# Abel equation

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. 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

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