Non-Trivial Self-Inverse Analytic Function In The Complex Plane

Asked Aug 06 '14 at 10:45

Active Aug 06 '14 at 11:54

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Let $f$ be a complex-valued function which is analytic for all values of $z$ in the complex plane and which satisfies $$f(f(z))= z $$ i.e. it is a self-inverse function. The trivial solutions for $f$ are clearly $f(z)=z$ and $f(z)=-z$. Are there any others?

Edit: There are more "trivial" examples: $f(z)=a-z$ for some constant $a$.

edited Aug 06 '14 at 11:54

asked Aug 06 '14 at 10:45

Asier Calbet

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This is a well studied problem on the Riemann sphere - namely the complex plane plus infinity. In that context, the only holomorphic, self-inverse functions are the Mobius functions, $z\rightarrow (az+b)/(cz+d)$. In the everywhere analytic case, $c\neq 0$ and you're left with your two examples. – Mark McClure Aug 06 '14 at 11:07
Is there a relatively easy proof of that I could understand? – Asier Calbet Aug 06 '14 at 11:10
user31373's answer to this question gives a proof based on Liouville's theorem that an injective, entire function is linear; the rest is easy. :) – Andrew D. Hwang Aug 06 '14 at 11:24
Is there a more elemental proof? – Asier Calbet Aug 06 '14 at 11:40

Non-Trivial Self-Inverse Analytic Function In The Complex Plane

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