Tetratophile Wrote:What is the implication of the growing quasi-linear part for the real or complex analytic extensions of those higher hyper-operations pentation, hexation, etc? Is it a good thing or a bad thing?
I haven't really looked at pentation much, except for the last link I gave above. But for tetration at least, the quasi-linear interval between (-1) and 0 is not linear in the more analytic/differentiable/holomorphic methods. Using natural iteration or Kouznetsov's method, the derivative \( \frac{d}{dx}({}^{x}e) \) is approximately 1.09176735, at both (-1) and 0, and since the "average slope" between those points is 1, the intermediate value theorem requires that the derivative is exactly 1 at least twice, and is less than 1 at some point in the interval (-1, 0). This numerical evidence is a strong indication that the quasi-linear interval you are talking about will not be linear for "holomorphic pentation" if such an extension exists. However, it would be interesting if this interval does actually get flatter with hexation, heptation, etc... but it would also be just as interesting if it gets less flat and starts oscillating wildly...
Andrew Robbins
