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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?

tox123
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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.

Charlotte Aten
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  • While that's a good point, and I will accept it (until a better answer comes along), reasonable satisfies: 1.) The new function is continuous 2.) ${}^yx$ 1= ${}^xy$, and other basic properties of the function – tox123 Jun 19 '16 at 22:24
  • The problem occurs already in the question of interpolating the fibonacci-numbers (http://math.stackexchange.com/questions/589841): a naive method suggests, that the interpolates go over the complex and another method suggests, that the interpolates go only through the reals. With iterated exponentiation with base $e$ a somehow natural method of interpolation to tetration gives complex numbers, and there is also a -somehow famous- solution by H. Kneser, which gives real numbers only when interpolated to real/fractional interpolation heights. The decision for this or that is not yet made... – Gottfried Helms Jun 21 '16 at 00:10