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Question:- Show that the image of the set $ S = {z \in \mathbb{C} | Im(z) \geq 0, |z|\geq 1} $ under the map $w = u+\iota v = z+ \frac{1}{z}$ is the upper half plane of $v\geq 0$

My approach Let $ z= re^{\iota \theta}$. Then $ w=re^{\iota \theta} + \frac{1}{r} e^{-\iota\theta} $ The mapping given in question is semicircle covering above half of $z$ plane, and the real axis.

Therefore i have taken three regions for mapping onto $w-plane$

case-1$$ z= x, x\geq1$$ Case-2$$z= -x, x\geq 1$$ Case-3 points on boundry of surface of the semi circle.

I understand the first two cases, but don't have any approch to third one. hints are highly appreciated

thankyou

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Points on the semicircle are of the form $e^{i\theta}$ with $0\leq \theta \leq \pi$. For $z=e^{i\theta}$ we have $z+\frac 1 z = e^{i\theta}+e^{-i\theta}=2\cos\, \theta$ and $\cos\, \theta$ takes all values between $-1$ and $+1$ so the image of the semicircle is the interval $[-2,2]$ of the real axis.

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