Cauchy's Conundrums

Any analytic function integrated on any closed contour with no singularities has integral value of 0 (Cauchy Integral Theorem).

An open set $\Omega \subset \mathbb{C}$ is said to be holomorphically conected if it is a conected open set and $\forall f \in \mathcal{H} (\Omega), \space \exists F \in \mathcal{H} (\Omega) \space / \space F '(z) = f(z) \space \forall z \in \Omega$ .

In this case, $\mathbb{C}$ is a holomorphically conected set and in these sets is fullfiled $\oint \gamma f(z) dz = 0, \space \forall f \in \mathcal{H} (\mathbb{C})$ being $\gamma$ a regular closed path.

Of course, in this case the function $f(z) = z^3$ is a holomorphic function in $\mathbb{C}$ and $\gamma$ is a regular closed path...