**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 *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*.

Read more about this topic: Abel Equation