Abel equation

This article is about certain functional equations. For ordinary differential equations that are cubic in the unknown function, see Abel equation of the first kind.

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⁻¹≡h, possibly satisfying additional requirements, such as α⁻¹(0) = 1.

The change of variables s^α(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(s^α(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).

See also: Iterated function § Abelian property and Iteration sequences

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)
• analytic solutions (Fatou coordinates) = approximation by asymptotic expansion of a function defined by power series in the sectors around parabolic fixed point

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

See also

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