03/12/2008, 09:20 PM

Just a short note, why this thread is called Bummer:

From a "proper" analytic iteration one would expect that it is analytic everywhere in the domain of the definition.

But we know that the regular iteration at a fixed point is the only analytic iteration that does not introduce a singularity at that fixed point (no oscillating first or higher order derivative when approachin the fixed point).

Conclusio:

There is no analytic iteration , , that is analytic at both fixed points for most , though for integer it is.

As every tetration (that satisfies , and analyticity) can be written as for some analytic iteration , this statement can be called a bummer as there is no "proper" analytic iteration.

From a "proper" analytic iteration one would expect that it is analytic everywhere in the domain of the definition.

But we know that the regular iteration at a fixed point is the only analytic iteration that does not introduce a singularity at that fixed point (no oscillating first or higher order derivative when approachin the fixed point).

Conclusio:

There is no analytic iteration , , that is analytic at both fixed points for most , though for integer it is.

As every tetration (that satisfies , and analyticity) can be written as for some analytic iteration , this statement can be called a bummer as there is no "proper" analytic iteration.