x↑↑x = -1

[sheldonison]

In fact Im having doubts about the carleman matrices approach due to issues such as singularities and possible non-uniqueness.

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I have not had time to analyze complex bases. I have my doubts that a unique solution exists, since the periods of the fixed points are different, but I'm waiting to make an analysis before I make any claims one way or the other.

Hey Jay,

Welcome back. The originaly Op asked about \( x\uparrow \uparrow x=i \), to which I answered that x~=2.6918099719192 + 0.62660048483655i is a solution. And from there, Tommy asked me to explain how complex base tetration works for base 3+i, so I think this has been a good thread, because I got to review some of the fascinating stuff Mike and I explored. Jay, you should see if you can generate the accelerated version of the intuitive method for base 3+i. See post #7, this thread, for some good equations and the Taylor series; I added the slog series too, and post #17, this thread for the tetcomplex.gp algorithm overview, and post #11, this thread for some pretty complex plane graphs for base=3+i, pictures. As you noted, the two pseudo periods are different in the upper and lower halves of the complex plane, as the function approaches the \( \alpha_1^{-1}(x) \) inverse Abel function in the upper half of the complex plane, and \( \alpha_2^{-1} \) in the lower half of the complex plane.

The conjecture is that if you start with Kneser's real valued tetration, and take the function which is the first derivitive of the \( f(b)=\frac{d}{dx}\text{tet}_b(x=0) dx \), then f(b) is an analytic function for the first derivative in terms of the base; This would also apply to any of the other Taylor series coefficients. Emperical results using the tetcomplex.gp pari-gp complex base program strongly support the conjecture.

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 from the primary fixed points, 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.