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

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

or

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

and shows non-trivial properties at the iteration.

Contents

[edit] 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 equation and perhaps, some additional requiremnts; for example, ~\alpha^{-1}(0)=1~.

[edit] History

Initially, the equation in the more general form [1] [2] was reported. Then it happens that even in the case of single variable, the equaiton 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]

[edit] Tetration

Equation of tetration is special case of equaiton of Abel, with ~f=\exp~.

[edit] Integer parameter

In the case of integer argument, the equation is just a recurrent procedure.

[edit] See also

[edit] References

  1. ^ 0.Abel (1827). "Correlative of the functional equation". Crelle's Journal 2: 389. 
  2. ^ A.R.Schweitzer (1903). "Theorems on functional equations". Bulletin des Sciences Mathématiques 27 (2): 31. 
  3. ^ G.Belitskii; Yu.Lubish (1999). "The real-analytic solutions of the Abel functional equations". Studia Mathematica 134 (2): 135–141. 
  4. ^ Jitka Laitochová (2007). "Group iteration for Abel’s functional equation". Nonlinear Analysis: Hybrid Systems 1, (1): 95–102. doi:10.1016/j.nahs.2006.04.002. 
  5. ^ G.Belitskii (1998). "THe Abel equation and total solvability of linear functional equaitons". Studia Mathematica 127: 81–89. 

[edit] also

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

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