using sinh(x) ?

[sheldonison]

\( \operatorname{TommySexp_e}(z,x)= \lim_{n \to \infty } \ln^{[n]} (\operatorname{2sinh}^{[z]}(\exp^{[n]}(x))) \)

.... lets work in base e for convenience.

if the x in my formula is set to the fixpoints L or L* we have

tommysexp(z,x) = tommysexp(z,L) = L

tommysexp(z,x) = tommysexp(z,L*) = L*

as is needed. (easy proof btw)

i wonder what the value - if converging ! - of 2sinh^[+ oo i](x) is.

in the simplest case , it converges to L.

so the fixpoints probably cause no problems for my formula.
....

Using x= the fixed points exp(e), L, L* is an interesting idea, but I doubt it really helps. The most immediate draw back is that we're looking for a real valued super function, with real values at the real axis, which is why I assumed x=0.
- Sheldon