# Difference between revisions of "Abel equation"

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<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 | ||

| author=G. Belitskii | | author=G. Belitskii|author2=Yu. Lubish | ||

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| title=The real-analytic solutions of the Abel functional equations | | title=The real-analytic solutions of the Abel functional equations | ||

| journal=[[Studia Mathematica]] | | journal=[[Studia Mathematica]] | ||

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<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 | | author=G. Belitskii|author2=Yu. Lubish | ||

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| title=The Abel equation and total solvability of linear functional equtions | | title=The Abel equation and total solvability of linear functional equtions | ||

| journal=[[Studia Mathematica]] | | journal=[[Studia Mathematica]] |

## Revision as of 19:18, 16 February 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 *s*^{α(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(*s*^{α(x)}) into Böttcher's equation, *F*(*f*(*x*)) = *F*(*x*)^{s}.

## History

Initially, the equation in the more general form
^{[1]}
^{[2]}
was reported. Then it happens that even in the case of single variable, the equation is not trivial, and requires special analysis
^{[3]}^{[4]}

In the case of 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,

## See also

## References

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- ↑ {{#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 }}