Personal Scratchpad

[jaydfox]

My scratchpad

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

Iterated logarithm is then shorthand:
\( \log_b^{\circ t}(z)\ \equiv\ \exp_b^{\circ ({\small -}t)}(z) \)

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

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

\(
\begin{eqnarray}
\mathcal{T}_{b}(t)
& = & \mathcal{T}_{[b,1]}(t) \\
& \equiv & {}^{t} b \\
\\[5pt]

\\
& \equiv & \exp_b^{\circ t}(1)
\end{eqnarray}
\)

\( \mathcal{E}_{[b,t]}(z)\ \equiv\ \exp_b^{\circ t}(z) \)

\(
\begin{eqnarray}
\mathcal{E}_{b}(z)
& = & \mathcal{E}_{[b,1]}(z) \\
& \equiv & b^z \\
\\[5pt]

\\
& \equiv & \exp_b^{\circ 1}(z)
\end{eqnarray}
\)

\( \mathcal{B}_{[t,z]}(b)\ \equiv\ \exp_b^{\circ t}(z) \)

\(
\begin{eqnarray}
\mathcal{B}_{t}(b)
& = & \mathcal{B}_{[t,1]}(b) \\
& \equiv & {}^{t} b \\
\\[5pt]

\\
& \equiv & \exp_b^{\circ t}(1)
\end{eqnarray}
\)

Generalizing what was discussed earlier, differentiation with respect to t:
\( \mathcal{T}_{[b,z]}^{'}(t)\ =\ \ln(b)\mathcal{T}_{[b,z]}(t)\mathcal{T}_{[b,z]}^{'}(t-1) \)

\( \mathcal{T}_{[b,z]}(t)\ =\ \frac{\mathcal{T}_{[b,z]}^{'}(t)}{\ln(b)\mathcal{T}_{[b,z]}^{'}(t-1)} \)