sqrt(exp)

[bo198214]

Why do you expect \( \sqrt{\exp} \) to have many fixed points?

As I already said, if \( \exp \) has a fixed point then also \( \exp^n \) has that fixed point for every integer \( n \), at least on some branch. So I would expect that it has that fixed point also for every real \( n \) on some branch.

Additionally for each singularity \( s \) of \( \text{slog} \) there should be singularities at \( \log(s)+2\pi i k \) (on some branch), because:
\( \text{slog}(s)=\text{slog}(\exp(\log(s)+2\pi i k))=\text{slog}(\log(s)+2\pi i k))+1 \)

A similar evidence was brought up by jdfox already
here and by Andrew here

Those singularities are also singularities of \( \sqrt{\exp} \) by
\( \sqrt{\exp}(z)=\text{sexp}(0.5+\text{slog}(z)) \)
(as long as \( \Re(\text{slog}(z))>-2 \))

Another question: Could you deduce analytically the coefficients in the first terms of the asymptotic expanstion of slog in civinity of the fixed point?

Asymptotic from where?