File:Penplot.jpg: Difference between revisions

Summary

Title / Description	plot of the natural pension ${\displaystyle \mathrm {pen} =\mathrm {pen} _{\mathrm {e} }}$, id set, pentation to base <maht>\mathrm e=\exp(1)\approx 2.71</math>, id set, pentation to base ${\displaystyle \mathrm {e} =\exp(1)\approx 2.71}$; the thik black curve shows ${\displaystyle y=\mathrm {pen} (x)}$; the thik black curve shows ${\displaystyle y=\mathrm {pen} (x)}$. The thin curves show the two asymptotics of pentation and the error ${\displaystyle \delta }$ of the linear approximation ${\displaystyle z\mapsto 1+z}$
Citizendium author & Copyright holder	Copyright © Dmitrii Kouznetsov. See below for licence/re-use information.
Date created	2014
Country of first publication	Japan, Germany
Notes	Pentation is described at TORI, http://mizugadro.mydns.jp/t/index.php/Pentation and also (In Russian) in the book [1]
Other versions	The image is borrowed from TORI, http://mizugadro.mydns.jp/t/index.php/File:Penplot.jpg
Using this image on CZ	Please click here to add the credit line, then copy the code below to add this image to a Citizendium article, changing the size, alignment, and caption as necessary. {{Image|Penplot.jpg|right|350px|Add image caption here.}}

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Description

Pentation pen is superfunction of tetration to the same base. Natural pentation is solution ${\displaystyle F}$ of the transfer equation

${\displaystyle F(z\!+\!1)=\mathrm {tet} {\big (}F(z){\big )}}$

constructed with regular iteration at the smallest real fixed point ${\displaystyle L}$ of tetration; ${\displaystyle L\approx -1.8503545290271812}$ is solution of equation

${\displaystyle L=\mathrm {tet} (L)}$

with additional condition ${\displaystyle F(0)=1}$.

The real-real plot ${\displaystyle y=\mathrm {pen} (x)}$ is shown with thick black curve.

The thin curves show approximations of pentation.

The red horizontal line shows the fixed point of tetration, ${\displaystyle y=L}$.

The thin blue curve shows the asymptotic of pentation at large negative values of the real part of the argument,

${\displaystyle y=L+\exp(k(x+x_{1}))}$

where ${\displaystyle k\approx 1.86573322821}$

and ${\displaystyle x_{1}\approx 2.24817451898}$

The thin green line shown the deviation from the linear approximation

${\displaystyle \mathrm {linear} (x)=1+x}$

The deviation is denoted as ${\displaystyle ~\delta (x)=\mathrm {pen} (x)-\mathrm {linear} (x)}$

In the range ${\displaystyle -2.1\!<\!x\!<\!1.1}$, the deviation is small, the linear approximation provides 2 correct significant digits. In order to make the deviation visible, it is scaled with factor 10, so, ${\displaystyle y=10\delta (x)}$ is plotted.

Properties of tetration are described in publications ^[2]

The regular iteration in construction of superfunction is described at TORI, http://mizugadro.mydns.jp/t/index.php/Regular_iteration and also in ^[3][4][5].

References