An explanation for this?

[sheldonison]

I was wondering if anyone had the means by which to reproduce this graph? And also, why this is not the accepted extension for tetration of rational values?

Sorry to be a newb and ask so many questions

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 \( \exp^{[-1]}(z)=\log(z) \)