Difference between revisions of "Abel equation"
en>Trappist the monk m (→History: Fix CS1 deprecated coauthor parameter errors (test) using AWB) |
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| coauthors= | | coauthors= | ||
| title=Theorems on functional equations | | title=Theorems on functional equations | ||
− | | journal= | + | | journal= Bull. Amer. Math. Soc. |
− | | volume= | + | | volume=19 |
| issue=2 | | issue=2 | ||
− | | pages= | + | | pages= 51-106 |
− | | year= | + | | year=1912 |
+ | | doi=10.1090/S0002-9904-1912-02281-4 | ||
}}</ref> | }}</ref> | ||
− | was reported. | + | 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 | ||
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| 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 | + | | title=The Abel equation and total solvability of linear functional equations |
| journal=[[Studia Mathematica]] | | journal=[[Studia Mathematica]] | ||
| volume=127 | | volume=127 | ||
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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]] | ||
− | |||
*[[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
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
- Functional equation
- Iterated function
- Schröder's equation
- Böttcher's equation
- Infinite compositions of analytic functions
References
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }} Studied is the Abel functional equation α(f(x))=α(x)+1
- ↑ {{#invoke:Citation/CS1|citation |CitationClass=journal }}
- ↑ Dudko, Artem (2012). Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets Ph.D. Thesis