05/02/2009, 06:14 PM
(This post was last modified: 05/03/2009, 02:34 AM by Base-Acid Tetration.)

(05/02/2009, 08:32 AM)andydude Wrote: So do you mean the following?

Yes. EDIT 5/2/09: actually f^^3 (x) == f^[f^[f]](x), not [[f]^f]^f(x). Along the same lines as "functional root" (a function which, iterated n times, gives f(x)), the "functional logarithm" can be defined so that for all n. So we should be able to define <EDIT: of course I shouldn't call this logarithm, but "iteration number", perhaps?> There's not much else that I think we can do, however, because functions don't behave quite like numbers. A function might have no fixed points, and superiterations of a function may not even be continuous. I just wanna know what areas of mathematics do they use when doing real-valued iteraitons?