Carlson's theorem and tetration

[bo198214]

I found the following easy uniqueness theorem that characterizes the regular tetrational of the base \( b = \eta = e^{1/e} \), and perhaps also the whole regular tetrational (with attracting fixed point) (though base-\( \eta \) is particularly interesting since it seems that both the regular and non-regular (i.e. Kneser's, etc.) method approach the same tetrational at this base.),

Hm, you mean that the super-exponential is bounded on the positive real axis by e and is imaginary periodic and hence is (exponentially) bounded on the right halfplane ...
This is realy a nice finding, Mike!

I guess it can be generalized to arbitrary regular superfunctions as they are always of the form \( \eta(\pm e^{\kappa z}) \) for some function \( \eta \) analytic at 0.