I'd like to find the image of the Unit square in the complex plane (vertices 0,1,i,i+1) under the Möbius Transformation $f(z)=\frac { 1 }{ z } $. I did plug in the values of the corner points to see on which point they get mapped.We have
$0\rightarrow \infty \quad 1\rightarrow 1\quad i\rightarrow -i\quad i+1\rightarrow \frac { 1 }{ i+1 } $ So the image would be the Unit square in the 4.Quadrant? But i think this is wrong because one needs to evaluate the lines connecting these points? How would i do this? Since im very new to complexe analysis some explanation on the line of thought would be appreciated.