File:Penplot.jpg

Summary

Title / Description	plot of the natural pension ${\displaystyle \mathrm {pen} =\mathrm {pen} _{\mathrm {e} }}$, id set, pentation to base ${\displaystyle \mathrm {e} =\exp(1)\approx 2.71}$, 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][6].

The picture is described in the book Суперфункции, In Russian ^[1].

C++ generator of curves

#include <math.h> #include <stdio.h> #include <stdlib.h> #define DB double #define DO(x,y) for(x=0;x<y;x++) #include <complex> typedef std::complex<double> z_type; #define Re(x) x.real() #define Im(x) x.imag() #define I z_type(0.,1.) #include "ado.cin" #include "fsexp.cin" #include "fslog.cin" z_type pen0(z_type z){ DB Lp=-1.8503545290271812; DB k,a,b; k=1.86573322821; a=-.6263241; b=0.4827; z_type e=exp(k*z); return Lp + e*(1.+e*(a+b*e)); } z_type pen7(z_type z){ DB x; int m,n; z=pen0(z+(2.24817451898-7.)); DO(n,7) { if(Re(z)>8.) return 999.; z=FSEXP(z); if(abs(z)<40) goto L1; return 999.; L1: ;} return z; } z_type pen(z_type z){ DB x; int m,n; x=Re(z); if(x<= -4.) return pen0(z); m=int(x+5.); z-=DB(m); z=pen0(z); DO(n,m) z=FSEXP(z); return z; } int main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d, cu,cd; FILE *o;o=fopen("penplo.eps","w"); ado(o,608,1008); fprintf(o,"404 204 translate\n 100 100 scale\n"); #define M(x,y) fprintf(o,"%8.4f %8.4f M\n",0.+x,0.+y); #define L(x,y) fprintf(o,"%8.4f %8.4f L\n",0.+x,0.+y); for(m=-4;m<3;m++) {M(m,-2)L(m,8)} for(n=-2;n<11;n++) {M( -4,n)L(2,n)} fprintf(o,"2 setlinecap 1 setlinejoin .004 W 0 0 0 RGB S\n"); DO(n,150){x=-4+.04*n;y=Re(pen7(x)); if(n==0) M(x,y)else L(x,y); if(y>8.)break;} fprintf(o,".02 W 0 0 0 RGB S\n"); DO(n,150){x=-2.2+.04*n;y=10.*(Re(pen7(x))-(1.+x)); if(n==0) M(x,y)else L(x,y); if(y>.3)break;} fprintf(o,".01 W 0 .5 0 RGB S\n"); DB L=-1.8503545290271812; DB K=1.86573322821; DB a=-.6263241; DB b=0.4827; DO(n,80){x=-4.+.04*n; DB e=exp(K*(x+2.24817451898)); y=L+e; if(n==0) M(x,y) else L(x,y); if(y>8.) break;} fprintf(o,".01 W 0 0 1 RGB S\n"); M(-4,L)L(0,L) fprintf(o,".01 W 1 0 0 RGB S\n"); DB t2=M_PI/1.86573322821; DB tx=-2.32; fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o); printf("pen7(-1)=%18.14f\n", Re(pen7(-1.))); printf("Pi/1.86573322821=%18.14f %18.14f\n", M_PI/1.86573322821, 2*M_PI/1.86573322821); system("epstopdf penplo.eps"); system( "open penplo.pdf"); }

Latex generator of labels

\documentclass[12pt]{article} \paperwidth 608px \paperheight 1008px \textwidth 1394px \textheight 1300px \topmargin -104px \oddsidemargin -90px \usepackage{graphics} \usepackage{rotating} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \newcommand \rmi {\mathrm{i}} \begin{document} {\begin{picture}(608,1008) \put(0,0){\ing{penplo}} \put(377,994){\sx{3.2}{$y$}} \put(377,895){\sx{3.2}{$7$}} \put(377,795){\sx{3.2}{$6$}} \put(377,695){\sx{3.2}{$5$}} \put(377,594){\sx{3.2}{$4$}} \put(377,494){\sx{3.2}{$3$}} \put(377,394){\sx{3.2}{$2$}} \put(377,294){\sx{3.2}{$1$}} \put(377,194){\sx{3.2}{$0$}} \put(358, 93){\sx{3.2}{$-1$}} \put(80,174){\sx{3.2}{$-3$}} \put(180,174){\sx{3.2}{$-2$}} \put(280,174){\sx{3.2}{$-1$}} \put(396,174){\sx{3.2}{$0$}} \put(496,174){\sx{3.2}{$1$}} \put(590,174){\sx{3.2}{$x$}} \put(242,406){\sx{3.6}{\rot{85}$y\!=\!L+\exp(k(x\!+\!x_1))$\ero}} \put(446,370){\sx{3.9}{\rot{70}$y\!=\!\mathrm{pen}(x)$\ero}} \put(8,236){\sx{3.3}{$y=10\,\delta(x)$}} \put(312, 9){\sx{3.2}{$y\!=\!L$}} \end{picture} \end{document}

References