Difference between revisions of "Abel equation"

From formulasearchengine
Jump to navigation Jump to search
en>Trappist the monk
en>Cuzkatzimhut
 
Line 43: Line 43:
| coauthors=
| coauthors=
| title=Theorems on functional equations
| title=Theorems on functional equations
| journal=[[Bulletin des Sciences Mathématiques]]
| journal= Bull. Amer. Math. Soc.
| volume=27
| volume=19
| issue=2
| issue=2
| pages=31
| pages= 51-106
| year=1903
| year=1912
| doi=10.1090/S0002-9904-1912-02281-4
}}</ref>
}}</ref>
was reported. Then it happens that even in the case of single variable, the equation is not trivial, and requires special analysis
was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis.
<ref name="U1">{{cite journal
<ref name="U1">{{cite journal
| url=http://matwbn.icm.edu.pl/ksiazki/sm/sm134/sm13424.pdf
| url=http://matwbn.icm.edu.pl/ksiazki/sm/sm134/sm13424.pdf
Line 67: Line 68:
| doi=10.1016/j.nahs.2006.04.002       
| doi=10.1016/j.nahs.2006.04.002       
| author=Jitka Laitochová
| author=Jitka Laitochová
| title =Group iteration for Abel’s functional equation }} Studied is the Abel functional equation α(f(x))=α(x)+1</ref>
| title =Group iteration for Abel’s functional equation}} Studied is the Abel functional equation α(f(x))=α(x)+1</ref>
   
   
In the case of linear transfer function, the solution can be expressed in compact form
In the case of a linear transfer function, the solution can be expressed in compact form.
<ref name="linear">{{cite journal
<ref name="linear">{{cite journal
| url=http://matwbn.icm.edu.pl/ksiazki/sm/sm127/sm12716.pdf
| url=http://matwbn.icm.edu.pl/ksiazki/sm/sm127/sm12716.pdf
| author=G. Belitskii|author2=Yu. Lubish
| author=G. Belitskii|author2=Yu. Lubish
| title=The Abel equation and total solvability of linear functional equtions
| title=The Abel equation and total solvability of linear functional equations
| journal=[[Studia Mathematica]]
| journal=[[Studia Mathematica]]
| volume=127
| volume=127
Line 88: Line 89:
and so on,
and so on,
:<math>\alpha(f_n(x))=\alpha(x)+n ~.</math>
:<math>\alpha(f_n(x))=\alpha(x)+n ~.</math>
Fatou coordinates represent solutions of Abel's equation, describing local dynamics of discrete dynamical system near a [[Classification_of_Fatou_components|parabolic fixed point]].<ref>Dudko, Artem (2012). [http://www.math.toronto.edu/graduate/Dudko-thesis.pdf  ''Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets''] Ph.D. Thesis </ref>


==See also==
==See also==
*[[Functional equation]]
*[[Functional equation]]
*[[Iterated function]]
*[[Iterated function]]
*[[Abel function]]
*[[Schröder's equation]]
*[[Schröder's equation]]
*[[Böttcher's equation]]
*[[Böttcher's equation]]
*[[Infinite compositions of analytic functions]]
==References==
==References==
<references/>
<references/>

Latest revision as of 15:14, 2 September 2014

Template:Dablink

The Abel equation, named after Niels Henrik Abel, is special case of functional equations which can be written in the form

or

and controls the iteration of Template:Mvar.

Equivalence

These equations are equivalent. Assuming that Template:Mvar is an invertible function, the second equation can be written as

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

For a function f(x) assumed to be known, the task is to solve the functional equation for the function α−1, possibly satisfying additional requirements, such as α−1(0) = 1.

The change of variables sα(x) = Ψ(x), for a real parameter Template:Mvar, brings Abel's equation into the celebrated Schröder's equation, Ψ(f(x)) = s Ψ(x) .

The further change F(x) = exp(sα(x)) into Böttcher's equation, F(f(x)) = F(x)s.

History

Initially, the equation in the more general form [1] [2] was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis. [3][4]

In the case of a linear transfer function, the solution can be expressed in compact form. [5]

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.,

and so on,


Fatou coordinates represent solutions of Abel's equation, describing local dynamics of discrete dynamical system near a parabolic fixed point.[6]

See also

References

  1. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  2. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  3. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  4. {{#invoke:Citation/CS1|citation |CitationClass=journal }} Studied is the Abel functional equation α(f(x))=α(x)+1
  5. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  6. Dudko, Artem (2012). Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets Ph.D. Thesis