 Solution

The Cauchy integral theorem for a star-shaped domain says: if $U$ is open and star-shaped and $F$ is holomorphic on $U$, then ${\int }_{\gamma }F(z)dz=0$ 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 $U$.