Difference between revisions of "Abel function"

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(Existence and unuqueness)
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The examples of the [[transfer function]]s, the [[superfunction]]s and the [[Abel functoons[[ $G$ are suggested in the [[Table of superfunctions]].
 
The examples of the [[transfer function]]s, the [[superfunction]]s and the [[Abel functoons[[ $G$ are suggested in the [[Table of superfunctions]].
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==Etymology==
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The [[Abel function]] and the [[Abel Equation]] are named after [[Neils Henryk Abel]]
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<ref>
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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
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</ref>.
   
 
==Superfunction and iterates of the transfer function==
 
==Superfunction and iterates of the transfer function==
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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)
 
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)
 
</ref>.
 
</ref>.
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==Keywords==
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[[ArcFactorial]],
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[[ArcTania]],
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[[ArcTetration]],
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[[AuZex]],
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[[AuTra]]
   
 
==References==
 
==References==

Revision as of 06:57, 1 December 2018

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)