Tetration below 1

[jaydfox]

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

Without going to complex values, I'd come to the same conclusion. A sinuisoidal wave would seem to make the most sense. For bases <= e^-e, we could just use a sine (cosine, same difference, I'm mostly referring to the base shape) wave. For bases between e^-e and 1, the upper and lower points converge exponentially, so we'd need an exponentially scaled sine wave. This unfortunately raises the question: do we make the integer values the peaks of the wave, or do we make them tangent to the exponential asymptotes? My personal preference is to make them tangent, but I haven't investigated. At any rate, while it seems like a potentially insightful path to pursue, I'm still leaning towards complex values.

I had spent a while pondering over it yesterday, then came up with essentially what Henryk has posted (with regards to his analogy of exponentiating bases less than 0, by having a "family" of exponential functions that return complex values for non-integers), so I think he's on the right track.