05/22/2015, 05:28 AM
(This post was last modified: 05/22/2015, 12:01 PM by sheldonison.)

(05/21/2015, 06:31 PM)marraco Wrote: Tetration base 1 is a constant function equal to 1, for x>-1, and has a discontinuity at -1. Or at least that's the solution if ... I wonder if there is a family of continuous solutions, and this one is the evolvent.

b=-1 has two primary fixed points. both repelling; I'm not sure what the other fixed points are.

-1,

0.266036599292773 + 0.294290021873387*I

Pairs of repelling fixed points can sometimes be used to build analytic complex valued solutions, with the property that tet(-1)=0. As you discovered, there also appears to be another family of solutions for 0<b<1, which is a damped oscillator type solution. So it quickly appears that there is no uniqueness what so ever... with an infinite number of interesting solutions possible.

I find it easier to work with the conjugate base, and analyze iterating the function:

instead of where

and this is a simple linear transformation from y to z

Then analyzing the function for iterating

is conjugate (or mathematically equivalent) to iterating the function

but this conjugate form is much simpler to work with and understand. k=0 is the parabolic case which corresponds to base k>0 corresponds to Kneser's real valued tetration solution, and corresponds to Marraco's bases between 0..1

And the conjugate value of k for b=-1 is

I have a series solution for the two fixed primary fixed points; http://math.eretrandre.org/tetrationforu...hp?tid=728 which turns out to have a nice Taylor series solution with and with rational coefficients. I am also in the process of debugging a very powerful generic slog/abel pari-gp program for iterating for arbitrary complex values of k. This bipolar Abel function may be unique, based on Henryk's proof, but this solution requires that the Abel function be analytic in a strip between the fixed points. For Marraco's damped oscillating solutions, the Abel function has singularities where the derivative of the sexp'(z)=0. I haven't yet generated any analytic solutions for Marraco's damped oscillating solutions for 0<b<1; that's also a longer term goal.

- Sheldon