If a function has an antiderivative, its complex integral around a circle will be $0$?

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I'm reading a proof of uniqueness of Laurent Series for a function. At page $4$, it says the following:

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Did you understand why the integrals vanish? It says that it's because they have antiderivatives within $N_{R_1}(z_0)$ (what is $N_{R_1}(z_0)$?)