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).

(04/14/2015, 04:48 PM)sheldonison Wrote: [ -> ][/code]

And here is a graph, showing an analytic super-function for 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

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