Continuum sum - a new hope

[sheldonison]

.... For regular iteration you need a fixed point.
Real fixed points exist only for base \( b<e^{1/e} \).
Of course you can do it also at a complex fixed point of \( e^x \) however then the resulting tetration is not real on the real axis (actually it is not even analytic).

Henryk,
Could you clarify what function you are referring to that is not analytic? I thought that regular tetration developed from the complex fixed point for base e, with complex values on the real number line, was analytic and entire. It must have something to do with the rest of this post....
- Sheldon