04/15/2009, 04:37 PM

04/17/2009, 03:36 AM

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

04/17/2009, 09:59 AM

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.