The <Cauchy integral theorem> for a <star-shaped domain> says: if $U$ is open and star-shaped and $F$ is holomorphic on $U$, then \b[$\int_\gamma F(z)\,dz=0$ for every closed contour $\gamma$ in $U$]. Equivalently, $F$ has a <complex antiderivative> on $U$. Star-shaped means that there is one point of $U$ from which every straight segment to another point of $U$ remains in $U$.

Solution

Past exam of the metric spaces and complex analysis course of the University of Oxford / 2017 / 3 / b / ii / Solution

 Solution (source code)