how to evaluate this complex integral

Question

$I=\int_{L} \frac{z}{\bar{z}}dz$, where L is $1\le|z|<2$, $Im(z)\ge 0$ traversed counterclockwise. thanks for helping.

$I=\int_{L} \frac{z}{\bar{z}}dz=\int_{L}{z^2\over |z|}dz$,

Answer 1

Let

$C_1:\,z=2e^{i\theta}~~;~~0\leq\theta\leq\pi$.

$C_2:\,z=t~~;~~-2\leq t\leq-1$.

$C_3:\,z=e^{i\theta}~~;~~0\leq\theta\leq\pi$.

$C_4:\,z=t~~;~~1\leq t\leq2$. \begin{eqnarray} I&=&\int_{L}\frac{z}{\bar{z}}dz\\ &=&\int_{C_1}+\int_{C_2}-\int_{-C_3}+\int_{C_4}\\ &=&\int_0^\pi e^{2i\theta}2ie^{i\theta}d\theta+\int_{-2}^{-1}dt-\int_0^\pi e^{2i\theta}ie^{i\theta}d\theta+\int_{1}^{2}dt\\ &=&\dfrac{-4}{3}+1-\dfrac{-2}{3}+1\\ &=&\dfrac{4}{3} \end{eqnarray}