Why bases 0<a<1 don't get love on the forum?

[sheldonison]

I had being looking what had being done with bases lower than 1, and found nothing.

I didn't expected to get oscillating/damped periodic functions. They also seem to converge to \( \\[25pt]

{\lim_{x\to \infty}{^xa=N_a} \Leftrightarrow {N_a}^{\frac{1}{N_a}}=a} \)


¿Does this hints that \( \\[20pt]

{^{i.x}a\,=\,^x{(\frac{1}{a})} \) ??

Well there is at least one post; http://math.eretrandre.org/tetrationforu...0&pid=6748 tetration base \( \exp(-e)\approx0.0660 \)

This is an example of a really difficult case, where you can't use Koenig's solution, since the multiplier at the fixed point is \( \frac{\exp(\2\pi i)}{5} \), which is an indifferent case with a rational multiplier. Otherwise, using Koenig's solution is fairly straightforward, but of course Koenig's solution is never the same as the complex base solution you would get using both fixed points, if you started with tetration for a real base greater than \( \eta=\exp(1/e) \), and then slowly changed the base in the complex plane, while avoiding the singularity at \( \eta \).

I think Mike and I and Henryk are the only people on the forum that have any understanding of complex base tetration for \( b<\eta \); not that we understand it that well. One of the interesting things is that for real bases; \( b<\eta \); that it (\( \text{Tet}_b(z);\; \) complex base tetration for base b) is not real valued! For bases close to eta, it is almost real valued, as the imaginary pseudo period for both fixed points gets arbitrarily large.