Alternate solution of tetration for "convergent" bases discovered

[mike3]

Now that we have a way to define continuum sums, we start with an initial guess \( F_0(z) \) to our tetrational function \( \tet_b(z) \), and then apply the iteration formula

\( F_{k+1}(z) = \int_{-1}^{z} \log(b)^w \exp_b\left(\sum_{n=0}^{w-1} F_k(n)\right) dw \)

and take the convergent as \( k \rightarrow \infty \).

Now the adaptation of this theory to construct a numerical algorithm is a little more complicated. I could post that, if you'd like it, in the Computation forum. But the above is the basic outline of the method.

thanks thats a lot better.

Yes. Though I accidentally left out a detail -- see the post again, I updated it.

1) is that similar or equal to my " using the sum hoping for convergence " thread ?

It should be.

2) i assume the intial guess needs to be a taylor , laurent or coo fourier series , in other words analytic. and also map R+ -> R+ , but then we have to start from an already tetration solution ?

In the abstract, it's just a holomorphic (multi-)function. But for the numerical algorithm, we'd use a (truncated) Fourier series.

so what is the advantage of this sum method solution ? afterall it is just a 1periodic wave transform of another sexp/slog.

Biggest advantage seems to be that it gives the widest range of bases I've seen for any tetration method.

what are its properties ?

Not sure exactly what you'd want to know. It seems to be similar to the "Cauchy integral" for bases outside and in some parts of the STR, but the weird part is that when it is used for \( 1 < b < e^{1/e} \), it gives the attracting regular iteration. ???