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 s^α(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(s^α(x)) into Böttcher's equation, F(f(x)) = F(x)^s.

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 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,

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

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

See also

• Functional equation
• Iterated function
• Schröder's equation
• Böttcher's equation
• Infinite compositions of analytic functions