 Solution

The chain rule gives $dF(\gamma (t))/dt=F^{\prime }(\gamma (t)){\gamma }^{\prime }(t)=f(\gamma (t)){\gamma }^{\prime }(t)$. The fundamental theorem of calculus therefore yieldsbecause the closed curve has equal endpoints. The existence of a single-valued holomorphic antiderivative on the whole neighborhood is the decisive hypothesis.