Real and complex tetration

[JmsNxn]

My approach to fractional tetration involves the complex numbers. Others have found fractional tetration constrained to the real numbers to be more natural. Which tetration methods require complex numbers and which only need real numbers? For example I consider Kneser as real based because fractional tetration of f(x) where \( f(f(x))=e^x \) is real even though the proof uses complex numbers.

I don't know of any standard of terms for this. But complex valued tetration would be what your tetration would be called. And real valued tetration would be a tetration that preserves real values.

A real valued tetration can still produce complex numbers, but only for complex arguments.

A complex valued tetration can produce complex numbers for real arguments.

With the exception that we ignore \((-\infty,-2]\), which produces complex values for Kneser; but that's okay because it's not analytic here.

I think a standard way to phrase this would be real-valued iteration, versus a complex valued iteration.

Your question is very hard to answer because there are uncountably infinitely many tetrations, lol. So to ask which are real valued and which are not is actually a very deep problem. Especially if you no longer ask for analycity, and solely talk about smooth solutions (on the real line).

It's widely conjectured the only real-valued tetration with \(b > \eta\) and holomorphic in the upper half plane is Kneser. This procedure fails within Shell-Thron \(e^{-e} \le b \le \eta\) and there are uncountably infinite real-valued tetrations--with uniqueness criterion depending on what qualities you want of your iteration.

Generally, the hardest part about tetration is A.) making it real valued, B.) seeing if it's the same real-valued solution as another/Uniqueness. Not sure what else can be said