Real and complex tetration

[Ember Edison]

1. No

We can pick a one periodic function theta(s) such that

sexp_1(s + theta(s)) = sexp_2(s)

where one of them is analytic and the other is only C^{oo}.

2. Depends what you consider mainstraim.

For the bases you mention we have 2 fixpoints , so we have 2 expansions with koenigs function at each fixpoint.
I recently added a way to unify the two fixpoints so that we have tetration between them , but not analytic at them and not beyond them (*).

And then we have kneser which uses the smallest (nonreal) fixpoint(s) of ln_b(z) = z.

Thank you very much.