sqrt(exp)

[Kouznetsov]

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.

Every fixed point of \( \sqrt{f} \) is fixed point of \( f \).
Some of fixed points of \( f \) can be also fixed points of \( \sqrt{f} \).

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

Yes, I see these singularities if I direct the cuts to the right hand side.
However, the higest fixed points of exp cannot be also those of \( \sqrt{exp} \), at least, in the branch, where the function is \( 2\pi\mathrm{i} \) periodic: the fixed points of exp are not equidistant.

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?

\(
\mathrm{slog}(z)=\frac{1}{L}\left( \log(z-L) + r - (z-L)a+(z-L)^2 B +\mathcal{O}(z-L)^3 \right)
\)
where \( r\approx 1.07796 - 0.94654 \mathrm{i} \),
\( a=0.5/(L-1) \) and so on.. \( L \) is fixed point of log.