File:Ater01.png

Figure 17.2 from page 240 of book «Superfunctions»^[1], 2020.
The same plot appears also in Рис.17.2 at page 245 of the Russian version «Суперфункции»^[2], 2014.

The figure shows the explicit plot of ArcTetration for various base \(b\): \[ y=\mathrm{ate}_b(x) \]

C++ generator of curves

/* Files GLxw2048.inc, f2048ten.inc, f4ten.cin, 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 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"
void ado(FILE *O, int X, int Y)
{      fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%');
       fprintf(O,"%c%cBoundingBox: 0 0 %d %d\n",'%','%',X,Y);
       fprintf(O,"/M {moveto} bind def\n");
       fprintf(O,"/L {lineto} bind def\n");
       fprintf(O,"/S {stroke} bind def\n");
       fprintf(O,"/s {show newpath} bind def\n");
       fprintf(O,"/C {closepath} bind def\n");
       fprintf(O,"/F {fill} bind def\n");
       fprintf(O,"/o {.1 0 360 arc C S} bind def\n");
       fprintf(O,"/times-Roman findfont 20 scalefont setfont\n");
       fprintf(O,"/W {setlinewidth} bind def\n");
       fprintf(O,"/RGB {setrgbcolor} bind def\n");}
 //#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);
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("ate10bx.eps","w");ado(o,134,134);
fprintf(o,"22 22 translate\n 10 10 scale\n");
fprintf(o,"2 setlinecap\n");
for(m=-2;m<12;m++){if(m!=0){M(m,-2)L(m,11)}} 
for(n= -2;n<12;n++){if(n!=0){M(-2,n)L(11,n)}}  fprintf(o,".006 W 0 0 0 RGB S\n");
M(-2,0)L(11.1,0) M(0, -2)L(0,11.1)             fprintf(o,".03  W 0 0 0 RGB S\n");
//M(0,M_E)L(11.,M_E)                             fprintf(o,".006  W 0 0 0 RGB S\n"); 
M(M_E,0)L(M_E,11.)                             fprintf(o,".006  W 0 0 0 RGB S\n"); 
fprintf(o,"1 setlinejoin 1 setlinecap\n");
//DO(m,400){x=-1.99+.008*m;y=Re(FIT1(log(10.),x)); if(y>11.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".03 W .5 0 .5 RGB S\n");

DO(m,500){x=-1.80+.02*m; y=Re(FIT1(log(2.),x)); if(y>11.4) break; if(m==0)M(y,x)else L(y,x)} fprintf(o,".03 W .5 0 .5 RGB S\n");
DO(m,500){x=-1.78+.02*m; y=Re(FIT1(log(1.9),x)); if(y>11.4) break; if(m==0)M(y,x)else L(y,x)} fprintf(o,".02 W 0 .5 0 RGB S\n");
DO(m,500){x=-1.77+.02*m; y=Re(FIT1(log(1.8),x)); if(y>11.33) break; if(m==0)M(y,x)else L(y,x)} fprintf(o,".02 W 0 .5 0 RGB S\n");
DO(m,500){x=-1.74+.02*m; y=Re(FIT1(log(1.7),x)); if(y>11.33) break; if(m==0)M(y,x)else L(y,x)} fprintf(o,".02 W 0 .5 0 RGB S\n");
DO(m,500){x=-1.72+.03*m; y=Re(FIT1(log(1.6),x)); if(y>11.33) break; if(m==0)M(y,x)else L(y,x)} fprintf(o,".02 W 0 .5 0 RGB S\n");
DO(m,500){x=-1.68+.04*m; y=Re(FIT1(log(1.5),x)); if(y>11.33) break; if(m==0)M(y,x)else L(y,x)} fprintf(o,".02 W 0 .5 0 RGB S\n");
DO(m,500){x=-1.64+.04*m; y=Re(FIT1(log(1.4),x)); if(x>11.33) break; if(m==0)M(y,x)else L(y,x)} fprintf(o,".02 W 0 .5 0 RGB S\n");
DO(m,500){x=-1.58+.04*m; y=Re(FIT1(log(1.3),x)); if(x>11.33) break; if(m==0)M(y,x)else L(y,x)} fprintf(o,".02 W 0 .5 0 RGB S\n");
DO(m,500){x=-1.52+.04*m; y=Re(FIT1(log(1.2),x)); if(x>11.33) break; if(m==0)M(y,x)else L(y,x)} fprintf(o,".02 W 0 .5 0 RGB S\n");
DO(m,500){x=-1.42+.04*m; y=Re(FIT1(log(1.1),x)); if(x>11.33) break; if(m==0)M(y,x)else L(y,x)} fprintf(o,".02 W 0 .5 0 RGB S\n");
//
DO(m,500){x=-1.65+.04*m; y=Re(FIT1(1./M_E,x)); if(x>11.2) break; if(m==0)M(y,x)else L(y,x)} fprintf(o,".03 W 0 0 .7 RGB S\n");
DO(m,500){x=-1.64+.04*m; y=Re(FIT1(log(sqrt(2.)),x)); if(x>11.2) break; if(m==0)M(y,x)else L(y,x)} fprintf(o,".03 W .8 0 0 RGB S\n");
DO(m,500){x=-1.873+.01*m; y=Re(FIT1(1.,x)); if(y>11.2) break; if(m==0)M(y,x)else L(y,x)} fprintf(o,".02 W 0 0 0 RGB S\n");
//
DO(m,501){x=-1.987+.01*m;y=Re(F4TEN(x));if(m==0)M(y,x)else L(y,x) if(y>11.4) break; }fprintf(o,".02 W .5 0 .5 RGB S\n");
//DO(m,39){x=-1.89+.098*m; y=Re(FSEXP(x));if(m==0)M(x,y)else L(x,y)} fprintf(o,".01 W 0 0 0 RGB S\n");
//DO(m,48){x=-1.82+.0973*m; y=Re(F2(x));       if(m==0)M(x,y)else L(x,y)} fprintf(o,".01 W 0 0 0 RGB S\n");
//DO(m,126){x=-1.72+.1*m; y=Re(F15(x));        if(m==0)M(x,y)else L(x,y)} fprintf(o,".01 W 0 0 0 RGB S\n");
//DO(m,130){x=-1.65+.1*m; y=Re(E1ETF(x));      if(m==0)M(x,y)else L(x,y)} fprintf(o,".01 W 0 0 0 RGB S\n");
//DO(m,130){x=-1.64+.1*m; y=Re(f21E(x));       if(m==0)M(x,y)else L(x,y)} fprintf(o,".01 W 0 0 0 RGB S\n");
//M(-1.998,-2)L(-1.992,-.01)L(-1,0)L(-.01,.01)L(0,1)L(.012,10.1)  fprintf(o,".02 W .4 0 .8 RGB S\n");
//M(-1.01,-2)L(-1,0)L(-.99,.99)L(0,1)L(10.97,1.01)        fprintf(o,".02 W .2 .4 0 RGB S\n");

