Abel equation

From formulasearchengine
Jump to navigation Jump to search


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


and controls the iteration of Template:Mvar.


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

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

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.


Initially, the equation in the more general form [1] [2] was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis. [3][4]

In the case of a 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.,

and so on,

Fatou coordinates represent solutions of Abel's equation, describing local dynamics of discrete dynamical system near a parabolic fixed point.[6]

See also


  1. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  2. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  3. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  4. {{#invoke:Citation/CS1|citation |CitationClass=journal }} Studied is the Abel functional equation α(f(x))=α(x)+1
  5. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  6. Dudko, Artem (2012). Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets Ph.D. Thesis