Let $\displaystyle f(z)=z+ \frac{1}{z}$.

What's the image of :

1)A circle with radius $\displaystyle r$ centered at $\displaystyle z=0$?

2)A circular semi-ring $\displaystyle \{ z=re^{i\theta} | 0 \leq \theta \leq \pi$ , $\displaystyle 1 \leq r \leq b \}$, $\displaystyle b \geq 1$?

What's the preimage by $\displaystyle f$ of the semi-plane $\displaystyle \{ w : w \geq 0 \}$?

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For now I'm only looking for help for the 1).

My attempt : I wrote $\displaystyle z$ as $\displaystyle re^{i \theta}$ to get $\displaystyle f(re^{i\theta})=r (\cos \theta + i \sin (\theta))+ r^{-1} (\cos \theta + i \sin \theta)^{-1}$. I don't recognize what it is. Especially the part $\displaystyle r^{-1} (\cos \theta + i \sin \theta)^{-1}$.

Any idea about how I could rewrite $\displaystyle f(re^{i\theta})$?