The "cheta" function

[jaydfox]

With further investigation of this function (and yes, I'm fairly certain it is a complex function, with no branch cuts, and I'm fairly certain without any singularities), ...

I mentioned before that the cheta function is indeed a true "function", i.e., it does not take on multiple possible values, depending on which branch of the function you are in. There are no branches!

Indeed, as far as I can surmise, it is an entire function, much like simple exponentiation. If that doesn't surprise you, consider for a moment that for all bases greater than eta, the superexponentiation function branches, much like a logarithm, and is therefore a multi-valued function.

To see that the cheta function is indeed an entire function, we need merely perform a simple thought experiment. Start with any complex number, excepting the tetrates of eta between 0 and e, inclusive (0, 1, eta, eta^eta, ..., e). Call this number \( x_0 \). (For clarity, that is "x naught" or "x sub zero", as opposed to the tetration of base zero).

Starting with \( x_0 \), begin taking logarithms, base eta. For simplicity, we will only use the principal branch, but we will generalize thereafter.

As we perform iterative logarithms, eventually, something remarkable will happen. The sequence of iterated logarithms will begin to converge to e, from the positive real direction. This might not be immediately obvious, and it is key to understanding the entirety of the cheta function. Take a moment to convince yourself that this is indeed the case.

We can now generalize to include using various branches of logarithms. Pick any complex number, except 0. Perform iterated logarithms, choosing arbitrary branches. Note: If e or an integer tetrate of eta was chosen, such as 1 or eta, etc., then a branch must be chosen that avoids 0 or e during the iterations!

Then, at some point, stop. You now have a new complex number, which will still comply with the original exceptions I noted. At this point, follow the principal branch, and as before, convergence on e from the positive real direction is assured.

Considering that all complex numbers (except 0) will eventually converge on e (from the same direction), regardless of whatever branching set we chose for the first few iterated logarithms (except where forced in cases to avoid 0 or e), we can now define cheta.

You see, as the iterated logarithms approach e, they will do so at a rate that seems to slow to a crawl. This is good, because it essentially guarantees that successive iterates will behave almost linearly near e. For practical purposes, this convergence is too slow to be useful, but for theoretical purposes it suffices. In the limit as the number of iterated logarithms goes to infinity, the relation becomes linear, and we can then use linear interpolation to define a function based on iteration count. This allows us to leave the domain of integer iterations and move into the domain of complex iterations. We then iteratively exponentiate our way back "out of the well", so to speak, until we get back to the starting point.

Take a moment to convince yourself that in so doing, we have guaranteed that we can continuously iterate exponentiation in base eta, sufficient to reach all complex numbers (except 0) from any starting point (except 0), using any particular branching system we want (except where a 0 or e would be generated by a logarithm).

So, it should also be clear, hopefully, that there aren't "multiple" cheta functions, based on different starting points. They are related by a simple linear transformation on the iteration count (adding an appropriate constant, in fact), regardless of the starting point or branching system desired.

And I should be careful, because when I say branching system, I don't mean branches in cheta. I mean, for example, that we might choose a starting point like e^2 (as I do), but desire that cheta(-1) be 2e + 2e*pi*i. I'll call this variant chmeta. Of course, chmeta(1) would still be e^e, but we would rightly expect that continuous iteration between chmeta(0) and chmeta(1) would stray off the real line. However, there is a complex constant k, such that chmeta(z+k) would be equal to cheta(z). Thus, it's not a different branch, just in a different part of the domain, which might not be immediately obvious.