Abel equation

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The Abel equation, named after Niels Henrik Abel, is special case of functional equations which can be written in the form


and controls the iteration of Template:Mvar.


These equations are equivalent. Assuming that Template:Mvar 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, possibly satisfying additional requirements, such as α−1(0) = 1.

The change of variables sα(x) = Ψ(x), for a real parameter Template:Mvar, 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.


Initially, the equation in the more general form [1] [2] was reported. Then it happens that even in the case of single variable, the equation is not trivial, and requires special analysis [3][4]

In the case of linear transfer function, the solution can be expressed in compact form [5]

Special cases

The equation of tetration is a special case of Abel's equation, with f = exp.

In the case of an integer argument, the equation encodes a recurrent procedure, e.g.,

and so on,

See also


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  4. {{#invoke:Citation/CS1|citation |CitationClass=journal }} Studied is the Abel functional equation α(f(x))=α(x)+1
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