06/09/2014, 10:26 AM
I am not convinced by the "2sinh"-stuff. A composition with inverse of 2sinh superfunction may not even be analytic (I think sheldonison did some work on related stuff involving tetrational base changes -- and maybe even 2sinh, I don't remember), and converges to a linear function plus a small 1-cyclic wobble. A small, 1-cyclic oscillation doesn't seem too good for "convexity"-related purposes -- something that wiggles a little about a line is not quite convex.
Also, with more testing it looks like that this method of repeated differentiation eventually teases out even tiny 1-cyclic wobbles applied to the gamma function, i.e. \( \Gamma(x) \theta(x) \) where \( \theta(x) \) is a small-amplitude 1-cyclic wobble with \( \theta(0) = 1 \). Just that it seems with tetration, you need to differentiate it a whole lot more (probably because the growth rate is soo much faster than the gamma function's). 32 derivatives was more than enough to tease out a wobble of amplitude \( 10^{-12} \).
Also, with more testing it looks like that this method of repeated differentiation eventually teases out even tiny 1-cyclic wobbles applied to the gamma function, i.e. \( \Gamma(x) \theta(x) \) where \( \theta(x) \) is a small-amplitude 1-cyclic wobble with \( \theta(0) = 1 \). Just that it seems with tetration, you need to differentiate it a whole lot more (probably because the growth rate is soo much faster than the gamma function's). 32 derivatives was more than enough to tease out a wobble of amplitude \( 10^{-12} \).