# Abel equation

The **Abel equation**, named after Niels Henrik Abel, is a type of functional equation which can be written in the form

or, equivalently,

and controls the iteration of f.

## Equivalence

These equations are equivalent. Assuming that α 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}≡*h*, possibly satisfying additional requirements, such as *α*^{−1}(0) = 1.

The change of variables *s*^{α(x)} = Ψ(*x*), for a real parameter s, 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}.

The Abel equation is a special case of (and easily generalizes to) the **translation equation**,[1]

e.g., for ,

- . (Observe
*ω*(*x*,0) =*x*.)

The Abel function *α*(*x*) further provides the canonical coordinate for Lie advective flows (one parameter Lie groups).

## History

Initially, the equation in the more general form [2] [3] was reported. Even in the case of a single variable, the equation is non-trivial, and admits special analysis.[4] [5][6]

In the case of a linear transfer function, the solution is expressible compactly. [7]

## 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,

## Solutions

- formal solution : unique (to a constant)[8] (Not sure, because if is solution, then , where , is also solution[9].)
- analytic solutions (Fatou coordinates) = approximation by asymptotic expansion of a function defined by power series in the sectors around parabolic fixed point[10]

- Existence : Abel equation has at least one solution on if and only if , where , n times.[11]

Fatou coordinates describe local dynamics of discrete dynamical system near a parabolic fixed point.

## See also

## References

- Aczél, János, (1966):
*Lectures on Functional Equations and Their Applications*, Academic Press, reprinted by Dover Publications, ISBN 0486445232 . - Abel, N.H. (1826). "Untersuchung der Functionen zweier unabhängig veränderlichen Größen x und y, wie f(x, y), welche die Eigenschaft haben, ..."
*Journal für die reine und angewandte Mathematik*.**1**: 11–15. - A. R. Schweitzer (1912). "Theorems on functional equations".
*Bull. Amer. Math. Soc*.**19**(2): 51–106. doi:10.1090/S0002-9904-1912-02281-4. - Korkine, A (1882). "Sur un problème d'interpolation",
*Bull Sci Math & Astron***6**(1) 228—242. online - G. Belitskii; Yu. Lubish (1999). "The real-analytic solutions of the Abel functional equations" (PDF).
*Studia Mathematica*.**134**(2): 135–141. - Jitka Laitochová (2007). "Group iteration for Abel's functional equation".
*Nonlinear Analysis: Hybrid Systems*.**1**(1): 95–102. doi:10.1016/j.nahs.2006.04.002. - G. Belitskii; Yu. Lubish (1998). "The Abel equation and total solvability of linear functional equations" (PDF).
*Studia Mathematica*.**127**: 81–89. - Classifications of parabolic germs and fractal properties of orbits by Maja Resman, University of Zagreb, Croatia
- R. Tambs Lyche,ÉTUDES SUR L'ÉQUATION FONCTIONNELLE D'ABEL DANS LE CAS DES FONCTIONS RÉELLES., University of Trondlyim, Norvege
- Dudko, Artem (2012).
*Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets*Ph.D. Thesis - R. Tambs Lyche,Sur l'équation fonctionnelle d'Abel, University of Trondlyim, Norvege