Reducing beta tetration to an asymptotic series, and a pull back

[JmsNxn]

That seems correct and logical (imo).

Originally I considered taking the lambertW function for approximations.
Not sure if that still relates much.

However a tiny remark.

division by tet(s) is not necc small , even for Re(s) large ; because tet(s) can be close to 0 !

This (chaos) complicates matters , despite perhaps still true ... not so trivially ...

regards

tommy1729

You're correct, Tommy.

But if \( \Im(s) = A \) is fixed, eventually \( \beta_\lambda(s) \to \infty \) as \( \Re(s) \to \infty \) And they aggregate to the orbit \( 0,1,e,e^e,e^{e^e}... \). So EVENTUALLY on each line it will be tiny. That's more so what I meant. But if we vary \( \Im(s) \) while we vary \( \Re(s) \) that's where the trouble happens.

Regards