I have posts strewn about the forum, discussing a "cheta" function. If, as is likely if you are not one of the handful of original members of this forum, you have no idea what the "cheta" function is, allow me briefly to explain. (If you are impatient, I get to the details in the second post, this first being background.)
The number \( e^{1/e} \) makes a particularly interesting base when discussing tetration. I nicknamed it \( \eta \), the greek letter eta, a letter which is similar in function to the letter e. Just as the constant e is a (perhaps the most) useful base for discussing and analyzing exponentiation, eta is a (and I once thought, the most) interesting base for discussing and analyzing tetration.
The name has stuck, and you will see \( e^{1/e} \) called eta throughout the forum.
When I first began studying tetration, I looked at it from the point of view of continuously iterating the exponentiation of a base. It made sense to me, then, to restrict myself to looking at \( {}^{n}b = \exp_b^{\circ n}(1) \), that is, starting from 1 and exponentiating n times, where the result is well-defined for non-negative integers n, and can even be fairly well defined for integers n >= -2. The problem, then, was to extend the solution to real (or at least rational) n.
For base eta, the tetrates as n goes to infinity approach e. That is, \( \lim_{k \to \infty} \left( {}^{k}\eta \right) = e \). (Note here that I used k instead of n, due to eta's unfortunate resemblance to the letter n.)
So what about "cheta"? Well, borrowing an idea from a 1991 paper by Peter Walker (referenced elsewhere on this site, and a valuable read), I developed a change of base formula for tetration. Unfortunately, this change of base formula works from the "top down", as opposed to working from the "bottom up".
By this, I mean that, rather than starting at 1 and exponentiating up, I would start at positive infinity and iteratively take logarithms down to 1 (or another suitable finite constant). Doing this in two different bases, I found an (almost) linear relationship, which was the basis of the change-of-base formula. Of course, this involves using limits to approximate the infinity, but it's provably valid (though I'm not sure I formalized the proof to Henryk's liking).
The problem for base eta is that, just as the infinitely iterated exponentials, starting at 1, asymptotically approaches e, so do the infinitely iterated logarithms, starting at infinity. Thus, this meant that my "change of base" formula was not for tetration per se, but rather for a specific variant of iterated exponentiation.
I realized that there were two separate "graphs", if you will, of iterated exponentiation base eta, so I called the upper one cheta, for "checked eta", written \( \check{\eta} \). (I suppose the lower one would be "heta", for "hatted eta", written \( \hat{\eta} \)?)
Hopefully it's clear what the check (and hat) mean: it indicates the direction of concavity/convexity, and hence which variant of superexponentiation we are dealing with.
And that, in a nutshell, is what cheta is.
The number \( e^{1/e} \) makes a particularly interesting base when discussing tetration. I nicknamed it \( \eta \), the greek letter eta, a letter which is similar in function to the letter e. Just as the constant e is a (perhaps the most) useful base for discussing and analyzing exponentiation, eta is a (and I once thought, the most) interesting base for discussing and analyzing tetration.
The name has stuck, and you will see \( e^{1/e} \) called eta throughout the forum.
When I first began studying tetration, I looked at it from the point of view of continuously iterating the exponentiation of a base. It made sense to me, then, to restrict myself to looking at \( {}^{n}b = \exp_b^{\circ n}(1) \), that is, starting from 1 and exponentiating n times, where the result is well-defined for non-negative integers n, and can even be fairly well defined for integers n >= -2. The problem, then, was to extend the solution to real (or at least rational) n.
For base eta, the tetrates as n goes to infinity approach e. That is, \( \lim_{k \to \infty} \left( {}^{k}\eta \right) = e \). (Note here that I used k instead of n, due to eta's unfortunate resemblance to the letter n.)
So what about "cheta"? Well, borrowing an idea from a 1991 paper by Peter Walker (referenced elsewhere on this site, and a valuable read), I developed a change of base formula for tetration. Unfortunately, this change of base formula works from the "top down", as opposed to working from the "bottom up".
By this, I mean that, rather than starting at 1 and exponentiating up, I would start at positive infinity and iteratively take logarithms down to 1 (or another suitable finite constant). Doing this in two different bases, I found an (almost) linear relationship, which was the basis of the change-of-base formula. Of course, this involves using limits to approximate the infinity, but it's provably valid (though I'm not sure I formalized the proof to Henryk's liking).
The problem for base eta is that, just as the infinitely iterated exponentials, starting at 1, asymptotically approaches e, so do the infinitely iterated logarithms, starting at infinity. Thus, this meant that my "change of base" formula was not for tetration per se, but rather for a specific variant of iterated exponentiation.
I realized that there were two separate "graphs", if you will, of iterated exponentiation base eta, so I called the upper one cheta, for "checked eta", written \( \check{\eta} \). (I suppose the lower one would be "heta", for "hatted eta", written \( \hat{\eta} \)?)
Hopefully it's clear what the check (and hat) mean: it indicates the direction of concavity/convexity, and hence which variant of superexponentiation we are dealing with.
And that, in a nutshell, is what cheta is.
~ Jay Daniel Fox