In all calculators avaliable online, none can calcate x^^n for n being a negative number below-2. It is indeed undefiened for all integers below-2, but not for all reals below. You just have to define tetration on a interval of lenght one, and taking the logs the right amount of times.
So I used my tetration method to visualize. Each independent parts are borders by vertical asymptotes which aren't shown, but the overall shape looks like as if it was spiraling around a fixed point, the fixed point Log(a+bi)=a+bi, which is interesting because asides from the asymptotes, it converges towards the fixed point the sams way a real function converges in some conditions with my method : spiralling around it in the complex plane (it's the case if f'(fixed point)<0).
So here the graph is 2^^x, red is the real part and blue the imaginary. I also put 2 lines to show the fixed point.
(06/01/2023, 02:59 PM)Shanghai46 Wrote: In all calculators avaliable online, none can calcate x^^n for n being a negative number below-2. It is indeed undefiened for all integers below-2, but not for all reals below. You just have to define tetration on a interval of lenght one, and taking the logs the right amount of times.
So I used my tetration method to visualize. Each independent parts are borders by vertical asymptotes which aren't shown, but the overall shape looks like as if it was spiraling around a fixed point, the fixed point Log(a+bi)=a+bi, which is interesting because asides from the asymptotes, it converges towards the fixed point the sams way a real function converges in some conditions with my method : spiralling around it in the complex plane (it's the case if f'(fixed point)<0).
So here the graph is 2^^x, red is the real part and blue the imaginary. I also put 2 lines to show the fixed point.
I believe this oscillating behaviour was already noticed in the appendix of this book by Gianfranco Romerio (GFR) (famous contributor here, and also on of the founder of the subject of hyperoperations as we know today) and Konstantin Rubtsov here (not available online):
Rubtsov, K. A., and G. F. Romerio. "Hyperoperations, for science and technology: New algorithmic tools for computer science." (2011): 185.
I remember me talking with GFR back in 2014/15 about this and how this behaviour is vagualy reminescent of the gamma function and of how we have an odd/even oscillation inbetween the asymptotes at negative integers.
This convergence to a complex number going to negative infinity seems surprising to me at first, but yeah, I'm not very much into this anymore, so maybe this was already discussed on the forum. What I'm sure was discussed was a somehow orthogonal convergence phenomena.
The convergence to a fixed point for odd-ranked hyperexponentiation going to negative infinity, is part of a more general pattern that links all the hyperoperations. Even-ranked hyperoperations seems always to have vertical ansymptotes way before negative infinity.
