fractional powers of function inversion (was: changing terminology)

[bo198214]

so you might want to consider fractional iterates of functional inversion operator inv[]?

slowly, slowly, I can not remember that we anywhere on the forum already considered extended iterates of an *operator*.

Perhaps you can make an infinite matrix that maps the powerseries of the input function to the powerseries of the output function. Then one could take matrix powers to define the fractional iterates, however I didnt see this anywhere done.
I dont even know whether inv is expressible as such a matrix.

but if we consider the half-iterate of inv, it will be a hopefully.

you mean i?

can we assume this f^a)^b = f^(ab) for most cases?

Yes I think so (for real a,b), it should be derivable from f^(a+b)=f^a o f^b, when taking into account cancelability, i.e. g^n=h^n should imply g=h for g and h being an iterate of f.

how are these complex iterate thingies numerically computed anyway?

for *functions* you just plug i into the formula
I.e. into a regular iteration formula.