File:Tet5loplot.jpg

Fig.19.3 from page 259 of of book «Superfunctions»^[1], 2020.

This picture is used also as Рис.19.2 at page 266 of the Russian version «Суперфункции»^[2], 2014.

The figure shows the graphical search for the real fixed points of tetration.

The curves show \(y=\mathrm{tet}_b(x)\) for various \(b\); line \(y\!=\!x\) is also drawn.

Notations:

\( \mathrm e\!=\!\exp(1)\!\approx\!2.71\) is base of the natural logarithm,

\(\eta=\exp(1/\mathrm e) \approx 1.41 \)

\(L_{\mathrm e,0}\approx -1.8503545290271812\) is the only real fixed point of the natural tetration, \(b\!=\!\mathrm e\)

\(\tau\!\approx\! 1.63532\) is crytical base; at base \(b\!=\!\tau\), the tetration has 2 real fixed points: Regular one at \(L_{\tau,0}\!\approx\! -1.7\) and exotic one \(L_{\tau,1}\!\approx\!3.087\)

At smaller base \(b\), function \(\mathrm{tet}_b\) has 3 regular real fixed points; in this sense, variety of supertetrations is richer, than that of superexponentials.

An \(b\) decreases, approaching the Henryk constant \(\eta=\exp(1/\mathrm e)\), the biggest fixed point runs to infinity; and at \(1\!<\!b\!\le\! \eta\), function \(\mathrm{tet}_b\) has two real regular fixed points.

C++ generator of curves

/* Files ado.cin and fit1.cin should be loaded in order to compile the code below:

#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
//using namespace std;
#include <complex>
typedef std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
//b=10
//#include "f4ten.cin"
#include "fit1.cin"
#include "ado.cin"
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
#define o(x,y) fprintf(o,"%6.4f %6.4f o\n",0.+x,0.+y);
int main(){ int j,k,m,n; DB p,q,t1,t3,u,v,w,x,y; z_type z,c,d;
FILE *o;o=fopen("tet5loplo.eps","w");ado(o,708,708);
fprintf(o,"204 204 translate\n 100 100 scale\n");
fprintf(o,"2 setlinecap\n");
for(m=-2;m<6;m++){if(m!=0){M(m,-2)L(m,5)}} 
for(n=-2;n<6;n++){if(n!=0){M(-2,n)L(5,n)}}  fprintf(o,".006 W 0 0 0 RGB S\n");
M(-2,0)L(5.1,0) M(0, -2)L(0,5.1)             fprintf(o,".01  W 0 0 0 RGB S\n");
M(0,M_E)L(1.,M_E)                             fprintf(o,".006  W 0 0 0 RGB S\n"); 
fprintf(o,"1 setlinejoin 1 setlinecap\n");
DO(m,300){x=-1.74+.02*m; y=Re(FIT1(log(1.7),x)); if(y>5.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n");
DO(m,300){x=-1.72+.03*m; y=Re(FIT1(log(1.63532),x)); if(y>5.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n");
DO(m,300){x=-1.72+.03*m; y=Re(FIT1(log(1.6),x)); if(y>5.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n");
DO(m,300){x=-1.68+.04*m; y=Re(FIT1(log(1.5),x)); if(x>5.1 || y>5.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n");
DO(m,300){x=-1.65+.04*m; y=Re(FIT1(1./M_E,x)); if(x>5.1) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".03 W 0 0 .7 RGB S\n");
DO(m,300){x=-1.64+.04*m; y=Re(FIT1(log(sqrt(2.)),x)); if(x>5.1) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".03 W .8 0 0 RGB S\n");
DO(m,340){x=-1.873+.01*m; y=Re(FIT1(1.,x)); if(y>5.) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 0 0 RGB S\n");
M(-2,-2)L(5,5)  fprintf(o,".01 W 0 0 0 RGB S\n");
x=3.087; M(x,0) L(x,x) L(0,x) fprintf(o,".001 W 0 0 0 RGB S\n");
DB Lt=-1.8503545290271812;
M(Lt,0) L(Lt,Lt) L(0,Lt); fprintf(o,".001 W 0 0 0 RGB S\n");
fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
       system("epstopdf tet5loplo.eps");
       system(    "open tet5loplo.pdf"); //mac
getchar(); system("killall Preview");// mac
}

Latex generator of labels

\documentclass[12pt]{article} 
\usepackage{geometry} % See geometry.pdf 
\geometry{letterpaper} % ... or a4paper or a5paper or ... ?? 
\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{hyperref}
\usepackage{rotating}
\usepackage[utf8x]{inputenc}
\usepackage[english,russian]{babel} %some packages are not used
\usepackage{color}
\definecolor{red}{rgb}{1,0.1,0.1}
\definecolor{black}{rgb}{0,0,0}
\definecolor{white}{rgb}{1,1,1}
\definecolor{yellow}{rgb}{1,.93,0}
\definecolor{bluedark}{rgb}{0,0,.87}
\paperwidth 712pt
\paperheight 716pt
\topmargin -98pt
\oddsidemargin -72pt
\textwidth 810pt
\textheight 870pt

\newcommand \sx {\scalebox}
\newcommand \ing {\includegraphics}
\newcommand \tet {\mathrm{tet}}
\newcommand \pen {\mathrm{pen}}
\newcommand \bC {\mathbb C}
\newcommand \fac {\mathrm {Factorial}}
\newcommand \rme {\mathrm e}
\newcommand \rmi {\mathrm i}
\newcommand \ds {\displaystyle}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}

\begin{document}
\parindent 0pt
{\normalsize
\begin{picture}(702,700)\put(0,0){\ing{tet5loplo}}
\put( 172,680){\sx{4}{$y$}}
\put( 172,590){\sx{4}{$4$}}
\put( 182,467){\sx{4}{e}}
\put( 172,390){\sx{4}{$2$}}
\put( 172,190){\sx{4}{$0$}}
\put( 194,164){\sx{4}{$0$}}
\put( 394,164){\sx{4}{$2$}}
\put( 594,164){\sx{4}{$4$}}
\put( 686,166){\sx{4}{$x$}}
\put(344,570){\sx{4}{\rot{80} $b\!=\! \rme$ \ero } }
\put(522,574){\sx{4}{\rot{66} $b\!=\!1.7$ \ero } }
\put(600,603){\sx{4}{\rot{64} $b\!=\!\tau$ \ero } }
\put(550,492){\sx{4}{\rot{42} $b\!=\!1.6$ \ero } }
%\put(574,422){\sx{4}{\rot{13} $b\!=\!1.5$ \ero } }
\put(580,438){\sx{4}{\rot{14} $b\!=\!1.5$ \ero } }
\put(592,400){\sx{4}{\rot{7} $b\!=\! \eta$ \ero}}
\put(574,345){\sx{4}{\rot{2}$b\!=\!\sqrt{2}$\ero}}
\put(130,504){\sx{4}{$L_{\tau,1}$}}
\put(500,170){\sx{4}{$L_{\tau,1}$}}
\put(-2,219){\sx{4}{$L_{\mathrm e,0}$}}
\put(202,6){\sx{4}{$L_{\mathrm e,0}$}}
\end{picture}}
\end{document}

References

Keywords

«[[]]», «Explicit plot», «Fixed point», «Natural tetration», «Tetration», «Pentation», «Superfinction», «Superfinctions»,