Personal Scratchpad

[jaydfox]

Notation for iteration of exponentiation:
\( \exp_b^{\circ t}(z) \)

Tetration is a special case:
\( {}^{t} b\ \equiv\ \exp_b^{\circ t}(1) \)

"Cleaner" notations to allow "primed" derivative notation:
\( \mathcal{T}_{[b,z]}(t)\ \equiv\ \exp_b^{\circ t}(z) \)
\( \mathcal{E}_{[b,t]}(z)\ \equiv\ \exp_b^{\circ t}(z) \)
\( \mathcal{B}_{[t,z]}(b)\ \equiv\ \exp_b^{\circ t}(z) \)

I rather would additionally introduce:

\(
\begin{align*}
\text{sexp}_b (t) &={}^tb = \mathcal{T}_{[b,1]}(t)\\
\text{spow}_t (b) &={}^tb = \mathcal{B}_{[t,1]}(b)\\
\text{slog}_b&=\text{sexp}_b^{\circ -1}
\end{align*}
\)

And we can recover the full 3 variable expression by means of \( \text{sexp}_b \) and its inverse \( \text{slog}_b \):

\( \exp_b^{\circ t}(z) = \text{sexp}_b(\text{slog}_b(z) + t) \)

Those notations are useful for bases above eta. However, for bases between 1 and eta, you cannot recover the full exp function, unless you extend the definition of slog to include the 2 additional domains (only 1 additional for eta). While I'm in favor of such an extension, I'm still working on defining the correct z value to start from, just as 1 is the "correct" value to start from for the principal domain.

Until we define good starting points, a full analysis of fractional iterates will require the more general "T" notation, where we can start from a location other than 1 and leave it at that. For example, for base sqrt(2), we could start at sqrt( or e, etc., to solve for the corridor between the asymptotes at 2 and 4, and we could start at 4.1 or 5 or 16, etc. to solve above the asymptote at 4. (BTW, 16 seems to me the best place. Essentially, exponentiate the upper and lower asymptotes. The order doesn't matter: 2^4=4^2, and this is true for any base between 1 and eta, so it seems pretty "unique" to me.)