tetration base conversion, and sexp/slog limit equations

[bo198214]

what does \( {\kappa_{a,b} \)
tell us about limit expression for sexp,
\( \text{sexp}_e(x) =
\lim_{b \to \eta^+}\text{ } \lim_{n \to \infty}
\text{ln(ln(ln(}\cdots
\text{sexp}_b (x + \text{slog}_b(\text{sexp}_e(n)))))) \)

This expression has mixed limits, one inside and one outside. The limits can be separated by the following manipulation:
\( \lim_{n \to \infty}
\ln^{\circ n}(\text{sexp}_b (x + \text{slog}_b(\exp^{\circ n}(0))) \)
\( \lim_{n\to\infty}\ln^{\circ n}(\text{sexp}_b(x+n+\text{slog}_b(\exp^{\circ n}(0))-n)) \)
\( \lim_{n\to\infty}\ln^{\circ n}(\exp_b^{\circ n}(\text{sexp}_b(x+c_{b,e})))=\kappa_{e,b}(\text{sexp}_b(x+c_{b,e})) \)

Now we can seperately investigate the behaviour of \( \kappa \) and of the linear approximation \( \text{sexp}_b \) and perhaps come to the conclusion that \( \kappa_{e,b}\circ \text{sexp}_b \) converges for \( b\to\eta \).

Though I doubt that it is analytic; each \( \kappa_{e,b}\circ \text{sexp}_b \) is a piecewise analytic function (provided that \( \kappa \) is analytic). Somehow it would be strange (or at least difficult to show) if a sequence of only piecewise analytic functions converges to an analytic function. Though it may well be that it is infinitely differentiable, if the jumps reduce to zero for \( b\to\eta \). From you description I would expect that \( b\to\eta \) makes \( \kappa_{e,b}\circ \text{sexp}_b \) smooth.

I've been an engineer for the last 25 years, but I'd like to take some classes in higher mathematics; maybe when I retire. I have a lot of catching up to do, to follow the posts on these forums. I understand Andy's matrix equations, and Jay's suggestions about the odd derivatives of the sexp extension, but I don't understand Dmitrii Kouznetsov's equations. And I don't understand much of the details of the complex plane graphs, aside from the radius of convergence.

Hey Sheldon, your are welcome! I really appreciate your contributions, your approach of \( b\to\eta^+ \) is completely new and enriches the forum. I hope it will work out!