f( f(x) ) = exp(x) solved ! ! !

[bo198214]

Pretending that there are no open questions does not help here.
Especially for you tommy I list them again:

1. Do the approximations of \( \exp^{1/2} \) converge? I.e. we have approximations of \( \exp \) these are the functions \( g_n(x)=\exp(x)-\exp(-n x^2) \). These converge pointwise to \( \exp \) (except at 0). Then we take the regular iteration of these functions \( f_n={g_n}^{1/2} \). The question is whether they converge to anything.
   
2. If they converge to \( f \), is \( f \) a differentiable or even analytic function?
   
3. If they converge to \( f \), does \( f \) really satisfy \( f(f(x))=\exp(x) \)?
   

As I already told some pictures would make a good start if proofs could not provided yet.