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



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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.