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 Personal Scratchpad jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 09/01/2007, 04:18 PM 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 \\ \vspace{5} \\ & \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 \\ \vspace{5} \\ & \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 \\ \vspace{5} \\ & \equiv & \exp_b^{\circ t}(1) \end{eqnarray}$ Generalizing what was discussed earlier, differentiation with respect to t: $\mathcal{T}_{[b,z]}^{'}(t)\ =\ \ln(b)\mathcal{T}_{[b,z]}(t)\mathcal{T}_{[b,z]}^{'}(t-1)$ $\mathcal{T}_{[b,z]}(t)\ =\ \frac{\mathcal{T}_{[b,z]}^{'}(t)}{\ln(b)\mathcal{T}_{[b,z]}^{'}(t-1)}$ ~ Jay Daniel Fox « Next Oldest | Next Newest »