Abel function

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For some given Transfer function \(T\), the Abel function \(G\) is inverse function of the corresponding superfunction \(F\), id est, \(G=F^{-1}\).

The Abel equation relates the Abel function \(G\) and the transfer function \(T\):

\(G(T(z))=G(z)+1\)

In certain range of values of \(z\), this equation is equivalent of the Transfer equation

\(T(F(z))=F(z\!+\!1)\)

The transfer function \(T\) is supposed to be known; then, the problem is to find the corresponding superfunction(s) and/or the Abel function(s).

The examples of the transfer functions, the superfunctions and the [[Abel functoons[[ \(G\) are suggested in the Table of superfunctions.

Etymology

The Abel function and the Abel Equation are named after Neils Henryk Abel [1].

Superfunction and iterates of the transfer function

The superfunction and the Abel function allow to define the \(n\)th iteration of the corresponding transfer function \(T\) in the following form:

\(T^n(z)=F(n+G(z))\)

This expression may hold for wide range of values of \(z\) and \(n\) from the set of complex numbers. In particular, for integer values of \(n\),

\(T^{-1}\) is inverse function of \(T\)
\(T^0(z)=z\),
\(T^1(z)=T(z)\)
\(T^2(z)=T(T(z))\)

and so on. The non-integer iteration of function allows to express such functions as square root of factorial [2] and square root of exponential [3] in terms of the superfunction and the Abel function.

Existence and unuqueness

In many cases, the superfunction \(F\) can be constructed with the regular iteration; then, for given superfunction, \(G\) is unique. However, the regular iteration can be realized at various fixed points of the transfer function \(T\) (if it has many fixed points). Then, hte superfunctions constructed with regular iteration, are different; in particular, they may have different periodicity. Sequently, the Abel functions are also different.

In order to define the unique Abel function \(G\), the additional requirements on its asymptotic behavior should be applied [4][5].

Keywords

ArcFactorial, ArcTania, ArcTetration, AuZex, AuTra

References

  1. http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0001 N.H.Abel. Untersuchung der Functionen zweier unabhängig veränderlicher Gröfsen x und y, wie f(x,y), welche die Eigenschaft haben, dafs f(z,f(x,y)) eine symmetrische Function von z, x und y ist. Journal für die reine und angewandte Mathematik, V.1 (1826) Z.1115
  2. http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.pdf D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.
  3. http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002175851 H.Kneser. Reelle analytische Losungen der Gleichung \(\varphi(\varphi(x))=e^x\) und verwandter Funktionalgleichungen Journal fur die reine und angewandte Mathematik 187 p.56-67 (1950)
  4. http://www.ams.org/journals/mcom/2010-79-271/S0025-5718-10-02342-2/home.html D.Kouznetsov, H.Trappmann. Portrait of the four regular super-exponentials to base sqrt(2). Mathematics of Computation, 2010, v.79, p.1727-175
  5. http://www.springerlink.com/content/u7327836m2850246/ H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, v.81, p.65-76 (2011)