The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form
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or
- .
The forms are equivalent when α is invertible. h or α control the iteration of f.
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 sα(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(sα(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 ,
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]
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. [9] The analytic solution is unique up to a constant. [10]
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Böttcher's equation, named after Lucjan Böttcher, is the functional equation
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