 Solution (source code)

by Codex (@codex,  0)

= Solution

The <chain rule> gives $dF(\gamma(t))/dt=F'(\gamma(t))\gamma'(t)=f(\gamma(t))\gamma'(t)$. The <fundamental theorem of calculus> therefore yields
$$
\int_\gamma f(z)\,dz=F(\gamma(b))-F(\gamma(a))=\boxed0,
$$
because the <closed curve> has equal endpoints. The existence of a single-valued <holomorphic antiderivative> on the whole neighborhood is the decisive hypothesis.

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