(12/25/2010, 07:06 PM)sheldonison Wrote: I believe this is a graph of \( \exp^{[c]}(z) \). So the line with c=0.5 would be the half iterate of the exp(z), which can be calculated as \( \exp^{[0.5]}(z)=\text{sexp}(\text{slog}(z)+0.5) \). For integer values of c, the equations are simpler. \( \exp^{[2]}(z)=\text{sexp}(\text{slog}(z)+2)=\exp(\exp(z)) \). And \( \exp^{[-1]}(z)=\log(z) \)
Sorry, I should've been more specific. I meant to ask for a formula independent of tetration; I'm assuming there's a power series of some kind defining 0<= q<=1\( \exp^{[q]}(x) \) Something that would reproduce this graph, since the linear model of tetration I've been using doesn't match up. Even a generalized power series for tetration would work, now that I think of it.