Base -1 - Printable Version +- Tetration Forum (https://tetrationforum.org) +-- Forum: Tetration and Related Topics (https://tetrationforum.org/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://tetrationforum.org/forumdisplay.php?fid=3) +--- Thread: Base -1 (/showthread.php?tid=1004) Pages: 1 2 Base -1 - marraco - 05/21/2015 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 $$\\[15pt] {^{x}1=\lim_{b \to 1} ^xb}$$. The oscillatory behavior of bases 0-1, and has a discontinuity at -1. Or at least that's the solution if $$\\[15pt] {^{x}1=\lim_{b \to 1} ^xb}$$... 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 00 corresponds to Kneser's real valued tetration solution, and $$\;\;k= \pi i + c\;\;$$ corresponds to Marraco's bases between 0..1 And the conjugate value of k for b=-1 is $$k=2.14472988584940 + 0.5\pi i$$ I have a series solution for the two fixed primary fixed points; http://math.eretrandre.org/tetrationforum/showthread.php?tid=728 which turns out to have a nice Taylor series solution with $$z=\sqrt{-2 k }\;\;$$ and with rational coefficients. I am also in the process of debugging a very powerful generic slog/abel pari-gp program for iterating $$z\mapsto \exp(z)-1+k$$ 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 0-1, and has a discontinuity at -1. Or at least that's the solution if $$\\[15pt] {^{x}1=\lim_{b \to 1} ^xb}$$... 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*IThey can be calculated with the Lambert W function, as $LambertW(-i\pi)*i\div\pi\forall n\in\mathbb{Z}$.