Английская Википедия:Abel equation

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The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form

<math>f(h(x)) = h(x + 1)</math>

or

<math>\alpha(f(x)) = \alpha(x)+1</math>.

The forms are equivalent when Шаблон:Mvar is invertible. Шаблон:Mvar or Шаблон:Mvar control the iteration of Шаблон:Mvar.

Equivalence

The second equation can be written

<math>\alpha^{-1}(\alpha(f(x))) = \alpha^{-1}(\alpha(x)+1)\, .</math>

Taking Шаблон:Math, the equation can be written

<math>f(\alpha^{-1}(y)) = \alpha^{-1}(y+1)\, .</math>

For a known function Шаблон:Math , a problem is to solve the functional equation for the function Шаблон:Math, possibly satisfying additional requirements, such as Шаблон:Math.

The change of variables Шаблон:Math, for a real parameter Шаблон:Mvar, brings Abel's equation into the celebrated Schröder's equation, Шаблон:Math .

The further change Шаблон:Math into Böttcher's equation, Шаблон:Math.

The Abel equation is a special case of (and easily generalizes to) the translation equation,

<math>\omega( \omega(x,u),v)=\omega(x,u+v) ~,</math>

e.g., for <math>\omega(x,1) = f(x)</math>,

<math>\omega(x,u) = \alpha^{-1}(\alpha(x)+u)</math>.     (Observe Шаблон:Math.)

The Abel function Шаблон:Math further provides the canonical coordinate for Lie advective flows (one parameter Lie groups).

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 Шаблон:Math.

In the case of an integer argument, the equation encodes a recurrent procedure, e.g.,

<math>\alpha(f(f(x)))=\alpha(x)+2 ~,</math>

and so on,

<math>\alpha(f_n(x))=\alpha(x)+n ~.</math>

Solutions

The Abel equation has at least one solution on <math>E</math> if and only if for all <math>x \in E</math> and all <math>n \in \mathbb{N}</math>, <math>f^{n}(x) \neq x</math>, where <math> f^{n} = f \circ f \circ ... \circ f</math>, is the function Шаблон:Mvar iterated Шаблон:Mvar times.

Analytic solutions (Fatou coordinates) can be approximated by asymptotic expansion of a function defined by power series in the sectors around a parabolic fixed point. The analytic solution is unique up to a constant.

See also

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