uniqueness

[tommy1729]

here are the equations that make half-iterate of exp(x) unique :

(under condition f(x) maps reals to reals and f(x) > x )

exp(x)

= f(f(x))

= D f(f(x)) = f ' (f(x)) * f ' (x)

= D^2 f(f(x)) = f '' (f(x)) * f ' (x)^2 + f ' (f(x)) * f '' (x)

= D^3 f(f(x)) = D^4 f(f(x))

Arent these equations valid for every half iterate of exp?

NO

of course not.

for example : the first case :

exp(x) = f ' (f(x)) * f ' (x)

now consider a solution that satisfies f(f(x)) = exp(x)

and let assume f ' (x) = 0 has a finite real zero at x = r1.

thus f ' (r1) = 0

exp( r1 ) = 0 * f ' (f(r1)) => ??????

you see , contradiction.


regards

tommy1729