Möbius-transformation $w=\frac{z-i}{z+i}$

Question

I'm struggling a bit with the Möbius transformation below.

Describe the image of the region under the transformation

b) the quadrant $x>0, y>0$ under $w=\frac{z-i}{z+i}$

My solution is so far:

1. Check that it is in fact a valid M.transformation with $ad-bc \neq 0$.
2. Calculate transformation of 3 points on the edge of quadrant, using points in a specific order:

$p_1=(i)$ $\Rightarrow w(p_1)=0$

$p_2=(0)$ $\Rightarrow w(p_2)=-1$

$p_3=(1)$ $\Rightarrow w(p_3)=-i$

At this point, i assumed it would be enough with 3 points, but when looking at the image i get and the answer it does not make sense how to end up with the answer.

Would anyone like to give me hint on how to proceed?

Answer 1

Hint (too long for a comment)

Rearrange to make $z$ the subject and write $w=u+iv$

You can apply the conditions $Re(z)>0$ and $Im(z)>0$ to obtain inequalities in terms of $u$ and $v$

Answer 2

Using the hints, check that $f(2i)=1/3$. Putting that together with $f(0)=-1, f(i)=0$, we see that the imaginary axis goes to the $x$-axis. Since $f(1)=-i$, the right half plane goes to the lower half plane.

But it's easy to see, and well-known, that the upper half plane goes to the unit disk.

Finally, the intersection of the unit disk and the lower half plane is the lower half disk.