# Abel equation

(Redirected from Abel function)

The Abel equation, named after Niels Henrik Abel, is special case of functional equations which can be written in the form

$f(h(x)) = h(x + 1)\,\!$

or

$\alpha(f(x))=\alpha(x)+1\!$

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−1, 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.

## History

Initially, the equation in the more general form [1] [2] was reported. Then it happens that even in the case of single variable, the equation is not trivial, and requires special analysis [3][4]

In the case of linear transfer function, the solution can be expressed in compact form [5]

## 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,

$\alpha(f_n(x))=\alpha(x)+n ~.$

Fatou coordinates represent solutions of Abel's equation, describing local dynamics of discrete dynamical system near a parabolic fixed point.[6]