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

[marraco]

I had just installed Sage, which includes PariGP, but still are stuck on excel, because I need to learn PariGP.

Thanks for your help.
I still chewing it. Since λ is negative, that means that his powers take complex values, then h(x) should be complex, and that's a mess with excel.

So, as practice, I tried with base b=2^⁽½⁾, and still trying to make it work.
The fixed point is 2, and λ=0,693147181

The blue line should be the tetration base b=1,414213562, and should be matching the cyan line (which I got with Excel).

[/code]
\( \text{superfunction}(z)=S\left( (-\lambda)^z \cdot \sin(\pi z)\right) \)
And here is a graph, showing an analytic super-function for \( {0.01}^z \) the Op might be interested in, from -10 to 10, where it starts out oscillating around the primary fixed point, and then converges to the two cycle that the Op noted. Of course, this isn't tetration; Tet(-2) is by convention a logarithmic singularity. And this function has no uniqueness, I could have just as easily used sine or cosine, or an infinite number of other 1-cyclic functions.

Maybe that curve is inverted?
That way, it would match my own solution, and would give ⁰b≈1 and ⁻¹b≈0




I got that solution adjusting the coefficients with Excel's Solver, for a Taylor series with 12 coefficients, (expanded around 0), subject to these restrictions:
a₀=1
⁻¹b=0
⁰b=1
\( ^xb=b^{\,{^{x-1}b}} \)

I got these coefficients:

Code:

a0    1
a1    -1,339425732123610E+00
a2    -3,822141746305190E+00
a3    4,850222934263920E+00
a4    5,354079775248650E+00
a5    -6,056488564613110E+00
a6    -2,938727974303330E+00
a7    2,550689528770280E+00
a8    4,116553795338360E-01
a9    1,989418887288240E-06
a10    1,291559624345350E-04
a11    -1,519921425061600E-07
a12    1,409996370575610E-08
a13    1,348473734742680E-10
a14    1,133187395821660E-11
a12    -3,860717643983480E-09