Tetration Forum
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:

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.