I am trying to calculate: $$\int_C \frac{z}{\bar{z}}dz$$ where C is the upper semicircles of the circles centred in (0,0) of radii 1,2 joined at their intersections with the real axis by the real segments [−2,−1],[1,2] (counter clockwise)
![the upper semicircles of the circles centred in (0,0) of radii 1,2 joined at their intersections with the real axis by the real segments [−2,−1],[1,2]](https://i.stack.imgur.com/Slmkb.jpg){kind=link}
$\bar{z}$ is the complex conjugate of z.
This is a basic problem from my mathbook. Their result is 0. However my calculation disagrees. I disected the contour into two line segments and two half circles. So I have four integrals:
1) $$\int_{-2}^{-1}\frac{x}{x}dx=1$$ 2) $$\int_{\pi}^{0}\frac{e^{it}}{e^{-it}ie^{it}}dt=\frac{2}{3}$$ 3) $$\int_{1}^{2}\frac{x}{x}dx=1$$ 4) $$\int_{0}^{\pi}\frac{e^{it}}{e^{-it}ie^{it}}dt=-\frac{4}{3}$$
When I add them up I get 4/3 which is not correct according to the book.The results in the book are usually correct so I want to ask if I do it right. (I can't use the Cauchy theorem since $\bar{z}$ is not holomorphic in a single point.)