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How to find the general form of a Möbius transform that maps the circle $S=\{z\in\mathbb{C}:|z|=r\}$ onto itself.

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Start with $r = 1$, and then scale, I would think. The case $r = 1$ is rather well known, since these are generated by a) the automorphisms of $\mathbb{D}$, b) the inversion $\frac{1}{z}$. – Daniel Fischer Jun 26 '13 at 23:20
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Following the hint by Daniel Fischer, the transformations end up being $$z\mapsto r^2 e^{i\alpha}\frac{z-a}{r^2-\bar a z}\quad \text{and}\quad z\mapsto e^{i\alpha}\frac{r^2-\bar a z}{z-a}$$ where $|a|<r$ and $\alpha\in \mathbb R$.

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