x↑↑x = -1

[jaydfox]

Real tetration is not unique since we also have a 1periodic wave.

So the question of uniqueness is imho not resolved.

regards

tommy1729

For a real base greater than eta, the Kneser solution is unique. Henryk established a condition somewhere, roughly equivalent to Kneser's solution, that the tetration function should be bounded as the imaginary part goes to +/- infinity. This is equivalent to solving for an slog that asymptotically goes to a logarithm at the primary fixed points (which, coincidentally, is how I accelerate convergence of the intuitive/matrix solution). So there is a simple uniqueness criterion, for real bases greater than eta.

Im aware of the uniqueness condition. But there is a difference between uniqueness and uniqueness condition !

In fact I proved the related TPID 4 !

http://math.eretrandre.org/tetrationforu...ght=TPID+4

At least that is how I interpret your uniqueness condition , correct me If I misunderstood.

Im suggesting that as for the real tetration , also the complex base tetration that agrees on both fixpoints has a 1periodic wave ( and still agrees on both fixpoints ).

Im here for quite a while.

Maybe I'm just misunderstanding what you mean by uniqueness. b^x is not the only solution to the functional equation f(z+1) = b*f(z). We can apply any 1-cyclic transform to z that we wish, e.g., b^(z+sin(2*pi*z)).

Would you consider b^z to be unique? If we can't agree to a unique solution for exponentiation, there's no point talking about a unique solution to tetration or any other (inverse) Abel function.

I'm suggesting that for real bases greater than eta, there is a solution for tetration that is as unique as exponentiation is.