Equation for function that computes iterated values
This article is about certain functional equations. For ordinary differential equations which are cubic in the unknown function, see
Abel equation of the first kind.
The forms are equivalent when α is
invertible. h or α control the
iteration of f.
Equivalence
The second equation can be written
Taking x = α−1(y), the equation can be written
For a known function f(x) , a problem 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 sα(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(sα(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,[1]
e.g., for ,
. (Observe ω(x,0) = x.)
The Abel function α(x) further provides the canonical coordinate for
Lie advective flows (one parameter
Lie groups).
Initially, the equation in the more general form
[2][3]
was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis.[4][5][6]
In the case of a linear transfer function, the solution is expressible compactly.[7]
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,
Solutions
The Abel equation has at least one solution on if and only if for all and all , , where , is the function fiteratedn times.[8]