I've searched this topic already and found similar question.

But it was not fully answered.

Actually I know the Möbius transformation of form

$\frac{z-a}{1-\bar a z}$ is the automorphism of unit disc and interchanges $0$ and $a$.

But if I want it to change two arbitrary points in the disc. The above mapping fails.

Then how does the mapping which satisfies the condition on the title look like?

I'd like to see the mapping itself. (I mean like $f(z)=z$)

  • 4
    $\begingroup$ Suppose your $2$ points are $a$ and $b$. Then you have the $2$ transformations which send $0 \rightarrow a$ and $0 \rightarrow b$. Compose one with the inverse of the other. $\endgroup$ – none Jun 11 at 17:39
  • $\begingroup$ There are two answers here. $\endgroup$ – Maxim Jun 12 at 19:25
  • $\begingroup$ @Maxim: Indeed, I somehow missed that. $\endgroup$ – Martin R Jun 12 at 21:22
  • 1
    $\begingroup$ @none That composition will map $a$ to $b$ but not $b$ to $a$ (or the other way around). $\endgroup$ – Maxim Jun 12 at 22:12