Real and complex tetration

[tommy1729]

WARNING: Religious topic discussed!!!

Is real tetration dependent on complex tetration? Could real tetration exist without complex tetration?

Very difficult. There exists Smooth real tetration; but it isn't analytic--and there are uncountably many of these. The trouble is, we can also make uncountably many real analytic tetrations by just adding a well behaved enough \(\theta\) mapping. By which; these wouldn't succumb to the complex uniqueness conditions though.

So No, real tetration is not dependent on complex tetration. It's just that complex tetration allows us the only confirmed construction--and allows us the uniqueness conditions we so love.

I want to confirm something.

1. Is \( R^\infty \) not enough to uniquely determine an real tetration?
2. Are there two (mainstream) real tetration sexp(base, height) defined in two different ways, one of which satisfies \( e^{-e} \leq b \leq e^{\frac{1}{e}} \) and the other satisfies \( b > 1 \), and the natural extension of the latter is Kneser?

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.

( * https://math.eretrandre.org/tetrationfor...p?tid=1652 )

and alot of methods that do not consider fixpoints ...

( my fav : the gaussian method and Peter Walkers method )

And ofcourse many open questions.


regards

tommy1729