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

[sheldonison]

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Here is tetration base a=0.01:
c₁ = 0,941488369
c₂ = 0,013092521
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The negative axis probably converges to a real function akin to a cosine.
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I need something better than excel.

Pari-gp is what you want. Let's assume you're only interested in Koenig's solution as opposed to the much more complicated complex base tetration solution. For your base b=0.1, you should be able to find a real valued fixed point, plus two complex conjugate repelling fixed points. For Koenig's solution, you'll only need the real valued fixed point, which is attracting for b=0.1. That's a good place to start. Figure out how the function behaves in the neighborhood of the fixed point, and what its periodicity is.... The problem with base exp(-e), is that the periodicity is 2, which is a really nasty case since it turns out there is no Koenig's solution... If you'll notice from the link, it took me 8 months to find a conjectured complex base solution, from my first post, to the post with the Taylor series for the complex base tetration solution.

But anyway, base b=0.1, find the fixed point, and find the multiplier \( \lambda \) at the fixed point, and from that the periodicity\( =\frac{2\pi i}{\ln(\lambda)} \); and that's a pretty darn good start, assuming you ever get that far .... The multiplier \( \lambda \) is defined where \( b^L=L \) and \( b^{L+\delta} \approx L + \lambda\delta \)

So then there is a formal Koenig solution that has \( S(\lambda z) = b^{S(z)}\;\;\;\exp_b^{\circ n} = S(\lambda^n) \). From that, you should be able to generate graphs, or a Taylor series, or whatever you like.