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Nowhere analytic superexponential convergence
(01/20/2011, 05:23 PM)sheldonison Wrote: 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.
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 or , I guess all these are nowhere analytic on the real line (and produce superexponentials).

(01/26/2011, 11:17 PM)tommy1729 Wrote: 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.


Messages In This Thread
RE: Nowhere analytic superexponential convergence - by bo198214 - 01/28/2011, 11:29 PM

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