I did this:
Assuming my Möbius transformation is some $\omega$ in terms of $z$, I want to work out a formula that gives me:
- 1) $\omega = 0$ when $z = i$
- 2) $\omega = \infty$ when $z = 0$
- 3) $\omega = -i$ when $z = \infty$
So I did this
1) Happens when $\omega = z - i$
2) Happens when $\omega = \dfrac{1}{z}$
3) Don't know when this happens
So, by combining the top two functions, I get my Möbius transformation to be $\omega = \dfrac{z - i}{z}$.
This was wrong. The answer is $\omega = \dfrac{-iz - i}{z}$ and it gets this in a really wierd way which I don't understand. The answers say:
Omitting the terms containing $\infty$ in
$ \dfrac{z - z_1}{z - z_3} \dfrac{z_2 - z_3}{z_2 - z_1} = \dfrac{\omega - \omega _1}{\omega - \omega _3} \dfrac{\omega _2 - \omega _3}{\omega _2 - \omega _1} $
We get
$ \dfrac{z - i}{-i} = \dfrac{\omega}{\omega + i} $
Can someone explain the answer to me, or show me another way of doing these types of questions please?