# 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

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

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

$\alpha ^{-1}(\alpha (f(x)))=\alpha ^{-1}(\alpha (x)+1)\,.$ 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 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   was reported. Then it happens that even in the case of single variable, the equation is not trivial, and requires special analysis 

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

## 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,

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