07/02/2011, 02:00 AM
(07/01/2011, 10:47 PM)sheldonison Wrote: Cherrina, Would you use the sexp(z) for the tetra-euler base, or the pent(z) for the tetra-euler base? b~1.6353245. In the pentation.gpp code, the genpen function uses the lower repelling fixed point (-1.6409), not the upper parabolic fixed point which is ~3.0885323. The parabolic fixed point would generate a similar but different pentation function for this base. Also, presumably one might be able to generate another pentation function going to infinity, from the parabolic fixed point. I haven't generated numerical approximations for either of those functions.
- Sheldon
The function sexp(z) is used for non-integer iterates of the exponential, which for the 'tetra-euler' base has no real fixed point. The non-integer iterates of exponential can then be used to construct operators between addition and multiplication along with corresponding means.
The function pent(z) can generate non-integer iterates of sexp(z) but they can't be analytic at the upper fixed point, so I just used the regular iteration at the lower fixed point of sexp(z).
There is also a possible function that rises to infinity based on iteration of sexp(z) but it wouldn't be 'pentation' in the sense that \( S(0) \not = 1 \) much like the 'cheta' function. Besides, I'm not totally sure about how to generate an analytic superfunction from a parabolic fixed point... From what I have read there is no way to construct an analytic non-integer iterate of a function that is analytic at a parabolic fixed point, is that correct?
