The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form

or

.

The forms are equivalent when α is invertible. h or α control the iteration of f.

The second equation can be written

Taking x = α⁻¹(y), the equation can be written

For a known function f(x) , a problem 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).

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]

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,

The Abel equation has at least one solution on if and only if for all and all , , where , is the function f iterated n times.^[8]

We have the following existence and uniqueness theorem^{[9]: Theorem B}

Let be analytic, meaning it has a Taylor expansion. To find: real analytic solutions of the Abel equation .

A real analytic solution exists if and only if both of the following conditions hold:

• has no fixed points, meaning there is no such that .
• The set of critical points of , where , is bounded above if for all , or bounded below if for all .

The solution is essentially unique in the sense that there exists a canonical solution with the following properties:

• The set of critical points of is bounded above if for all , or bounded below if for all .
• This canonical solution generates all other solutions. Specifically, the set of all real analytic solutions is given by

Analytic solutions (Fatou coordinates) can be approximated by asymptotic expansion of a function defined by power series in the sectors around a parabolic fixed point.^[10] The analytic solution is unique up to a constant.^[11]

• Functional equation
  • Schröder's equation
  • Böttcher's equation
• Infinite compositions of analytic functions
• Iterated function
• Shift operator
• Superfunction

• M. Kuczma, Functional Equations in a Single Variable, Polish Scientific Publishers, Warsaw (1968).
• M. Kuczma, Iterative Functional Equations. Vol. 1017. Cambridge University Press, 1990.