06/01/2023, 02:59 PM

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.

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.

Regards

Shanghai46

Shanghai46