Functional power - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Functional power (/showthread.php?tid=1158) Functional power - Xorter - 03/11/2017 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. RE: Functional power - Catullus - 07/11/2022 What about functional addition? How would that work? Or, how about functional zeration or functional negative rank hyper-operations? RE: Functional power - Daniel - 07/11/2022 (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:[tex]\mathbb{What \: about \: functional \: addition?}[/tex] [tex]\mathbb{How \: would \: that \: work?}[/tex] [tex]\mathbb{Or,\: How \: about \: functional \: zeration,\: or \: negative \: rank \: hyper-operations?}[/tex] [tex]\mathbb{Please\:remember\:to\:stay\:hydrated.}[/tex] [tex]\mathbb{Sincerely:Catullus}[/tex] 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. RE: Functional power - Catullus - 07/11/2022 Okay, I changed it.