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
:
or
:
.
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
:
Taking , the equation can be written
::
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:
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Music Albums
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...
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,
:
e.g., for
,
:
. (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.,
:
and so on,
:
Solutions
The Abel equation has at least one solution on
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
and all
,
, where
, 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
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See also
*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 ...
**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 ...
**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 ...
*Infinite compositions of analytic functions
In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the ...
*Iterated function
In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times. The process of repeatedly applying the same function is ...
*Shift operator
In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function
to its translation . In time series analysis, the shift operator is called the lag operator.
Shift o ...
* Superfunction
References
Niels Henrik Abel
Functional equations