11/29/2015, 03:03 PM
(This post was last modified: 11/29/2015, 03:03 PM by sheldonison.)

Given half iterates, f(f(x))=exp(x), and g(g(x))=2sinh(x), which uniquely satisfy the Op's relationship, then the obvious conjecture from my post would be that only one of the two (f,g) can be analytic. If one of them is analytic, then the other is infinitely differentiable and nowhere analytic, so that the other does not converge to its Taylor series anywhere. This is kind of an interesting conjecture; that Tommy might be inclined to agree with based on earlier posts in this thread (see Tommy's post#61 for example). As far as "dubious reasons" go, I was merely answering Tommy's question, "both be analytic or both not.", as well as searching for cool mathematical truths.

- Sheldon