Mapping Argand Plane to Upper Half Plane

[d2j2003]

Homework Statement

find linear fractional transformation from D={z:|Arg z| < $\alpha$}, $\alpha≤$$\pi$ to the upper half plane

Homework Equations

The Attempt at a Solution

The problem I am having here what exactly D is.. (visualizing it) D is just z such that |Arg z|≤$\pi$ right? so wouldn't that just be the entire complex plane? If we consider the Argument from 0 to $\pi$ and from 0 to -$\pi$ since |-$\pi$|=$\pi$ Is this correct??

 

[d2j2003]

anyone??

 

[Chaos2009]

I believe what this is saying is that you first select $\alpha \leq \pi$ and then form $D := \left\{ z : \left| \mathit{arg} \ z \right| < \alpha \right\}$. This would not be the entire complex plane. I believe for say $\alpha = \frac{\pi}{2}$ would look like $D = \left\{ z : \mathit{real} \ z > 0 \right\}$.

 

[d2j2003]

right but then if $\alpha$=$\pi$ wouldn't it just be the entire plane? since if it is $\pi$/2 then it is half of the plane..