Mizugadro, pentation, Book

[sheldonison]

Happy year 2015! I have several news.

1. I could not recover my previous institute site at tori.ils.uec.ac.jp
But now I have clone at http://mizugadro.mydns.jp/t
Is it seen from other countries?

2. I have constructed the natural pentation (superfunction of tetration)...
http://mizugadro.mydns.jp/t/index.php/Pentation

Hi Dimitrii,

Thanks for getting your tetration site back online. The graphs, as usual, are really nice. I thought I would limit my comments to your Pentation wiki; although I now see that the pdf has the same material. There is pari-gp pentation program posted here on eretrandre; I haven't played with it for awhile. The results match your results, for the period, and the fixed points, and the slope at the fixed point for pentation.

Where are the singularities of pentation? You wrote: "Pentation is holomorphic at least in the part of the complex plane, while the real part of the argument does not exceed some constant. For b=e, this constant is about −2.5"; I calculated the first singularity is -2.31527062760141 +/- 1.68383807835630i, so that would also match. One thing that I have puzzled over (but haven't posted) is that for pentation in the complex plane, is if pent(z) is a negative integer less than -1, then pent(z+1) is a singularity. This seems to imply that as real(z) increases, there are an infinite number of singularities arbitrarily close to the real axis, as pent(z) grows arbitrarily large.

You mention that tetration works for b>1. See the post by Nuinho, also see my post#28, on what he calls the super-euler number, b=1.6353244967, 1.6353244967 ^^^ oo ~= 3.0885549441, since sexp(z) has an upper real valued parabolic fixed point for this base=1.635324496, where pent(infinity) goes to a constant ...

Then for this base, and bases smaller, there are multiple real valued fixed points for Tet(z)=z. In my pari-gp code, sexpupfixed will generate this peculiar pentation base.