Alternate solution of tetration for "convergent" bases discovered

[tommy1729]

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.

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

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 ?

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

what are its properties ?


thanks for the reply.

( btw i believe in the continuum sum )

tommy1729