Carlson's theorem and tetration

[sheldonison]

Hi.

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.)

Mike,

When I was working with eta, I noticed the simmilarity between \( \text{sexpUpper}_\eta(z) \) and \( \text{sexpLower}_\eta(z) \). As I remember, in the complex plane, sexpLower(z) looked a lot like sexpUpper(z+i), especially as imag(i) increased. I'll try to dig up an old contour graph.... Anyway, there are two solutions at base eta, one, the regular superfunction developed from the fixed point of "e", that goes to infinity as real(z) increases. This is the sexpUpper(z) function which has no singularities, and real(z)>e for all z at the real axis. The sexpLower(z) function of eta has a singularity at sexp(z,z=-2), and approaches "e" as real(z) increases to infinity. What you are pointing out is that the sexpLower(z) is a conformal/Kneser map of the sexpUpper(z) which is pretty cool, since it can also be developed from the attracting fixed point at +real infinity=e. Neat!

This conformal mapping of one solution into thte other would only work for eta, since bases<eta, the z+theta(z) between the two solutions no longer decays to zero at +I*infinity, but would be a Laurent series if wrapped around the unit circle.

.... The modifications also provide the theorem characterizing the regular tetrationals for \( 1 < b < \eta \).

I wonder if Carlson's theorem can be used to say anything about bases >eta?
- Sheldon