winding number of $\gamma$ and point exterior to $\gamma$

Question

$$ \begin{align} n(\gamma,z_0)=\frac{1}{2\pi i}\int_\gamma\frac{1}{z-z_0}dz . \end{align} $$

Is it safe to say that $n(\gamma,z)=0,\forall z \in \mathbb{C}\backslash \gamma^*$

Answer 1

No. This is not true.

However, $n(\gamma,z_0)=0$, for all $z_0$ in the (unique) unbounded connected component of $\mathbb C\smallsetminus \gamma$.