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Let $\gamma$ be the unit circle. Find a Möbius transformation that transforms $\gamma$ onto $\gamma$ and transforms 0 to $\frac{1}{2}$.

I think it's quite easy to find a Möbius transformation that transforms $\gamma$ onto $\gamma$, since Möbius transformation always maps circle to circle or line. We can actually choose three points on $\gamma$ than solve the Möbius transformation by pluging in numbers. But when we also need $0 \rightarrow \frac{1}{2}$, I have a problems find such map.

And to check the “onto” fashion, can we simply say that the inverse is also a Möbius transformation?

Thanks for the help!

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  • $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$ – Did Dec 28 '17 at 22:52
  • $\begingroup$ We can actually choose three points on γ than solve the Möbius transformation by pluging in numbers No, because you don't know what those points transform to, only that their transforms will lie on the unit circle. See the previous comment, and also here for a start. $\endgroup$ – dxiv Dec 29 '17 at 2:17
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    $\begingroup$ @dxiv Thanks, got it already~ $\endgroup$ – Nan Dec 29 '17 at 2:41

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