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.}}

Licensing

   

This media, Penplot.jpg, is licenced under the Creative Commons Attribution 3.0 Unported License

You are free: To Share — To copy, distribute and transmit the work; To Remix — To adapt the work.
Under the following conditions: Attribution — You must attribute the work in the manner specified by the author or licensor (but not in any way that suggests that they endorse you or your use of the work).
For any reuse or distribution, you must make clear to others the licence terms of this work (the best way to do this is with a link to this licence's web page). Any of the above conditions can be waived if you get permission from the copyright holder. Nothing in this licence impairs or restricts the author's moral rights.
Read the full licence.

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].

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=-.62632418; b=0.4827; 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"); /* DO(n,60){x=-4+.04*n; DB e=exp(K*(x+2.24817451898)); y=L+e*(1.+e*(a)); if(n==0) M(x,y) else L(x,y); if(y>8.||y<-2.) break;} */ M(-4,L)L(0,L) fprintf(o,".01 W 1 0 0 RGB S\n"); /* DO(n,60){x=-4+.04*n; DB e=exp(K*(x+2.24817451898)); y=L+e*(1.+e*(a+e*b)); if(n==0) M(x,y) else L(x,y); if(y<-2.) break;} 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"); } [edit]

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(12,0){\ing{24}} %\put(12,0){\ing{penma}} \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(560,510){\sx{3.6}{\rot{84}$y\!=\!L+\exp(k(x\!+\!x_1))$\ero}} %\put(532,708){\sx{3.6}{\rot{86}$y\!=\!\mathrm{pen}(x)$\ero}} \put(446,370){\sx{3.9}{\rot{70}$y\!=\!\mathrm{pen}(x)$\ero}} %\put(366,236){\sx{2.3}{$y\!=\!10(\mathrm{pen}(x)\!-\!1\!-\!x)$}} %\put(416,239){\sx{3.4}{$y=10\,\delta(x)$}} \put(8,236){\sx{3.3}{$y=10\,\delta(x)$}} %\put(366, 8){\sx{3.3}{$L$}} \put(312, 9){\sx{3.2}{$y\!=\!L$}} \end{picture} \end{document}

References