Evaluating a complex integral over a half-ring I need to integrate the $z/\bar z$ (where $\bar z$ is the conjugate of $z$) counterclockwise in the upper half ($y>0$) of a donut-shaped ring. The outer circle is $|z|=4$ and the inner circle is $|z|=2$. 
My method:
$z/\bar z = e^{2i\theta}$ - which is entire over the complex plane.
So with respect to $d\theta$, we get the integral $re^{i3\theta} d\theta$ which, we can then evaluate at r=4 (from pi to 0) and r=2 from (0 to pi)
Two questions:
1) As integrating in the counterclockwise direction, surely I shouldn't be getting a negative number?
2) Via the deformation theorem, as the function is holomorphic on both circles and the region between them, should I not be getting 0? 

In the ccw direction, there are 4 contributions to this integral:
$$\begin{align}\oint_C dz \frac{z}{z^*} &= 4 \int_0^{\pi} d\theta \: e^{i 3 \theta} -  2\int_0^{\pi} d\theta \: e^{i 3 \theta} + \int_{-4}^{-2} dt + \int_2^4 dt\\ &= \frac{-8}{3 i} + \frac{4}{3 i} + 2 + 2 \\ &= 4 + i\frac{4}{3} \end{align} $$
The fact that this is $\ne 0$ has something to do with the fact that $z^*$ is not holomorphic within the integration region.
