$$\oint_\Gamma z^{3/2} dz, \quad \text{where } \Gamma : |z| = 1$$
It should meet the requirements of the Theorem of Cauchy-Goursat so it should be $0$. But when I do it:
$$ \oint_0^{2\pi}e^{3i\theta/2} ie^{i\theta}d\theta = \oint_0^{2\pi}ie^{5i\theta/2}d\theta = -4/5$$
Please tell us what is your definition of $z^{3/2}$ so that it's a continuous function on that path of integration?
You might as well just ask your question with $z^{1/2}$ too: the square root function is more basic and exhibits the same issue.