I have a question with this proof. Let's see the proof and the result first.
Result: If $ \gamma $ is any closed path in $ C_{ \ne 0} $ then $ \frac{1} {2\pi i}\int\limits_\gamma \frac{dz}{z} \in {\Bbb Z} $
Proof: Let $f$ be a primitive of $ dz/z $ along $ \gamma $ Then $ \int\limits_\gamma \frac{dz} z = f\left( b \right) - f\left( a \right) $ where $[a,b]$ parameterizes $\gamma$ . Since $\gamma(a) = \gamma(b) $, this difference is just the difference between two branches of $\log z$, hence of the form $ 2\pi i n$
I don't understand how it uses the fact that $\gamma(a) = \gamma(b)$.