Fractional calculus and tetration

[sheldonison]

My extension \( F \) is also the sole extension that is bounded by \( |F(z)| < C e^{\alpha |\Im(z)| + \rho|\Re(z)|} \) where \( \rho, \alpha, C \in \mathbb{R}^+ \) and \( \alpha < \pi/2 \).

The regular iteration for bases \( 1<b<\eta \) satisfies that, as it is periodic and bounded in the right halfplane.
What complex bases does it work for? Does it work for base eta?

The Schroeder equations that give the formal solution for attracting (and repelling) fixed points do not work for the parabolic case, base eta=e^(1/e). See Will Jagy's post on mathstack. I have written pari-gp program that implements Jean Ecalle's formal Abel Series, Fatou Coordinate solution for parabolic points with multiplier=1; this is an asymptotic non-converging series, with an optimal number of terms to use. To get more accurate results, you may iterate f or \( f^{ -1} \) a few times before using the Abel series.