File:Penplot.jpg

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 ${\displaystyle y=\mathrm {pen} (x)}$
Citizendium author & Copyright holder	Copyright © Dmitrii Kouznetsov. See below for licence/re-use information.
Date created	2014
Country of first publication	Japan, Germany
Notes	The thin curves show the two asymptotics of pentation and the error of the linear approximation
Other versions	http://mizugadro.mydns.jp/t/index.php/File:Penplot.jpg
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Description

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

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

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.