Real and complex tetration

[JmsNxn]

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?

I'd just like to add to Tommy's correct answer.

Say I take a function which is infinitely differentiable on \(\mathbb{R}\)... let's call it \(f\).

Then \(f\) is unique because it has unique derivatives everywhere on \(\mathbb{R}\).

Now let's take a nowhere holomorphic function \(g : \mathbb{R} \to \mathbb{R}\); such that \(g :\mathbb{C}_{\Im(z) > 0} \to \mathbb{C}_{\Im(z)> 0}\) is holomorphic. Let's assume that \(g(\mathbb{R}) = 0\). Well then; if we try to move away from the real line with \(f\), then \(f+g\) is also a candidate (this is the same thing as the theta discussion). So we lose uniqueness.

This is pretty much the trouble with the beta method outside of the Shell-Thron region--and why for \(b = e\) we're actually NOWHERE holomorphic, despite finding infinitely differentiable solutions.

This is one of those moments where you have to remember that COMPLEX DIFFERENTIABILITY > DIFFERENTIABILITY. Once we lose complex differentiability; a lot of tetration can be pretty bananas. Especially; if say; I try to approximate a complex differentiable \(F(s)\) using \(f(x)\); near \(x \in \mathbb{R}\); \(f(x) \approx F(s)\), but maybe it's actually \(f(x) + g(s) \approx F(s)\)... where now we have an equivalence class of \(g\) which may or may not actually make a holomorphic tetration...

Shit's such a fucking head ache. Tommy's right though, just thought I'd add my two cents.