09/03/2010, 12:11 PM
another thing worth mentioning ,
wheither or not f(x) = f(g(x)) has singularities , poles or finite radius and how those behave.
it seems that if one solution f(x) is not entire , then neither is another.
and the other way around.
( this follows from singularities of the abel functions , and if im not mistaken thus only poles can form a counterexample , BUT abel functions never have poles ... if this implies that f(x) doesnt have poles is not yet clear to me ... )
we can express g(x) such that f(x) = f(g(x)) has (an) (only?) entire solution(s) :
- its basicly a fixpoint argument of course -
f(x) = f(exp(a(x)) + x) with a(x) an entire function.
so this equation has a special place.
tommy1729
wheither or not f(x) = f(g(x)) has singularities , poles or finite radius and how those behave.
it seems that if one solution f(x) is not entire , then neither is another.
and the other way around.
( this follows from singularities of the abel functions , and if im not mistaken thus only poles can form a counterexample , BUT abel functions never have poles ... if this implies that f(x) doesnt have poles is not yet clear to me ... )
we can express g(x) such that f(x) = f(g(x)) has (an) (only?) entire solution(s) :
- its basicly a fixpoint argument of course -
f(x) = f(exp(a(x)) + x) with a(x) an entire function.
so this equation has a special place.
tommy1729