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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

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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

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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

.

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.