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+--- 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.