slog_b(sexp_b(z)) How does it look like ?

[sheldonison]

Hi sheldon

the pseudoperiodic property has been on my mind, but still I havent figured things out as much as I desire.

Hi Tommy,

There are certainly more details, and probably other cases where the slog(sexp(z))<>z, where it is not 1-cyclic.

What is it about slog(2sinh(z)) ??
Does this relate to things said before ?

I was thinking along the lines that there are a whole family of analytic iterated functions that grow super-exponentially, such that in the limit as z gets larger,
\( \lim_{z \to \infty} \text{slog}(f^z)=\text{slog}(f^{z+1})-1 \)

For example,iterated gamma function also falls into this category, as does 2sinh^z, as does sinh^z, and other bases for tetration. The conjecture is that for these functions, taking the limiting behavior of the slog(f^z)-z is a repeating 1-cyclic function, that is c infinity and nowhere-analytic, which I find fascinating.