x↑↑x = -1

[sheldonison]

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.)

I think a quote from Tommy best sum's up my answer:

But then what did mike and sheldon compute !??

There are equations to show the mathematical equivalence of the theta mappings used in kneser.gp for real bases to Kneser's Riemann mapping, assuming convergence; I posted the relevant equations below. Mike had this idea that Tetration solutions involving both fixed points could be calculated and gave some examples. So in the tetcomplex.gp program, I extending the theta mappings in kneser.gp to complex bases. Obviously there is no Riemann mapping for complex bases, but empirically there are a pair of theta mappings, which lead to a complex base solution. Lacking a Riemann mapping, this is much less robust mathematically, at least until we come up with a theoretical framework. But from a practical standpoint, it works. I published a Taylor series for base 3+I accurate to 13 decimal digits in post#7. I loaded up tetcomplex.gp with "\p 173", and "init 3+I". And after 34 iterations, it damn sure acts like a complex tetration solution for base=3+I, accurate to a little over 86 decimal digits. Each iteration improved precision by roughly 8.4 binary bits, until it reached the limit of the precision for the Abel functions and their inverses.

\( \alpha(\text{tet}_b(z)) = z + \theta(z)\;\;\;\theta(z) = \sum_{n=0}^{\infty} a_n (\exp(2\pi i z))^n\;\; \) formal Abel function of tet(z) equals z+theta(z)
For real bases, the z+theta(z) mapping is mathematically equivalent to Kneser's Riemann mapping. Kneser's Riemann mapping function would be generated from the \( a_n \) coefficients of the \( \theta(z) \) mapping as follows.

The Riemann mapping starts with, \( z + \theta(z)\; \) which is modified too \( \exp(2\pi i (z + \theta(z))) \) and then we map \( z \mapsto \frac{\ln(z)}{2\pi i} \)

\( \text{RiemannMapping}(z)= \exp(2\pi i \times( \frac{\ln (z)}{2\pi i} + \sum_{n=0}^{\infty} a_n z^n)) \)
\( \text{RiemannMapping}(z)= z \times \exp( \sum_{n=0}^{\infty} 2\pi i a_n z^n)\;\; \) which generates Kneser's Riemann mapping from Sheldon's theta(z) coefficients