bo198214 Wrote:Quote:however , im not sure that f(f(x)) = exp(x) is not unique by analyticity alone ?
care to explain ?
Now if we have one analytic solution \( F \) then we have a lot of other analytic solutions given by
\( F_\theta(x):=F(x+\theta(x)) \) for any 1-periodic analytic function \( \theta \) (prove that \( F_\theta(z+1)=\exp(F_\theta(x)) \) too!)
If we make the amplitude of those \( \theta \) sufficiently small, then \( x+\theta(x) \) is strictly increasing and \( F_\theta \) is too.
Finally:
\( f_\theta(x):=F_\theta\left(\frac{1}{2}+F^{-1}_\theta(x)\right) \)
is another analytic solution of \( f(f(x))=F(x) \).
i had expected such a reply.
but i disagree.
let F( real ) map to reals.
and let f( real ) map to reals.
assuming those are satisfied ,
i feel that F(x+1) = exp(F(x))
should also satisfy F(x+1/2) = f(F(x))
where f(f(x)) = exp(x) , if it wants to be tetration.
in general F(x+a) = f_a(F(x))
where f_a satisfies f_a(((... a times ...(x)))) = exp(x)
should be satisfied.
furthermore the inverse of F might be multivalued !!
but that doesnt mean f(x) is all the possible results of F(1/2 + invF(x))
taking that into account , its probably clear that i will only accept examples of 2 distinct analytic solutions f(x) that map all reals to a subset of reals and satisfy f(f(x)) = exp(x).
im not trying to be difficult.
but this is important.
working with F(x) seems like overkill , instead i focus on f : f(f(x)) = exp(x).
( as an analogue : there are multiple functions that satisfy f(x+1) = e*f(x) but that doesnt mean there are multiple functions that are a solution to exp( log(x) + 1 ) ( being e*x ) )
regards
tommy1729