I'm trying to solve my first Möbiustransformation, and im struggling abit.
Let $f(z)=\frac{2z}{z+2}$ and calculate what the transformation will be using $0$, $\pm2$, $\infty$, $\pm(1+i)$.
a) the x-axis plus $\infty$
I know, by definition that circles maps to circles or lines and lines to circles or lines. I also know that if the transformation contains $\infty$ its a line and that Möbiustransformations contains it's angles.
So far i've done
$f(0)=0$, $f(\infty)=2$
which gives me two points on the x-axis.
My idea new would be to try to find a third point which then, i guess, would make it obvious if it maps a circle or a line.
So i then tried with $f(1+i)=\frac{2+2i}{3+i}$ which dont really gives me a clear point on the plane.
My next idea would be to work with the idea that Möbiustransformations contains its angles.
Would someone like to help me out and give me hint?
Thanks!