x↑↑x = -1

[sheldonison]

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 \( \;\alpha_2(b^z)=\alpha_2(z)+1 \)

So the conjecture is if Tet(z) is the bipolar complex analytic tetration function, then in the upper half of the complex plane,
\( \alpha_1(\text{tet}_b(z)) = \alpha_1(\text{tet}_b(z+1))+1 \)
\( \alpha_1(\text{tet}_b(z)) = z + \theta_1(z) \)
\( \text{tet}_b(z)=\alpha_1^{-1}(z+\theta_1(z))\;\;\;\theta_1(z) = \sum_{n=0}^{\infty} a1_n \exp(2\pi i z)^n \)

And in the lower half of the complex plane
\( \alpha_2(\text{tet}_b(z)) = \alpha_2(\text{tet}_b(z+1))+1 \)
\( \alpha_2(\text{tet}_b(z)) = z + \theta_2(z) \)
\( \text{tet}_b(z)=\alpha_2^{-1}(z+\theta_2(z))\;\;\;\theta_2(z) = \sum_{n=0}^{\infty} a2_n \exp(-2\pi i z)^n
\)

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 \( \exp(1/e)+\epsilon \)

If you load up tetcomplex. and init(3+I), you get the following. The Taylor series is accurate to a little better than 13 decimal digits inside a circle of radius=0.85.

Code:

base           3 + I
  fixed point    0.324536386411255842495384 + 1.00148180609593401223717*I
  fixed point2   -0.0976763924712228440288378 - 1.39768129114469821694239*I
  Pseudo Period  4.01614979115113383262106 + 0.603487054263112033534298*I
  Pseudo Period2 4.02129656594167739617471 - 1.51597598902672933652589*I

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}

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