Nowhere analytic superexponential convergence

[bo198214]

I found some surprisingly close similarities in the behavior of tommysexp and the base change function, and may have made some progress on why both are probably nowhere analytic functions.
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The similarities between the two are somewhat striking, and lead to some new ideas to explore about nowhere analytic superexponential convergence.

Ya, actually this method would also work for a lot of other functions than \( \exp(x)-1 \) or \( 2\sinh(x) \), I guess all these are nowhere analytic on the real line (and produce superexponentials).

i agree that the base change and my sinh method function are probably - without extensions - Coo but not complex analytic.

but the point is my sinh method is Coo and REAL - analytic.

Look : since log log ... exp exp ... (z) is only REAL - analytic and not complex analytic , but we can extend log log ... exp exp ... (z) simply to id(z) BECAUSE it is REAL -analytic , and then it BECOMES complex - analytic.

Tommy, it seems you are not familiar with the definitions. "Real-analytic" means analytic at a certain interval of the real axis and the function returning real values there. "Analytic" at a point means there is a powerseries development with a non-zero convergence radius. If this is the case then there is disk around this point in the complex plane where the function is analytic/holomorphic.

A term like "complex-analytic" does not exist. When one says "analytic" there must be including a statement of the domain, where it is analytic. A function that is analytic in the whole complex plane is called entire.

So "nowhere analytic" on the real axis means: all the powerseries developments at the real axis have 0 convergence radius, which indicates a really strange function.