Abel equation

The second equation can be written :\alpha^{-1}(\alpha(f(x))) = \alpha^{-1}(\alpha(x)+1)\, . Taking , the equation can be written ::f(\alpha^{-1}(y)) = \alpha^{-1}(y+1)\, . For a known function , a problem is to solve the functional equation for the function , possibly satisfying additional requirements, such as . The change of variables , for a real parameter , brings Abel's equation into the celebrated Schröder's equation, . The further change into Böttcher's equation, . The Abel equation is a special case of (and easily generalizes to) the translation equation, :\omega( \omega(x,u),v)=\omega(x,u+v) ~, e.g., for \omega(x,1) = f(x), :\omega(x,u) = \alpha^{-1}(\alpha(x)+u).     (Observe .) The Abel function further provides the canonical coordinate for Lie advective flows (one parameter Lie groups). ==History==

Initially, the equation in the more general form was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis. ==Special cases==

The equation of tetration is a special case of Abel's equation, with . In the case of an integer argument, the equation encodes a recurrent procedure, e.g., :\alpha(f(f(x)))=\alpha(x)+2 ~, and so on, :\alpha(f_n(x))=\alpha(x)+n ~. ==Solutions==

The Abel equation has at least one solution on E if and only if for all x \in E and all n \in \mathbb{N}^*, f^{n}(x) \neq x, where f^{n} = f \circ f \circ ... \circ f, is the function iterated times. We have the following existence and uniqueness theorem Let h: \R \to \R be analytic, meaning it has a Taylor expansion. To find: real analytic solutions \alpha: \R \to \C of the Abel equation \alpha \circ h = \alpha + 1. Existence A real analytic solution \alpha exists if and only if both of the following conditions hold: • h has no fixed points, meaning there is no y \in \R such that h(y) = y. • The set of critical points of h, where h'(y) = 0, is bounded above if h(y) > y for all y, or bounded below if h(y) for all y. Uniqueness The solution is essentially unique in the sense that there exists a canonical solution \alpha_0 with the following properties: • The set of critical points of \alpha_0 is bounded above if h(y) > y for all y, or bounded below if h(y) for all y. • This canonical solution generates all other solutions. Specifically, the set of all real analytic solutions is given by \{\alpha_0 + \beta\circ \alpha_0 | \beta : \R \to \R \text{ is analytic, with period 1}\}. Approximate solution Analytic solutions (Fatou coordinates) can be approximated by asymptotic expansion of a function defined by power series in the sectors around a parabolic fixed point. The analytic solution is unique up to a constant. ==See also==