The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form or The forms are equivalent when α is invertible.
h or α control the iteration of f. The second equation can be written Taking x = α−1(y), the equation can be written For a known function f(x) , a problem is to solve the functional equation for the function α−1 ≡ h, possibly satisfying additional requirements, such as α−1(0) = 1.
The change of variables sα(x) = Ψ(x), for a real parameter s, brings Abel's equation into the celebrated Schröder's equation, Ψ(f(x)) = s Ψ(x) .
, The Abel function α(x) further provides the canonical coordinate for Lie advective flows (one parameter Lie groups).
Initially, the equation in the more general form [2] [3] was reported.
Even in the case of a single variable, the equation is non-trivial, and admits special analysis.
[4][5][6] In the case of a linear transfer function, the solution is expressible compactly.
In the case of an integer argument, the equation encodes a recurrent procedure, e.g., and so on, The Abel equation has at least one solution on
be analytic, meaning it has a Taylor expansion.
To find: real analytic solutions
exists if and only if both of the following conditions hold: The solution is essentially unique in the sense that there exists a canonical 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.
[10] The analytic solution is unique up to a constant.