Tetration below 1

[andydude]

The interval b in (0, 1) is a very difficult region to do tetration over. The best that I have been able to do is something similar to my matrix equations, only instead of being automated, I solve each step of the way manually. From my manual solutions (i.e. using a 4-th degree polynomial, changing coefficients as necessary) I have found that tetration over the critical interval (-1<x<0) can be approximated by \( {}^xb \approx \cos(\frac{\pi}{2}x)^2 \) where 0<b<1 and -1<x<0, is much more sinusoidal than with higher bases (where usually \( {}^xb \approx x + 1 \) for example). One of the reasons is that \( {}^{-2}b = -\infty \) for b>1 and \( {}^{-2}b = +\infty \) for b<1. In order for the curve to get the "direction" in needs to go in, it needs to be more sinusoidal, than linear as with b=e.

For more about the domain and the line where x=-2, see my previous post
http://math.eretrandre.org/tetrationforu...287#pid287
about the domain of real-valued (and possibly real-analytic) tetraiton.

Andrew Robbins