The next nagging point is that we need a uniqueness criterion (this is not the same as the "uniqueness" criteria that we often discuss on this forum). After all, the negative iteration of exponentiation (i.e., the logarithm) has branches. This is a problem in general when dealing with discussions of superexponentiation in any base.
Luckily, it is possible to further rewrite the definition of the cheta function so that branches are an impossibility. This can be done by only allowing "forward" iterations, so to speak:
\( \check{\eta}(z) = \lim_{k \to \infty} \exp_{\eta}^{[\circ k+z]} \left( \log_{\eta}^{[\circ k]} \left( e^{e} \right) \right) \)
In doing this, k+z will always have a positive real part, so we needn't ever worry about logarithms, and hence about branches or singularities. (What about the imaginary part? Couldn't that lead to singularities? Well, let's wait and see...)
For each of the k iterations of the logarithm (in the limit), we can choose any branch we want, and each choice gives us a different variant of the cheta function. Interestingly, all these variations are related to each other, but more on that later.
Intuitively, then, we would simply choose all iterations of the logarithm to use the principal branch.
** Note: the brackets in the "functional iteration" operator are there for clarity: compare \( \exp_{\eta}^{[\circ k+z]}(\ldots) \) and \( \exp_{\eta}^{\circ k+z}(\ldots) \)
On an amusing sidenote, the \( \circ k \) can be read as "OK" (oh kay) or "Circle K", both of which have meanings in the USA (the former is an idiomatic term which most English speakers, even non-native ones, would know; the second is the name of a gas station chain). I prefer "circle K", as I'm less likely to think somebody means a gas station when discussing math.
Luckily, it is possible to further rewrite the definition of the cheta function so that branches are an impossibility. This can be done by only allowing "forward" iterations, so to speak:
\( \check{\eta}(z) = \lim_{k \to \infty} \exp_{\eta}^{[\circ k+z]} \left( \log_{\eta}^{[\circ k]} \left( e^{e} \right) \right) \)
In doing this, k+z will always have a positive real part, so we needn't ever worry about logarithms, and hence about branches or singularities. (What about the imaginary part? Couldn't that lead to singularities? Well, let's wait and see...)
For each of the k iterations of the logarithm (in the limit), we can choose any branch we want, and each choice gives us a different variant of the cheta function. Interestingly, all these variations are related to each other, but more on that later.
Intuitively, then, we would simply choose all iterations of the logarithm to use the principal branch.
** Note: the brackets in the "functional iteration" operator are there for clarity: compare \( \exp_{\eta}^{[\circ k+z]}(\ldots) \) and \( \exp_{\eta}^{\circ k+z}(\ldots) \)
On an amusing sidenote, the \( \circ k \) can be read as "OK" (oh kay) or "Circle K", both of which have meanings in the USA (the former is an idiomatic term which most English speakers, even non-native ones, would know; the second is the name of a gas station chain). I prefer "circle K", as I'm less likely to think somebody means a gas station when discussing math.
~ Jay Daniel Fox