Abel equation

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.