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| &lt; \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 &gt; 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..