f( f(x) ) = x tommy1729 Ultimate Fellow Posts: 1,384 Threads: 338 Joined: Feb 2009 04/15/2009, 04:37 PM 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 andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 04/17/2009, 03:36 AM (This post was last modified: 04/17/2009, 03:38 AM by andydude.) 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 bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 04/17/2009, 09:59 AM (This post was last modified: 04/17/2009, 10:00 AM by bo198214.) 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. « Next Oldest | Next Newest »