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Functional power - Printable Version

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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:

When N is an integer, it is trivial, just look:


We have rules for it, like these ones:

But for instance:

(Also functional tetration exists.)
My theory is that if we can get an explicit formula for with x and N, then N is extendable to any total function.
For example:

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:  

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?
[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]



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RE: Functional power - Catullus - 07/11/2022

Okay, I changed it.