04/30/2009, 10:41 PM
andydude Wrote: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.
It should be possible to choose the "natural" versions of the higher operations so that they are approximately linear on the intervals in question, since (e ^k^ x) + 1 = e ^k-1^ (e ^k^ x) = e ^k^ (x+1) would be approximately true by induction, and we always have e ^k^ 0 = 1, and e ^^ x = x+1 approximately on [-1, 0]. The degree of approximation might decay a bit at each level, I suppose.
None of the functions could be exactly linear, of course; only a linear function can have a linear segment and still be holomorphic.
Finally, this behaviour is both a good thing and a bad thing, I think. It is good in that the higher functions will have larger radii of convergence, but bad in that they must have increasingly more fiddly power series coefficients, if they cancel so cancel closely on one interval, but increase so rapidly outside it.
