Abel equation

The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form
 * $$f(h(x)) = h(x + 1)$$

or
 * $$\alpha(f(x)) = \alpha(x)+1$$.

The forms are equivalent when $α$ is invertible. $h$ or $α$ control the iteration of $f$.

Equivalence
The second equation can be written
 * $$\alpha^{-1}(\alpha(f(x))) = \alpha^{-1}(\alpha(x)+1)\, .$$

Taking $x = α^{−1}(y)$, the equation can be written
 * $$f(\alpha^{-1}(y)) = \alpha^{-1}(y+1)\, .$$

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 $s^{α(x)} = Ψ(x)$, for a real parameter $s$, brings Abel's equation into the celebrated Schröder's equation, $Ψ(f(x)) = s Ψ(x)$.

The further change $F(x) = exp(s^{α(x)})$ into Böttcher's equation, $F(f(x)) = F(x)^{s}$.

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 $ω(x,0) = x$.)

The Abel function $α(x)$ 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. In the case of a linear transfer function, the solution is expressible compactly.

Special cases
The equation of tetration is a special case of Abel's equation, with $f = exp$.

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 $f$ iterated $n$ times.

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.