Abel Equation - Equivalence

Equivalence

These equations are equivalent. Assuming that α is an invertible function, the second equation can be written as

Taking, 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, 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.

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