The Abel equation, named after Niels Henrik Abel

Niels Henrik Abel

Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...

, is special case of functional equation

Functional equation

In mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...

s which can be written in the form
or
and shows non-trivial properties at the iteration.

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) .

History

Initially, the equation in the more general form
was reported. Then it happens that even in the case of single variable, the equation is not trivial, and requires special analysis
In the case of linear transfer function, the solution can be expressed in compact form

Special cases

Equation of tetration

Tetration

In mathematics, tetration is an iterated exponential and is the next hyper operator after exponentiation. The word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration. Tetration is used for the notation of very large numbers...

is special case of Abel's equation, with . In the case of integer argument, the equation is just a recurrent procedure.