f( f(x) ) = x - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: f( f(x) ) = x (/showthread.php?tid=270) f( f(x) ) = x - tommy1729 - 04/15/2009 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 RE: f( f(x) ) = x - andydude - 04/17/2009 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 RE: f( f(x) ) = x - bo198214 - 04/17/2009 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.