ib / Paper 1 / 3F (Complex Analysis or Complex Methods)

The Cauchy integral formula states that if $f$ is holomorphic on a neighborhood of a positively oriented simple closed contour $C$ and its interior, then for $w$ inside,For the given counterclockwise circle, divide the rational function:The contour integral of $z$ is zero because it has a primitive, while both points $\pm i$ lie inside the circle. Applying the Cauchy integral formula to the constant function $1$ at each point givesThe sign reverses if the contour orientation is reversed.