Continuum sum - a new hope

[sheldonison]

Or are these singularities only on some "branches" of the Abel function, so there's other branches in which there are no singularities there, thus yielding values for where the integer-height tetration sequence occurs? (which would make more sense)

I guess so. But indeed I didnt have a closer look at other branches of the Abel function, I only vaguely remember that JayD had a "spider-web" picture of it, with slightly overlapping ends of two branches. (but couldnt find it again, the search function of the forum software is really weak.)

I'm more familiar with the graph of the superfunction than the abel function. There will be a place where the superfunction(z) = 2pi*i
and iterating f(z+1)=e^f(z) from that point,
superfunction(z+1,z+2,z+3,z+4)=1,e,e^e,e^(e^e) ....
The only value the superfunction never takes on is zero. But, the section of the superfunction relevant for Reimann mapping (and the abel function) is the contour line where imag f(z)=pi*i, whose corresponding real values range from +/- infinity. This pi*i contour is mapped to line segment between z=-3 and z=-2 for the real valued sexp function, which has singularities at z=-3 and z=-2, and has imag f(z)=pi*i between -3 and -2.
- Sheldon