Abel function

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].

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)