Abel equation

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

${\displaystyle f(h(x))=h(x+1)}$

or

${\displaystyle \alpha (f(x))=\alpha (x)+1}$.

The forms are equivalent when α is invertible. h or α control the iteration of f.

Equivalence

The second equation can be written

${\displaystyle \alpha ^{-1}(\alpha (f(x)))=\alpha ^{-1}(\alpha (x)+1)\,.}$

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

${\displaystyle f(\alpha ^{-1}(y))=\alpha ^{-1}(y+1)\,.}$

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 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,^[1]

${\displaystyle \omega (\omega (x,u),v)=\omega (x,u+v)~,}$

e.g., for ${\displaystyle \omega (x,1)=f(x)}$,

${\displaystyle \omega (x,u)=\alpha ^{-1}(\alpha (x)+u)}$.     (Observe ω(x,0) = x.)

The Abel function α(x) further provides the canonical coordinate for Lie advective flows (one parameter Lie groups).

History

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.,

${\displaystyle \alpha (f(f(x)))=\alpha (x)+2~,}$

and so on,

${\displaystyle \alpha (f_{n}(x))=\alpha (x)+n~.}$

Solutions

The Abel equation has at least one solution on ${\displaystyle E}$ if and only if for all ${\displaystyle x\in E}$ and all ${\displaystyle n\in \mathbb {N} }$, ${\displaystyle f^{n}(x)\neq x}$, where ${\displaystyle f^{n}=f\circ f\circ ...\circ f}$, is the function f iterated n times.^[8]

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]

See also

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

References

This page was last edited on 21 December 2023, at 17:27