The Abel equation, named after

Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...

, is a type of

functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...

of the form :

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

or :

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

. The forms are equivalent when is

invertible In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...

. or control the

iteration Iteration is the repetition of a process in order to generate a (possibly unbounded) sequence of outcomes. Each repetition of the process is a single iteration, and the outcome of each iteration is then the starting point of the next iteration. ...

of .

Equivalence

The second equation can be written :

\alpha^(\alpha(f(x))) = \alpha^(\alpha(x)+1)\, .

Taking , the equation can be written ::

f(\alpha^(y)) = \alpha^(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 Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

parameter , brings Abel's equation into the celebrated

Schröder's equation Schröder's equation, named after Ernst Schröder, is a functional equation with one independent variable: given the function , find the function such that Schröder's equation is an eigenvalue equation for the composition operator that send ...

, . The further change into

Böttcher's equation Böttcher's equation, named after Lucjan Böttcher, is the functional equation ::F(h(z)) = (F(z))^n where * is a given analytic function with a superattracting fixed point of order at , (that is, h(z)=a+c(z-a)^n+O((z-a)^) ~, in a neighbour ...

, . 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^(\alpha(x)+u)

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

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

s).

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 In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common. Under the definition as rep ...

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 ~,

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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

for all

x \in E

and all

n \in \mathbb

f^(x) \neq x

, where

f^ = f \circ f \circ ... \circ f

, is the function iterated {{mvar, n times. Analytic solutions (Fatou coordinates) can be approximated by

asymptotic expansion In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to ...

of a function defined by

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...

in the sectors around a parabolic fixed point The analytic solution is unique up to a constant.Classifications of parabolic germs and fractal properties of orbits by Maja Resman, University of Zagreb, Croatia
/ref>

References

Niels Henrik Abel Functional equations

Equivalence

History

Special cases

Solutions

See also

References