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I'm introducing myself to Complex analysis and Möbius transformations and I read that Möbius transformations map circles and lines to circles and lines.

Are there any other functions that are not Möbius transformations but they can map circles to circles?

If I know that $f(z)$ maps a circle to another circle, can I assume that $f(z)$ is a Möbius transformation?

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  • $\begingroup$ What if $f$ is not continuous? I'm sure you can find some counterexample. $\endgroup$ – user223391 Aug 21 '16 at 20:17
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    $\begingroup$ $f(z)=\overline{z}$ $\endgroup$ – sometempname Aug 21 '16 at 20:18
  • $\begingroup$ @sometempname That's an answer. =) $\endgroup$ – Pedro Tamaroff Aug 21 '16 at 20:19
  • $\begingroup$ Do you require that every circle maps to a circle, or just that there exists one circle that is mapped to a circle? $\endgroup$ – mrf Aug 22 '16 at 7:21
  • $\begingroup$ @mrf the latter one $\endgroup$ – mzdravkov Aug 22 '16 at 18:42
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To elaborate on sometempname's comment: if $f(z)=\overline{z}$, then for the circle $|z-a|=r$, we have $$|f(z)-\overline{a}|=r,$$ so the image of a circle is a circle. Similarly, the image of a line is a line, so this will have the desired property but not be a Mobius transform.

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I presume you talk about analytic maps. But even then you may take products of Möbius transformations which also maps $S^1=\{|z|=1\}$ to itself (1-1). Such transformations are called: Blaschke products

If you do not require 1-1 then you also have maps like $z\mapsto z^p$ and if you require analyticity only in a neighborhood of $S^1$ there are many more.

On the other hand a map that always maps any circle or line to a circle or a line is either a Möbius transformation (whence meromorphic) or a Möbius transformation composed with complex conjugation. Perhaps this is more what you are after... (and a proof is not that difficult)

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  • $\begingroup$ $z^2$ maps the circle $|z-1|=1$ to a cardioid. $\endgroup$ – sometempname Aug 21 '16 at 20:28
  • $\begingroup$ True, I only considered the last question and thought of the standard unit circle. $\endgroup$ – H. H. Rugh Aug 21 '16 at 20:33

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