tetration exp(z)-1+k

[sheldonison]

The tangent superfunction provides an excellent piecemeal definition of tetration, and when it replaces the linear approximation, the resulting sexp(z) approximation has a continuous first and second derivative. This works for all tetration bases.

There are many nonlinear approximations that give a continuous first and second derivative.
Why is this preferred ? Is there a uniqueness condition ?

ok, fixed the typos. I'm still inventing this post on the fly; as I said in my first post, "I plan on developing these ideas over the next several weeks or months ..."
The next step would be to look at the Abel function of \( g(z) = k+\frac{2z}{2-z} \; \), as well as to look at how the superfunction of g(z) changes as k rotates around 0 in the complex plane, and compare it to the closely related Abel function of \( \exp(z)-1+k \). The long term goal which this post may lead to would be to put some kind of theoretical base behind this complex plane tetration post. For tetration for real bases>eta, we have Kneser's Riemann mapping, and I can show that the \( z+\theta(z) \) mapping is equivalent. But for complex base tetration, there is no Riemann mapping; Henryk's post discusses the Abel function for complex perturbations. So the conjecture is that there is a quasi-conformal mapping between the Abel function of g(z), and the Abel function of \( \exp(z)-1+k \), and that the pretty pictures and pari-gp code for complex base tetration is computing the quasi-conformal mapping between the tangent superfunction, and complex base tetration. It would be nice to be able to say if these bipolar theta mappings converge, than they are computing the complex Abel function. More later when I have time. These ideas are still under construction, and it takes me time to think it through.