# Abel equation

The Abel equation, named after Niels Henrik Abel, is a type of functional equation which can be written in the form

$f(h(x))=h(x+1)$ or, equivalently,

$\alpha (f(x))=\alpha (x)+1$ and controls the iteration of   f.

## Equivalence

These equations are equivalent. Assuming that α is an invertible function, the second equation can be written as

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

$f(\alpha ^{-1}(y))=\alpha ^{-1}(y+1)\,.$ For a function f(x) assumed to be known, the task is to solve the functional equation for the function α−1h, 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 further change F(x) = exp(sα(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

• formal solution : unique (to a constant) (Not sure, because if $u$ is solution, then $v(x)=u(x)+\Omega (u(x))$ , where $\Omega (x+1)=\Omega (x)$ , is also solution.)
• analytic solutions (Fatou coordinates) = approximation by asymptotic expansion of a function defined by power series in the sectors around parabolic fixed point
• Existence : Abel equation has at least one solution on $E$ if and only if $\forall x\in E,\forall n\in \mathbb {N} ,f^{(n)}(x)\neq x$ , where $f^{(n)}=f\circ f\circ ...\circ f$ , n times.

Fatou coordinates describe local dynamics of discrete dynamical system near a parabolic fixed point.