 Complex antiderivative

A holomorphic function $F$ on a domain with complex derivative $F^{\prime }=f$. Its contour integral satisfies ${\int }_{\gamma }f(z)dz=F(\gamma (b))-F(\gamma (a))$ by the chain rule along each smooth contour segment. In particular every closed contour has integral zero. Having local complex antiderivatives need not provide a single one on a domain with holes.