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[tommy1729]

How does tetration work for base 3 + i ?
...

Start with a base>eta, and rotate counterclockwise till you get to 3+i, or any other base. Lets say we have a complex tetration for b=3+i.
\( \text{tet}_b(0)=1 \)

As we rotated, we follow the two fixed points, L_1, L_2, which started out as L and L*. 1 refers to the upper half of the complex plane 2 refers to the lower half.
\( L_1=b^{L_1}\;\;\; \) fixed point, upper
\( L_2=b^{L_2}\;\;\; \) fixed point, lower
\( \lambda_1 = \ln(L_1)\;\;\; \) derivative at the fixed point, upper
\( \lambda_2 = \ln(L_2)\;\;\; \) derivative at the fixed point, lower

Let as assume the most straightforward case, that both \( |\lambda_1|>1\; \) and \( |\lambda_2|>1\; \), so both fixed points are repelling. This is true at the real axis for bases>exp(1/e). On the ShellThron boundary, \( |\lambda_1|=1 \), and the Schroeder function does not converge if \( \lambda_1^n=1 \) for some integer n. For some mysterious reason, complex tet(z) is still defined and analytic for these bases; see this question on math overflow. But it is easier to start by assuming that the magnitude of both lambda's is greater than 1.

\( S_1(b^z) = \lambda_1(S_1(z))\;\;S_1(L_1)=0\;\;\; \) formal Schroeder equation, upper
\( S_2(b^z) = \lambda_2(S_2(z))\;\;S_2(L_2)=0\;\;\; \) formal Schroeder equation, lower
\( \alpha_1(z) = \ln(S_1(z))/\ln(\lambda_1)\;\;\; \) formal Abel function, upper \( \;\alpha_1(b^z)=\alpha_1(z)+1 \)
\( \alpha_2(z) = \ln(S_2(z))/\ln(\lambda_2)\;\;\; \) formal Abel function, lower \( \;\alpha_2(b^z)=\alpha_2(z)+1 \)

Ok. So i use the koenigs function to solve the schroeder equation and from that I compute the abel function.

But that gets me 2 abel functions ...

What to do with them ?
I cannot simply use one for the upper plane and one for the lower plane , can i ??

So the conjecture is if Tet(z) is the bipolar complex analytic tetration function, then in the upper half of the complex plane,
\( \alpha_1(\text{tet}_b(z)) = \alpha_1(\text{tet}_b(z+1))+1 \)
\( \alpha_1(\text{tet}_b(z)) = z + \theta_1(z) \)

Should that not be \( \alpha_1(\text{tet}_b(z)) = \alpha_1(\text{tet}_b(z+1))-1 \) ?

And in the lower half of the complex plane
\( \alpha_2(\text{tet}_b(z)) = \alpha_2(\text{tet}_b(z+1))+1 \)
\( \alpha_2(\text{tet}_b(z)) = z + \theta_2(z) \)

Now I still have 2 abel functions. Not 1.
And I have neither tet_b(z) or theta(z).
This equation just shows how tet_b(z) and theta(z) are related.
Since I have the abel function I can just plug in some theta(z) and get some tet_b(z).

But still 2 of those and I see no reason to prefer some theta resp tet_b over another.

Why dont we just take the inverse of the abel as superfunction(s) ?

hmm.

I see no motivation or uniqueness condition.

And picking one function for the upper and one for the lower seems partially arbitrary ?
Yeah ok that matches the fixpoints at imaginary infinity , but there are others too right.

How do we merge the two theta functions ??

We do merge , to avoid ill defined parts of the complex plane right ?

If both \( |\lambda_1|>1\; \) and \( |\lambda_2|>1\; \), then the conjectured tetration solution will have \( \theta_1(z) \) analytic in the upper half of the complex plane, with a singularity at integers at the real axis, and decaying to a constant as imag(z) goes to +infinity. Also, \( \theta_2(z) \) is analytic in the lower half of the complex plane, with a singularity at integers at the real axis, and decaying to a constant as imag(z) goes to -infinity. Then tet(z) decays to L1 as imag(z) goes to +infinity, and decays to L2 as imag(z) goes to -infinity. The pseudo periodicity in the upper half of the complex plane will be

\( \frac{2\pi i}{\ln(\ln(L_1))}\; \) as tet(z) decays to \( \;\alpha_1^{-1}(z) \)

And in the lower half of the complex plane the pseudo periodicity will be

\( \frac{2\pi i}{\ln(\ln(L_2))}\; \) as tet(z) decays to \( \;\alpha_2^{-1}(z) \)

The tetcomplex.gp program calculates the two theta(z) functions which can it turns out can be represented with an analytic Taylor series, \( \theta_1(z), \; \theta_2(z), \) as well as a Taylor series for tet(z) centered at z=0, where the tet(z) Taylor series has a radius of convergence of 2, since there is a singularity at z=-2. These three analytic functions, along with the inverse Abel functions, allows calculating tet(z) anywhere in the complex plane. The tet(z) taylor series at z=0 is important, since it allows getting accurate numeric results at the real axis, where the two theta(z) functions have a singularity, and it is difficult to get accurate results for tet(z) at the real axis using just the theta(z) functions, even though the three analytic representations are equal to each other at the real axis. Unlike real tetration, there is no Riemann mapping equivalent for determining the theta(z) functions, and all I can do is conjecture based on numeric evidence, so tetcomplex is fairly weak from a theoretical point of view. Moreover, as you rotate further you get to the ShellThron boundary, and there the Schroder function doesn't even converge, and then you continue on to where the upper |lambda_1|<1, and you have an attracting fixed point, which further complicates things... Things get even more complicated at the real axis for bases<exp(1/e), and even more complicated as you rotate further, and apparently run up against an analytic barrier the second time you cross the ShellThron boundary. My intent was to publish the tetcomplex.gp code and the results in the computation section of this forum because I don't have the background to make them mathematically rigorous; my best guess is that the techniques used in parabolic implosion might help using Ecalle cylinders to develop a rigorous approach for bases in the neighborhood of \( \exp(1/e)+\epsilon \)

ah so you do claim the 2 theta function agree on the real line.

But then how can they be different ??
By schwarz reflection and analytic continuation it seems weird that 2 analytic functions can agree on the real line , yet be different off the real line.

In short
1) I do not get the motivation and uniqueness.
2) I do not understand the fact that there are 2 functions.
3) I do not know how the theta is computed.

So maybe we should map " something " to the real line with a riemann mapping.

Maybe that something contains : 0,1,b,b^b,b^(b^b),...
However we need to map continu flow to the real line, not just some points.

So maybe the question is : can a riemann mapping merge 2 functions into 1 ?
If so , its probably harder than kneser.

I do not immediately see another possible explication or way.
But I assume there is since I have so many questions and probably misunderstandings about this.

Thanks for your swift reply.

Though Im still in a state of confusion and skepticism.
Maybe temporarily.

regards

tommy1729