# Möbius transformation from a unit disc to its complement

I am trying to find all möbius transformations that map $$D_{1}(0)$$ to its complement.

My Idea: I am trying to find 3 mappings say $$T_1, T_2,T_3$$ where

1. $$T_1$$ maps $$D_1(0)$$ to $$K(0,1)$$ "specific"
2. $$T_2$$ maps $$K(0,1)$$ to itself $$K(0,1)$$ "arbitrary"
3. $$T_3$$ maps $$K(0,1)$$ to $$[D_1(0)]^c$$ "again specific"

And then $$f=T_3 \circ T_2 \circ T_1$$ will be that arbitrary map which I want to find...

Now I know how to map $$D_1(0)$$ to $$K(0,1)$$, so I'll find $$T_1$$, I also have $$T_2$$ the arbitrary one.. I just need to find $$T_3$$...

My question is " is there any such Möbius transformation exist? If yes then what will be that ?"

Note: A möbius transformation is a map of the form $$T(z) = \frac {az+b}{cz+d}$$, where $$a,b,c,d \in \Bbb C$$ such that $$ad-bc \neq 0$$. And here by $$K(0,1)$$ my means unit circle and by $$D_1(0)$$ my means unit disc.

• Please don't use inappropriate tags. :) – Ted Shifrin Apr 14 '20 at 22:20
• Sorry sir, but face the difficulty while studying the complex analysis course.... So it seems right for me.... Sorry for inconvenience – Badshah Khan Apr 15 '20 at 9:21
• These tags are fine. I erased differential geometry and complex geometry, which were not appropriate. – Ted Shifrin Apr 15 '20 at 17:02

given such a map $$\phi$$, $$\frac{1}{\phi(z)}:\mathbb{D}\to \mathbb{D}$$ is an automorphism of the disk , and thus is of the form $$e^{i\vartheta}\frac{z-\alpha}{1-\overline{\alpha}z}$$ (with $$\vartheta\in [0,2\pi], \alpha\in \mathbb{D}$$).
This gives us $$\phi(z)=e^{i\theta}\frac{1-\overline{\alpha}z}{z-\alpha}$$
Conversely, every map of this form maps $$\mathbb{D}$$ to its complement