Real and complex tetration

[Daniel]

WARNING: Religious topic discussed!!!

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