finding f(x) for f(f(x))=g(x) where f(x) and g(x) are continuous
I've been doodling around and I was pondering the solutions to the recursive function and I've got three things I'm working on. If you guys want I can put down my scratch but I think the only things I've come up with so far are pretty obvious.
- Let then an implicit formula of can be found with the Lambert W function.
- If g(x) is a bijection, so is f(x)
- For , for , for and that for the above formula on the nth recursion change the square root to the nth root.
I'm working on a formula for and finding but my algebra is not letting it work the other way given .
For n=1 of course it's trivial and for n<0 and n is an integer there is another class of solutions. in fact the given solution above is not unique and has a solution for .
- All of must be surjections.
- All of of closed form are of the two forms stated above. (I've checked most other elementary functions I can think of and they won't fly)
If there anything wrong here that you can point out so i don't end upi spinning my wheels? Is there any study of recursive continuous functions I should be aware of?