Functional power

[Xorter]

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.

[Catullus]

What about functional addition? How would that work?
Or, how about functional zeration or functional negative rank hyper-operations?

[Daniel]

\( \mathbb{What \: about \: functional \: addition?} \) 

\( \mathbb{How \: would \: that \: work?} \)

\( \mathbb{Or,\: How \: about \: functional \: zeration,\: or \: negative \: rank \: hyper-operations?} \)

\( \mathbb{Please\:remember\:to\:stay\:hydrated.} \)

\( \mathbb{Sincerely:Catullus} \)

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]

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[Catullus]

Okay, I changed it.