x↑↑x = -1

[tommy1729]

SO we do not have uniqueness ??

Or are there branch issues again ?

And how weird would it be to have a proof of non-uniqueness without a proof of existance.

So for complex base tetration, uniqueness would stem from the analytic continuation from Kneser's solution, which would also answer Tommy's question about uniqueness, since Henryk Trapmann has a published proof for the uniqueness of slog for Kneser's solution, but there are multiple branches in the pseudo period of tetration, and you can get these equations, but they should all be equivalent, since real Tetration is unique.

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.

But I'm not convinced that the Kneser solution works for complex bases. I can't imagine how to construct it, because I can't help but imagine a discontinuous first derivative at the endpoints of our initial section of the real line. (For reference, I've constructed the Kneser solution explicitly for base e, so I know how to imagine it for a real base 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.

I too do not believe in a kneser solution for complex bases.
Not sure if taking derivatives with respect to the base IS EQUIVALENT to kneser for complex bases ?

If it is equivalent , I think I can prove with the riemann mapping that the method is doomed to fail.

Yet I also do not believe that taking the derivative with respect to the base or (if different) complex kneser is EQUIVALENT to the equation with 2 theta functions.

SO as far as Im concerned they might be 3 distinct methods that might or might not fail.

It seems , but I could be wrong , that sheldon feels all 3 are equivalent and they work.

I had the idea that we need a special type of series expansions for these theta functions ; that combination of singularities , analytic for non-integer real and periodic is something that is not well described naturally by fourier series nor by Taylor series.
Also these annoying singularities make it hard to use theorems from complex analysis.

The question came to me wheither or not an analogue uniqueness criterion like TPID4 could apply to complex base tetration agreeing on both fixpoints. Or complex bases tout court.

Notice I dropped the idea of pseudoperiodicity , I feel it overcomplicated the equations.
I have not considered loosing the property of pseudoperiodicity seriously yet.

In the context of complex base tetration agreeing on fixpoints , I think the singularities are problematic for andrew robbins method if we focus on integers. If we use half-integers I think we have issues with the radius of convergeance.

Welcome back mr fox

regards

tommy1729