complex and decimal tetration

Question

So recently in the blog post on tetration, it talked about tetration with nice clean powers (calling them these because I don't know the right term). But how does it work when given a complex power? How about a decimal power? Or even just a negative power? And one final yet somewhat unrelated question: can you use some sort of method to reverse tetration by using tetration?

Answer 1

This is discussed on the Wikipedia page for tetration. Your question is somewhat ill-defined, in that we can trivially extend any function $f:\mathbb{N}\to\mathbb{N}$ to a new function $g:\mathbb{C}\to\mathbb{C}$ such that $g(z)=f(z)$ when $z\in\mathbb{N}$ and $g(z)=0$ otherwise.

Now you might say that's silly, but now you need to tell me what critera a proposed such function $g$ must satisfy in order to be a "reasonable" analog over $\mathbb{C}$. That is what is discussed on the wiki page.