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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.