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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 .
And that each solution of is strictly decreasing with a fixed point.
You can read further on this subject (keyword Babbage equation, involution) in
Kuczma, Iterative functional equations.