Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 4 / a / Solution

A sufficient contour setting for both results is a positively oriented, simple, closed, piecewise smooth contour $C$, with the function holomorphic on an open neighbourhood of $C$ and its entire interior. The Cauchy integral theorem then statesIf $z_0$ is strictly inside $C$, the Cauchy integral formula statesThe contour must not pass through $z_0$. Positive orientation means anticlockwise; reversing it reverses the sign of the contour integral. The holomorphy assumption throughout the interior excludes hidden singularities there.