sqrt(exp)

[bo198214]

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

So it remains still open whether the fixed point of \( \log(z)+2\pi i \) is a fixed point of \( \sqrt{\exp} \) on some branch.

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

That looks ingenious, Dmitrii! As for my first observation, this is an extension of Andrew's slog method - which does not work at fixed points - to fixed points. It seems that the coefficients can be computed directly without limits! This approach is also similar to the one of regular iteration where the Abel function is the logarithm at the fixed point plus some power series. As my time is currently really limited, I hope I can make a more concrete reply (or better a new thread as it is no more this topic) tomorrow.