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

[tommy1729]

For base exp(1/e), there have been many posts that the half-iterate of \( \text{sexp}_\eta(z) \) is not analytic at z=e.

wow euh ...

for starters i think you meant something else from what you actually said.

i do not believe you consider half-iterates of any superfunction.

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.

that is because e is the fixpoint of eta^z and it has f ' (e) = 1.

since f(z) is not analytic at e , its taylor series is only an approximation.

just like with e^z - 1.