# Tetration Forum

Full Version: f( f(x) ) = x
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f(x) =/= x

f(f(x)) = x

exp(f(x)) = f(exp(x))

give examples of Coo f(x) satisfying all the above 3 equations at once.

regards

tommy1729
tommy1729 Wrote:give examples of Coo f(x) satisfying all the above 3 equations at once.

The only examples of those kinds of functions (roots of the identity function), that I can think of are not Coo, or piecewise-defined.

PS. Does the complex conjugate count?

Andrew Robbins
tommy1729 Wrote:f(x) =/= x

f(f(x)) = x

exp(f(x)) = f(exp(x))

give examples of Coo f(x) satisfying all the above 3 equations at once.

It may be interesting for you that there is no continuous solution of $f(f(f(x)))=x$.
And that each solution of $f(f(x))=x$ is strictly decreasing with a fixed point.
You can read further on this subject (keyword Babbage equation, involution) in
Kuczma, Iterative functional equations.