Functional power Xorter Fellow   Posts: 93 Threads: 30 Joined: Aug 2016 03/11/2017, 10:22 AM Let f and g be total functions (so e. g. C -> C) and N and M be complexes. Then (f o g)(x) and f o a = f(a) are so-called functional multiplications. But the interesting thing is the following: functional power: $f^{oN} = f o f o ... o f (N-times)$ When N is an integer, it is trivial, just look: $f^{o0} = x$ $f^{o1} = f$ $f^{o2} = f o f$ $f^{o3} = f o f o f$ ... $f^{o-1} = f^{-1}$ We have rules for it, like these ones: $(f^{oN}) o (f^{oM}) = f^{o N+ M}$ $(f^{oN})^{oM} = f^{o N M}$ $f o (f^{oN}) = (f^{oN}) o f = f^{o N+1}$ But for instance: $(f^{oN}) o (g)^{oN} != (f o g)^{oN}$ (Also functional tetration exists.) My theory is that if we can get an explicit formula for $f^{oN}$ with x and N, then N is extendable to any total function. For example: $(2x)^{oN} = 2^N x N := log_2 (x) (2x)^{o log_2 (x)} = x^2$ And in the same way, theoritacelly you could do the same with all the functions. But how? My concept is that by Carleman matrices. Xorter Unizo Catullus Fellow   Posts: 213 Threads: 47 Joined: Jun 2022 07/11/2022, 01:50 AM (This post was last modified: 07/11/2022, 10:43 PM by Catullus.) What about functional addition? How would that work? Or, how about functional zeration or functional negative rank hyper-operations? Please remember to stay hydrated. ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\ Daniel Long Time Fellow    Posts: 279 Threads: 94 Joined: Aug 2007 07/11/2022, 03:36 AM (This post was last modified: 07/11/2022, 03:37 AM by Daniel.) (07/11/2022, 01:50 AM)Catullus Wrote: $\mathbb{What \: about \: functional \: addition?}$  $\mathbb{How \: would \: that \: work?}$ $\mathbb{Or,\: How \: about \: functional \: zeration,\: or \: negative \: rank \: hyper-operations?}$ $\mathbb{Please\:remember\:to\:stay\:hydrated.}$ $\mathbb{Sincerely:Catullus}$ Catullus, are you trolling us? Do you have an issue with writing readable posts? Why didn't you try something readable in code and actually displaying tex, like the following? Code:$$\mathbb{What \: about \: functional \: addition?}$$ $$\mathbb{How \: would \: that \: work?}$$ $$\mathbb{Or,\: How \: about \: functional \: zeration,\: or \: negative \: rank \: hyper-operations?}$$ $$\mathbb{Please\:remember\:to\:stay\:hydrated.}$$ $$\mathbb{Sincerely:Catullus}$$ This is a great community if you choose to be a part of it. But that requires more than just posting on the Tetration Forum. It means being considerate of the other members and working to communicate as effectively as possible. In exchange, you will find the members here to be responsive to your questions and posts. Please feel free to respond here or by private message. Daniel Catullus Fellow   Posts: 213 Threads: 47 Joined: Jun 2022 07/11/2022, 06:03 AM (This post was last modified: 07/11/2022, 10:44 PM by Catullus.) Okay, I changed it. Please remember to stay hydrated. ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\ « Next Oldest | Next Newest »

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