Natural pentation

Natural pentation compared to other ackermanns

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\!=\!\mathrm{pen}(x)} and its approximations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u\!+\!\mathrm i v=\mathrm{pen}(x\!+\!\mathrm i y)} in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y} plane

Pentation appears as 5th ackermann.

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b>1} , Pentation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{pen}_b} specific superfunction of tetration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{tet}_b} , constructed with regular iteration at its lowest real fixed point. Pentation is denoted with symbol pen;

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{pen}_b(z)=A_{b, 5}(z)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{b,5}} is Fifth ackermann to base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} . .

If the base of tetration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b\!=\!\mathrm e\!=\!\exp(1)\!\approx\! 2.71} , then the tetration is called natural tetration and the corresponding pentation is called Natural pentation. Namely this case is considered on this article.

The first 5 ackermanns (addition of e, multiplication by e, exponent, tetration, pentation) are shown in the top figure at right.

Natural pentation is compared to its asymptotics at the second figure.

The complex map of the natural pentation is shown in the last figure.

General properties

Pentation satisfies the transfer equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{pen}_b(z\!+\!1) = \mathrm{tet}_b\Big( \mathrm{pen}_b(z)\Big)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{tet}_b} is tetration to base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} . The additional condition, common for all ackermanns is assumed,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{pen}_b(0)=1}

Pentation is holomorphic at least in the part of the complex plane, while the real part of the argument does not exceed some constant. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b\!=\!\mathrm e} , this constant is about Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2.5}

The range of holomorphism of pentation includes alto the real axis.

For the real base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b>1} , pentation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{pen}_b} is real–holomorphic,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{pen}_b(z^*)=\mathrm{pen}_b(z)^*}

Pentation has the exponential asymptotic at large negative values of the real part of the argument,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{pen}_b(z) = L + \varepsilon + O(\varepsilon^2)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} is fixed point of tetration, it is solution of equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=\mathrm {tet}_b(L)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon=\exp(k (z-X))} ,

parameters Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} and $X$ depend on the base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} . For base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b\!>\!1} , the increment Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is positive. This increment is determined by the derivative of tetration at its lowest real fixed point, and pentation is periodic function; the period Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P=2 \pi \mathrm i /k} . For real base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b\!>\!1} , the increment is positive.

Properties of the natural pentation

The base Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} of pentation is indicated as subscript, but it it is e, the subsctipt can be omitted, as in the case of tetration, arctetration, exponential and logarithm. The natural pentation is superfunction of natural tetration tet .

Natural pentation has the following properties:

The limiting value at minus infinity is fuxed point of the natural tetration, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L\!=\!L_{\mathrm e, 4,0} \!\approx\! -1.8503545290271812}

Increment at negative values of the real part is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k\approx 1.86573322821}

Period Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P=2\pi \mathrm i /k \approx 3.36767615657879\, \mathrm i}

The period is pure imaginary, so, the map shown in the figure reproduces at the translations along the imaginary axis. A little bit more than two periods are covered by the range of the complex map at right.

Natural pentation has the countable set of logarithmic singularities. The cut lines divide the right hand side of the complex plane to the set of "independent" strips of width Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} . In addition, there are many branch points at the right hand side of the map in vicinity of the real axis; and these branch points also reproduce at the translations for period Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} . There is no way to direct these cut lines to the left, because infinite amount of singularities are in the right hand side of the complex map.

Along the real axis, at minus infinity, the function pen grows exponentially from its asymptotic value Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L\approx -1.8503545290271812}

The graphic passes through points with coordinates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-3,-1)} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-1,0)} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,1)} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1,\mathrm e)} .

In vicinity of the range Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2.1 < z <1.1} , natural pentation can be approximated with linear function, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{pen}(z)\approx 1\!+\!z} ; this approximation gives of order of 2 correct decimal digits. Error of this approximation can be characterised with function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta(z)=\mathrm{pen}(z)-(1\!+\!z)} . In order to make it visible, it is scaled with factor 10, id est, curve Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\!=\!10\delta(x)} is plotted with thin curve in the intermediate figure.

At larger positive values of the argument, pentation shows very fast growth, much faster than that of superfactorial or that of natural tetration.

Visually, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{pen}(2)} looks as infinity, and, perhaps, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{pen}(3)} can be considered a "numerical approximation" of the fake Mizugadro number. Tetration has singularity and the branch point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2} , and this singularity determines the set of singularities in vicinity of the real axis.

Construction and evaluation of tetration, pentation and highest ackermanns may be good excersice for testing and debugging of the automatic procedure for calculation of superfuncions and non-integer iterates of holomorphic functions. Up to year 2014, such an automatic procedure is not yet implemented, and the superfunctions are built-up semi-manually; with human control of the overlapping of various representations and manual check of the residual at the substitution of various approximations into the transfer equation.

References

The first version of this article is adopted from TORI, http://mizugadro.mydns.jp/t/index.php/Pentation

Keywords

Ackermann, Pentation, Superfunction, Tetration, Transfer equation