Difference between revisions of "ArcTetration"

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[[File:Ater01.png|500px|thumb|$\mathrm{ate}_b(x)$ versus $x$ for various $b$]]
[[File:B271a.png|500px|thumb|[[Complex map]] of ArcTetration: $u\!+ \!\mathrm i v \!=\! \mathrm{ate}(x\!+\!\mathrm i y)~$ for base $b\!=\!\mathrm e$]]
'''ArcTetration''' $\mathrm{ate}$ is inverse function of [[tetration]]; it is [[Abel function]] of the [[exponential]].
==Basic properties of ArcTetration==
For real values of the argument, $\mathrm{ate}_b(x)$ is plotted versus $x$ at various valies of $b\!>\!1$.
Being the inverse function of [[tetration]], the ArcTetration $\mathrm {ate}_b$ to base $b$ satisfies the relations
: $\mathrm{ate}_b(\mathrm{tet}_b(z))=z$
: $\mathrm{tet}_b(\mathrm{ate}_b(z))=z$
at least in some ranges of values of $z$.
The ArcTetration satisfies the [[Abel equation]]
: $\mathrm{ate}_b(b^z)=\mathrm{ate}_b(z) +1$
for the [[exponential]] as the [[transfer function]]. In this sense, ArcTetration is an [[Abel function]] of the exponential.
Roughly, the arctetration counts, how many times the logarithm should be taken of a value before the value becomes unity.
For base $b\!=\!\mathrm e \!\approx\! 2.71$, the natural ArcTetration is presented in figure at right with the [[complex map]]. For $f\!=\!\mathrm{ate}(x+\mathrm i y)$, the isolines
$u\!=\!\Re(f)$ and
$v\!=\!\Im(f)$ are drawn in the $x$,$y$ plane.
==Terminology and cuts==
In some publications, the term [[super-logarithm]] is used instead of '''arctetration'''.
Such a notation may cause confusion, because ArcTetration is not a [[superfunction]] of [[logarithm]].
For this reason, in [[TORI]] the name ArcTetration is used.
According to the definition of [[Abel function]], ArcTetration is Abel function of the exponential. However, the additional requirement that
$\mathrm{tet}_b(\mathrm{ate}_b(z))\!=\!z~$ at least in some vicinity of the [[real axis]] specifies that
and determines its asymptotic properties; in particular, its behavior in vicinity of the fixed points of the exponential, which are the [[branch point]]s of $\mathrm{ate}_b$. The cut line(s) for ArcTetrational are chosen to be "horisontal" (assuming that the complex plane at the screen is vertical), they go from a fixed point of exponential to the left, to the negative values of the real part of the argument, keeping the imaginary part. In particular, for $1\!<\!b\!<\!\exp(1/\mathrm e)$, the fixed points of $\exp_b$ are real, and the ArcTetration $\mathrm{ate}_b$ has the only one cut line, that goes to $-\infty$ along the real axis.
While the tetration is [[superfunction]] of the [[exponential]], the arctetration is the [[Abel function]].
==Uniqueness of ArcTetration==
The requirement on the behavior of ArcTetration as the argument approaches the [[fixed point]]s of the corresponding exponential provide the uniqueness of this function.
For $b\!>\!\exp(1/\mathrm e)$, the conditions of the uniqueness are determined by the requirement of bi-holomorphism in certain range that extends from one fixed point of the exponential to another fixed point
<ref name="uniabel">
H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. [[Aequationes Mathematicae]], v.81, p.65-76 (2011)
==Numerical evaluation==
For real base $b\!>\!1$, the ArcTetration can be evaluated with numerical inversion of [[tetration]].
For $b=\mathrm e$, which refers to the "natural tetration", the 14-digit complex(double) C++ implementation is available
<ref name="vladie">
D.Kouznetsov. Superexponential as special function. [[Vladikavkaz Mathematical Journal]], 2010, v.12, issue 2, p.31-45.
Together with [[tetration]] $\mathrm{tet}$, the ArcTetration determines the [[iteration of functions|fractional iteration]] of the exponential. The $c$th iteration of $\exp_b$ is defined as follows:
:$ {\exp_b}^c = \mathrm {tet}_b( c + \mathrm{ate}_b(z))$
At $~0\!<\!c\!<\!1~$, such a function may be useful for description of processes that grow faster than any polynomial but slower than any exponential. Such functions greatly extend the ability of holomorphic fitting of dependences with non-trivial (or just unknown) asymptotic behavior.
Also, the ArcTetration can be used for the numerical representation of huge numbers, that cannot be stored in the conventional floating-point form (mantissa, logarithm). Instead of to store a huge number $N$, one may store just $\mathrm{ate}(N)$. Due to the fast growth of [[tetration]], such a representation is not precise at all, but it greatly extends the range of real numbers that are still distinguishable from infinity. However, the arithmetical functions should be implemented in such a way, that no conversion of the number into the conventional (floating point) form is required at the intermediate steps.
Up to year 2011, no commercial application of ArcTetration have been reported.
[[tetration]], [[Abel function]], [[Abel equation]], [[Transfer equation]], [[Transfer function]]
[[Category:Holomorphic function]]
[[Category:Articles in English]]

Revision as of 06:58, 1 December 2018

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