Complex to real tetration

[Daniel]

Does complex tetration based on a limit point approaching imaginary infinity completely dampen the oscillations giving real tetration?

[JmsNxn]

From what best I can understand of this question, the answer would be a sort of yes and no.

Any real valued tetration that is outside of Shell-thron, should have a very regular structure at \(\pm i\infty\). The trouble is, this is kind of a chicken and egg scenario. Is this regularity the reason it is real valued, or is it just real valued and this regular structure is coincidental.

You can settle this by remembering all we need is for \(\overline{F(z)} = F(\overline{z})\), for the function \(F\) to be real valued. And requiring this, sort of forces \(F(i\infty) = \overline{F(-i\infty)}\). So by asking that this happens for very large \(z\) near plus/minus imaginary infinity, then yes this forces the entire function to be real valued.

It's hard to answer this question, as it's a tad vague. But I hope that helps.