Imaginary iterates of exponentiation

[Ivars]

First, some observations. The ith iteration of natural exponentiation of base e has two primary fixed points at \( 0.318131505... \pm 1.337235701...i \). If we momentarily call the ith iteration "imaginary exponentiation" and the -ith iteration "imaginary logarithm", the the upper fixed point is an attracting fixed point for imaginary exponentiation, and the lower fixed point is a repelling fixed point. For imaginary logarithms, the upper fixed point is repelling and the lower is attracting.

Hi Jaydfox,

This may be relevant:
Omega constant =0.5671432904097838729999686622 is defined by 1=Omega*(e^Omega):

So selfroot of Omega:

Omega^(1/Omega) = e^ln(Omega^(1/Omega) = e^((1/Omega)*(ln(Omega))=e^((1/Omega)*(-Omega)) =e^(-1)=1/e=0,367879441

Infinite tetration of selfroot of Omega:

h(Omega^(1/Omega))=h(1/e) = -W(-ln(1/e))/(ln(1/e))= -W(1)/-1=Omega=0,56714329=-ln(Omega),

Square superroot of (Omega^1/Omega) :

ssrt(Omega^(1/Omega) = ln(1/e)/W(ln(1/e))= -1/W(-1)= -1/(-0.318131505204764 +- 1.337235701430689*I) = 0.16837688705553+-0.707755195958823*I.

W(-1) = 0.318131505204764 +- 1.337235701430689*I

So why not call it Omegation?

Ivars