Abel function

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.

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,

e.g., for ${\displaystyle \omega (x,1)=f(x)}$,

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

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