Abel equation

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

$f(h(x))=h(x+1)$

or

$\alpha (f(x))=\alpha (x)+1$.

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

Equivalence

The second equation can be written

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

Taking Template:Math, the equation can be written

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

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

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

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

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

$\omega (\omega (x,u),v)=\omega (x,u+v),$

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

$\omega (x,u)={\alpha }^{-1}(\alpha (x)+u)$.     (Observe Template:Math.)

The Abel function Template: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 Template:Math.

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

$\alpha (f(f(x)))=\alpha (x)+2,$

and so on,

$\alpha (f_n(x))=\alpha (x)+n.$

Solutions

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

We have the following existence and uniqueness theoremTemplate:Pg

Let $h:ℝ\to ℝ$ be analytic, meaning it has a Taylor expansion. To find: real analytic solutions $\alpha :ℝ\to ℂ$ of the Abel equation $\alpha \circ h=\alpha +1$.

Existence

A real analytic solution $\alpha$ exists if and only if both of the following conditions hold:

• $h$ has no fixed points, meaning there is no $y\in ℝ$ such that $h(y)=y$.
• The set of critical points of $h$, where $h^{\prime }(y)=0$, is bounded above if $h(y)>y$ for all $y$, or bounded below if $h(y)<y$ for all $y$.

Uniqueness

The solution is essentially unique in the sense that there exists a canonical solution ${\alpha }_0$ with the following properties:

• The set of critical points of ${\alpha }_0$ is bounded above if $h(y)>y$ for all $y$, or bounded below if $h(y)<y$ for all $y$.
• This canonical solution generates all other solutions. Specifically, the set of all real analytic solutions is given by

$$\{{\alpha }_0+\beta \circ {\alpha }_0|\beta :ℝ\to ℝ\text{ is analytic, with period 1}\}.$$

Approximate solution

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

References

• M. Kuczma, Functional Equations in a Single Variable, Polish Scientific Publishers, Warsaw (1968).
• M. Kuczma, Iterative Functional Equations. Vol. 1017. Cambridge University Press, 1990.