Personal Scratchpad

[bo198214]

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 \( \text{slog}_b \):

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