# Is a function that maps circles to circles necessarily a Möbius transformation?

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?

• What if $f$ is not continuous? I'm sure you can find some counterexample.
– user223391
Aug 21 '16 at 20:17
• $f(z)=\overline{z}$ Aug 21 '16 at 20:18
• @sometempname That's an answer. =) Aug 21 '16 at 20:19
• Do you require that every circle maps to a circle, or just that there exists one circle that is mapped to a circle?
– mrf
Aug 22 '16 at 7:21
• @mrf the latter one Aug 22 '16 at 18:42

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.
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.
• $z^2$ maps the circle $|z-1|=1$ to a cardioid. Aug 21 '16 at 20:28