Abel equation

Abel equation is special case of functional equations which can be written in the form

$f(h(x))=f(x+1)\,\!$

or

$\alpha (f(x))=\alpha (x)+1$

and shows non-trivial properties at the iteration

Equivalence

In some sense, these equaitons are equivalent. Assuming that $~\alpha ~$ is invertible function, the second equation can be written as

$\alpha ^{-1}(\alpha (f(x)))=\alpha ^{-1}(\alpha (x)+1)$

Assuming that $~x=\alpha ^{-1}(t)$, the equation can be written as

$f(\alpha ^{-1}(t))=\alpha ^{-1}(t+1)$

Function $~f~$ can be assumed to be known, then it may be suggested to find function $~\alpha ^{-1}~$ that satisfies the equaiotn and perhaps, some additional requireemnts; for example, $~\alpha ^{-1}(0)=1~$.

History

Initially, the equation in the more general form was reported. Then it happens that even in the case of single variable, the equaiton is not trivial, and requires special analysis

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

tetration

Equation of tetration is special case of equaiton of Abel, with $~f~$=\exp~</math>.

Integer parameter

In the case of integer argument, the equaiton is hyst a recurrent procerure.

See also

• functional equation

also

Crelle's Journal, volume 1 (1826), pages 11-15