# Difference between revisions of "Abel equation"

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

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

or

${\displaystyle \alpha (f(x))=\alpha (x)+1\!}$

and controls the iteration of Template:Mvar.

## Equivalence

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

${\displaystyle \alpha ^{-1}(\alpha (f(x)))=\alpha ^{-1}(\alpha (x)+1)\,.}$

Taking x = α−1(y), the equation can be written as

${\displaystyle f(\alpha ^{-1}(y))=\alpha ^{-1}(y+1)\,.}$

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 Template:Mvar, 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.,

${\displaystyle \alpha (f(f(x)))=\alpha (x)+2~,}$

and so on,

${\displaystyle \alpha (f_{n}(x))=\alpha (x)+n~.}$