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]
• analytic solutions (Fatou coordinates) = approximation by asymptotic expansion of a function defined by power series in the sectors around parabolic fixed point^[9]

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

See also

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

References