03/11/2017, 10:22 AM

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.

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.