paradox, accurate taylor series half iterate of eta not analytic at e

[sheldonison]

if i get it correctly :

the uppersuperfunction of eta^z is called cheta(z)

the lowersuperfunction of eta^z is called sexp(z)

you claim both cheta(z) and sexp(z) are not analytic at z = e.

No, not the superfunctions, of course is sexp analytic at e, its analytic on \( (-2,\infty) \) dont you remember?

We talk about the half iterate of \( \eta^x \), i.e. the function \( f \) such that
\( f(f(x))=\eta^x \). Though Sheldon's wording was indeed a bit misleading.
And even the explanation \( \eta^{0.5}(z) \) is not really clear, because \( \eta \) is not a function but a constant.
Better may be the expression \( \exp_\eta ^{0.5}(z) \) where \( \exp_\eta(z)=\eta^z \).

Sorry for the confusing wording. It seems like I made a few serious typos. I tried to fix the mistakes. As far as cheta as the upper superfunction for eta, that is a shorthand name that Jay invented. I try to occasionally throw in a detailed description. Then sexp for base eta follows the definition of sexp(0)=1.