How would tetration work for non integer numbers.

Question

Can you even do these and how would you do them? How does tetrations algebraically work? $$^{.5}x=?$$ $$^{-1}x=?$$ $$^ix=?$$ Is there such a number like e that converges? $$^xd=(some/equation/with/x)$$ What would the inverse be?

Can it be represented by some algebraic way over e?

The wikipedia article on this has some info about extending tetration to other real numbers, it even has the answer to the second line. — SquirtleSquad, Jul 19 '16 at 19:32
Have you checked out http://tetration.org/Tetration/index.html ? — , Jul 30 '20 at 16:44
For the $ ,^i x$-question you may also see https://mathoverflow.net/q/71429/7710 — Gottfried Helms, Oct 29 '20 at 19:47
I believe Knesers Tetration settles this: https://en.wikipedia.org/wiki/Tetration#Complex_heights — Sidharth Ghoshal, Dec 22 '22 at 21:58

score 3 · Answer 1 · edited Apr 13 '17 at 12:19

This topic is still open, and I would suggest you read the following links.

As the first link mentions, there is an entire community like this one dedicated solely to the tetration.

Tetration: Extension to Real Heights

The last link explains how to get to negative whole heights, like for your second example, and how to approximate what real values would be, as per your first example.

Not much is known for the third example.

score 0 · Answer 2 · answered Oct 03 '23 at 17:45

Kneser’s tetration requires going into the complex plane. Furthermore it is not clear how Kneser chooses one fixed point over another. There is a purely real number solution at The Fourth Operation, which to understand requires only single variable calculus and the rudiments of real analysis. (Note that it uses a suffix subscript for the iteration operand instead of a prefix superscript that the questioner above uses.)

How would tetration work for non integer numbers.

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