In my course in complex analisys i was given following problem:
Find the image of set $D$ under the map $f$ if $D=\{z \in \mathbb C| 0 < Re(z) < 1\}$ and $f=\frac{z-i}{z-1}$.
The problem is that I don't quite understand how to describe the image. I am supposed to use some geometric properties of Möbius transformations like preservation of angles, but I can't see how.
Any help will be appreciated. Thanks!
The set $D$ is an open strip of the complex plane whose left and right boundaries are described, respectively, by $\{z=z_r+iz_i\in\mathbb{C}:z=iz_i\}$ and $\{z=z_r+iz_i\in\mathbb{C}:z=1+iz_i\}$. You should look at the image of those lines under $f$ and then fill in in between since Möbius transformations take lines and circles to lines and circles and fill in everything in between by continuity.