The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form
f(h(x))=h(x+1)
\alpha(f(x))=\alpha(x)+1
The second equation can be written
\alpha-1(\alpha(f(x)))=\alpha-1(\alpha(x)+1).
Taking, the equation can be written
f(\alpha-1(y))=\alpha-1(y+1).
For a known function, a problem is to solve the functional equation for the function, possibly satisfying additional requirements, such as .
The change of variables, for a real parameter, brings Abel's equation into the celebrated Schröder's equation, .
The further change into Böttcher's equation, .
The Abel equation is a special case of (and easily generalizes to) the translation equation,[1]
\omega(\omega(x,u),v)=\omega(x,u+v)~,
\omega(x,1)=f(x)
\omega(x,u)=\alpha-1(\alpha(x)+u)
The Abel function 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]
The equation of tetration is a special case of Abel's equation, with .
In the case of an integer argument, the equation encodes a recurrent procedure, e.g.,
\alpha(f(f(x)))=\alpha(x)+2~,
\alpha(fn(x))=\alpha(x)+n~.
The Abel equation has at least one solution on
E
x\inE
n\inN
fn(x) ≠ x
fn=f\circf\circ...\circf
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.[9] The analytic solution is unique up to a constant.[10]