tetration base conversion, and sexp/slog limit equations

[sheldonison]

Bo,

Thanks for your reply.

How do you make use of this increased linearity to define \( \text{sexp}_b \) (\( b \) little bigger thean \( \eta \))?
Just in the usual way by \( \text{sexp}(x+1)=\exp_b(\text{sexp}(x)) \) and \( \text{sexp}(x-1)=\log_b(\text{sexp(x)) \)?

Yes, I'm using these equations. In particular, I did a lot of work on the critical section, with most of my results posted here. I rediscovered the same critical section that Jay discovered for any arbitrary base b larger than \( \eta \), but I also extended the results by finding the equations for a smooth 2nd and 3rd derivative for the \( \text{sexp}_b \) function for the critical unit section from \( \text{slog}_b(\text{log}_b(e)) \text{ to slog}_b(\text{log}_b(\text{log}_b(e))) \). The link shows how that a linear approximation has a continuous first derivative, since the 1st derivative is equal on both endpoints of the critical section, and I also posted the equations for a continuous 2nd derivative (3rd order approximation), and a continuous 3rd derivative (4th order approximation).

The wobble turned out to be quite a bit larger than the error term for my estimates,

How do you obtain the error term?

The data in this posts used b=1.485 (mostly an arbitrary number larger than \( \eta \text{ sexp}_{1.485}(e) \text{ is about }6.5 \)). To improve convergence, I used a 3rd order polynomial estimate for the critical section (2nd order smooth derivative), and used the delta to the 4th order polynomial approximation (with a 3rd order smooth derivative) as the error term. I also tried a second base a little smaller than the first, and got similar results consistent to within about \( 1x10^{-5} \) in the conversion to base e.

I see, the proper tetrational should become more and more linear when the base approaches \( \eta \) from above, and is characterized by this demand.
....
Here I see some difficulties to put that mathematically. You want to define tetration only for one base, but "approaching" means that you have to define tetration at least for a sequence of bases.
But perhaps this is just a question of exactness. The more exact the tetration for arbitrary bases should be the closer to \( \eta \) one have to choose the base of the initial tetration, something in that direction.

Lots of difficulties. The easy part is defining a sequence that approaches \( \eta \), and showing that the sequence itself has continually reducing relative error terms in the critical section, and by extrapolation, all of the sexp curve.

But then for each element in the sequence, the base conversion constant must be calculated. That requires a sequence. Then one must show the base conversion factor converges to a value. I think the base conversion factor can be shown to converge to a value for each term in the sequence, but that's not the same as showing it converges to the same value as b approaches \( \eta \). Then one must show that the resulting tetration curve converting from base b as it approaches \( \eta \) converges when it is converted to another arbitrary tetration base curve (it may not converge).