Solution
= Solution
Take the unit circle $\gamma(t)=e^{it}$, $0\le t\le2\pi$, in the <punctured complex plane>. Direct parameter integration gives
$$
\int_\gamma\frac{dz}{z}=\int_0^{2\pi}e^{-it}ie^{it}\,dt=2\pi i\ne0.
$$
A global <holomorphic antiderivative> would make this integral zero by the previous part. Thus \b[there is no such antiderivative on the punctured plane]. Local primitives can exist without combining into one single-valued function on this domain.