# 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.
We have the following existence and uniqueness theorem$^{:TheoremB}$
Let $h:\mathbb {R} \to \mathbb {R}$ be analytic, meaning it has a Taylor expansion. To find: real analytic solutions $\alpha :\mathbb {R} \to \mathbb {C}$ of the Abel equation α∘h=α+1{\textstyle \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 $y\in \mathbb {R}$ 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 $h(y)<y$ 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 $h(y)<y$ for all $y$.
* This canonical solution generates all other solutions. Specifically, the set of all real analytic solutions is given by
$\{\alpha _{0}+\beta \circ \alpha _{0}|\beta :\mathbb {R} \to \mathbb {R} {\text{ is analytic, with period 1}}\}.$

### 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.