05/01/2009, 10:22 AM
(This post was last modified: 05/01/2009, 10:55 AM by Base-Acid Tetration.)
BenStandeven Wrote: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.
Kinda what I was thinking too. yeah, you are right that no non-linear holomorphic function can be linear anywhere. That's why I think the apparently increasing "linear part" may be a bad thing for holomorphic extensions. I have a hunch that a holomorphic solution might be a function that oscillates or even have nonreal answers in its non-integer values...I don't know. Anyway, the requirement should be that at least in the most intuitive definition we should have b[k]n=n+1 exactly for -k+3 < any NATURAL n < 0, to keep b[k](x-1) = [k-1]log_b b[k]x true.
