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 Functional super-iteration and hierarchy of functional hyper-iterations Base-Acid Tetration Fellow Posts: 94 Threads: 15 Joined: Apr 2009 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? ${}^{\circ 2}f(x) = f^{f(x)}(x)$ ${}^{\circ 3}f(x) = f^{f^{f(x)}(x)}(x)$ ${}^{\circ 4}f(x) = f^{f^{f^{f(x)}(x)}(x)}(x)$ 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" $\sqrt[n]{f}(x),$ (a function which, iterated n times, gives f(x)), the "functional logarithm" can be defined so that $\operatorname{flog}_f (f^n(x)) = n$ for all n. So we should be able to define ${}^{\circ0} f(x) = \operatorname{flog}_f [{}^{\circ 1} (f^1)] (x) = 1.$ 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? « Next Oldest | Next Newest »

 Messages In This Thread Functional super-iteration and hierarchy of functional hyper-iterations - by Base-Acid Tetration - 05/02/2009, 02:23 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/02/2009, 07:48 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by andydude - 05/02/2009, 08:32 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by andydude - 05/02/2009, 09:23 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by tommy1729 - 05/02/2009, 12:22 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/02/2009, 01:14 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by andydude - 05/02/2009, 06:38 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/02/2009, 06:14 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/02/2009, 07:27 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/03/2009, 01:08 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/03/2009, 03:25 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/03/2009, 07:21 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/03/2009, 08:13 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/03/2009, 09:45 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/03/2009, 10:27 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/04/2009, 01:06 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/04/2009, 07:56 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/04/2009, 08:04 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/04/2009, 08:50 PM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by Base-Acid Tetration - 05/04/2009, 08:57 PM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by bo198214 - 05/04/2009, 09:01 PM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by Base-Acid Tetration - 05/04/2009, 09:06 PM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by bo198214 - 05/04/2009, 09:12 PM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by Base-Acid Tetration - 05/12/2009, 02:29 AM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by bo198214 - 05/12/2009, 07:11 AM

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