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| {{dablink|This article is about certain functional equations. For ordinary differential equations that are cubic in the unknown function, see [[Abel equation of the first kind]].}}
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| The '''Abel equation''', named after [[Niels Henrik Abel]], is special case of [[functional equation]]s which can be written in the form
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| :<math>f(h(x)) = h(x + 1)\,\!</math> | |
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| or
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| :<math>\alpha(f(x))=\alpha(x)+1\!</math>
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| and controls the iteration of {{mvar|f}}.
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| ==Equivalence==
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| These equations are equivalent. Assuming that {{mvar|α}} is an [[invertible function]], the second equation can be written as
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| :<math> \alpha^{-1}(\alpha(f(x)))=\alpha^{-1}(\alpha(x)+1)\, .</math>
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| Taking {{math|''x'' {{=}} ''α''<sup>−1</sup>(''y'')}}, the equation can be written as
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| ::<math>f(\alpha^{-1}(y))=\alpha^{-1}(y+1)\, .</math>
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| For a function {{math|''f''(''x'')}} assumed to be known, the task is to solve the functional equation for the function {{math|''α''<sup>−1</sup>}}, possibly satisfying additional requirements, such as {{math|''α''<sup>−1</sup>(0) {{=}} 1}}.
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| The change of variables {{math|''s''<sup>''α''(''x'')</sup> {{=}} Ψ(''x'')}}, for a real parameter {{mvar|s}}, brings Abel's equation into the celebrated [[Schröder's equation]], {{math|Ψ(''f''(''x'')) {{=}} ''s'' Ψ(''x'')}} .
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| The further change {{math|''F''(''x'') {{=}} exp(''s''<sup>''α''(''x'')</sup>)}} into [[Böttcher's equation]], {{math|''F''(''f''(''x'')) {{=}} ''F''(''x'')<sup>''s''</sup>}}.
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| ==History==
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| Initially, the equation in the more general form
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| <ref name="abel">{{cite journal
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| | url=http://gdz.sub.uni-goettingen.de/ru/dms/load/img/?PPN=PPN243919689_0001&DMDID=dmdlog6
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| | author= Abel, N.H.
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| | coauthors=
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| | title= Untersuchung der Functionen zweier unabhängig veränderlichen Größen x und y, wie f(x, y), welche die Eigenschaft haben, ...
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| | journal=[[Journal für die reine und angewandte Mathematik]]
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| | volume=1
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| | pages=11–15
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| | year=1826
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| }}</ref>
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| <ref name="s">{{cite journal
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| | url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.bams/1183421988&view=body&content-type=pdf_1
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| | author=A. R. Schweitzer
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| | coauthors=
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| | title=Theorems on functional equations
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| | journal=[[Bulletin des Sciences Mathématiques]]
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| | volume=27
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| | issue=2
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| | pages=31
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| | year=1903
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| }}</ref>
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| was reported. Then it happens that even in the case of single variable, the equation is not trivial, and requires special analysis
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| <ref name="U1">{{cite journal | |
| | url=http://matwbn.icm.edu.pl/ksiazki/sm/sm134/sm13424.pdf
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| | author=G. Belitskii
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| | coauthors=Yu. Lubish
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| | title=The real-analytic solutions of the Abel functional equations
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| | journal=[[Studia Mathematica]]
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| | volume=134
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| | issue=2
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| | pages=135–141
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| | year=1999
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| }}</ref><ref name="j">{{cite journal
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| | journal= Nonlinear Analysis: Hybrid Systems
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| | volume=1
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| | issue=1
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| | year=2007
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| | pages=95–102
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| | doi=10.1016/j.nahs.2006.04.002
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| | author=Jitka Laitochová
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| | title =Group iteration for Abel’s functional equation }} Studied is the Abel functional equation α(f(x))=α(x)+1</ref>
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|
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| In the case of linear transfer function, the solution can be expressed in compact form
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| <ref name="linear">{{cite journal
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| | url=http://matwbn.icm.edu.pl/ksiazki/sm/sm127/sm12716.pdf
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| | author=G. Belitskii
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| | coauthor=Yu. Lubish
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| | title=The Abel equation and total solvability of linear functional equtions
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| | journal=[[Studia Mathematica]]
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| | volume=127
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| | year=1998
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| | pages=81–89
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| }}</ref>
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| ==Special cases==
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| The equation of [[tetration]] is a special case of Abel's equation, with {{math|''f'' {{=}} exp}}.
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| In the case of an integer argument, the equation encodes a recurrent procedure, e.g.,
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| :<math>\alpha(f(f(x)))=\alpha(x)+2 ~,</math>
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| and so on,
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| :<math>\alpha(f_n(x))=\alpha(x)+n ~.</math>
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| ==See also==
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| *[[Functional equation]]
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| *[[Iterated function]]
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| *[[Abel function]]
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| *[[Schröder's equation]]
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| *[[Böttcher's equation]]
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| ==References==
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| <references/>
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| [[Category:Niels Henrik Abel]]
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| [[Category:Functional equations]]
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