(06/21/2010, 01:45 PM)bo198214 Wrote:(06/16/2010, 11:09 AM)mike3 Wrote: Yet it seems to agree with regular iteration in \( b \in (1, e^{1/e}] \),
According to my newest computations, intuitive iteration (I would like this to be the official name, deprecating Abel-matrix iteration) does not coincide with regular iteration. What does that mean for your theses?
It'd mean that regular iteration could still have a natural boundary, and so still not be the "right" tetrational (which I'd expect to work with all \( b \in [1, \infty) \)).
It would be interesting to see if there was some way to plot the intuitive iteration on the complex \( z \)-plane for a base in \( [1, e^{1/e}) \). The behavior there may (should?) differ dramatically from the regular iteration. You might not even need to plot the tet function, just the slog function, which has formulas to extend to the whole plane.
Another question: is this intuitive iteration thing the same as what you get from solving the equations given by substituting a Taylor series with unknown coefficients for slog in \( \mathrm{slog}_b(b^z) = \mathrm{slog}_b(z) + 1 \) and solving the resulting system of linear equations? If so, then maybe one could also try coming at this from that direction: try plugging the coefficients for the regular slog into those equations and then seeing that they do not satisfy them (being linear, they have only one solution. If this is not \( \mathrm{rslog} \), then it should fail to satisfy them.). Just construct the Taylor series for \( \mathrm{rslog}_b(b^z) \) and see if it is the same as that for \( \mathrm{rslog}_b(z) + 1 \) for a base \( b \in [1, e^{1/e}) \).
And what method did you use to do that computation?