  M(-2,-1.998)L(-.01,-1.992)L(0,-1)L(.01,-.01)L(1,0)L(10.1,.012)  fprintf(o,".02 W .4 0 .8 RGB S\n");
  M(-2,-1.01)L(0,-1)L(.99,-.99)L(1,0)L(1.01,11.97)        fprintf(o,".02 W .2 .4 0 RGB S\n");
fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
       system("epstopdf ate10bx.eps");
       system(    "open ate10bx.pdf"); //mac
getchar(); system("killall Preview");// mac
}

Latex generator of labels

\documentclass[12pt,oneside]{book}
\usepackage{cmap}
\usepackage{geometry}
%1,192 × 1,096
\paperwidth  960pt
\paperheight 798pt
\usepackage[utf8]{inputenc}
\usepackage[T2A]{fontenc}
\usepackage[english]{babel}
\usepackage{latexsym,amsmath,amssymb,amsbsy,graphicx}
\usepackage{rotating}
\usepackage{hyperref} 
\usepackage{wrapfig}
\renewcommand\baselinestretch{1.2}
\textwidth 150mm
%\oddsidemargin 5mm
%\topmargin -18mm
\oddsidemargin -96px
\topmargin -80px
\textwidth 1130px
\textheight 1200px
\pagestyle{plain}
\large
\usepackage{color}
\definecolor{rose}{rgb}{1,.9,1}
\definecolor{bottom}{cmyk}{0,1,1,0}

\renewcommand\thesection{\arabic{section}}
\addtolength{\skip\footins}{8pt} % Does it do anything?
\usepackage[figurename=Fig.]{caption}
\multlinegap10pt
\newcommand \be {\begin{eqnarray}}
\newcommand \ee {\end{eqnarray} }
\newcommand \sx {\scalebox}
\newcommand \rme {{\rm e}}      %makes the base of natural logarithms Roman font 
\newcommand \rmi {{\rm i}}      %imaginary unity is always roman font
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing \includegraphics
\begin{document}

%\sx{3.23}{\begin{picture}(155,130)
\sx{6}{\begin{picture}(155,130)
\put(0,0){\ing{ate10bx}}
\put( 17,130){\sx{.6}{$y$}}
\put( 14,120){\sx{.6}{$10$}}
\put( 17,100){\sx{.6}{$8$}}
\put( 17,80){\sx{.6}{$6$}}
\put( 17,60){\sx{.6}{$4$}}
\put( 17,40){\sx{.6}{$2$}}
\put( 17,20){\sx{.6}{$0$}}
\put( 20.4,16){\sx{.6}{$0$}}
\put( 40.4,16){\sx{.6}{$2$}}
\put( 47.8,16.2){\sx{.6}{e}}
\put( 60.4,16){\sx{.6}{$4$}}
\put( 80.4,16){\sx{.6}{$6$}}
\put(100.4,16){\sx{.6}{$8$}}
\put(119,16){\sx{.6}{$10$}}
\put(128.6,16.2){\sx{.6}{$x$}}
\put(31.8,123){\sx{.35}{\rot{90}$b\!\rightarrow\!1$\ero}}
\put(36.4,120){\sx{.35}{\rot{90}$b\!=\!1.2$\ero}}
\put(40.5,120){\sx{.35}{\rot{90}$b\!=\!1.4$\ero}}
\put(44.8,120){\sx{.33}{\rot{90}$b\!=\!\sqrt{2}$\ero}}
\put(48.6,109){\sx{.38}{\rot{88}$b\!=\!\exp(1/\rme)$\ero}}
\put(106,124.5){\sx{.47}{\rot{10} $b\!=\!1.5$ \ero } }
\put(106,79){\sx{.45}{\rot{8} $b\!=\!1.6$ \ero } }
\put(106,64.2){\sx{.45}{\rot{7} $b\!=\!1.7$ \ero } }
\put(106,56.6){\sx{.45}{\rot{6} $b\!=\!1.8$ \ero } }
\put(106,50.4){\sx{.45}{\rot{5} $b\!=\!1.9$ \ero } }
\put(106,44.2){\sx{.5}{\rot{4} $b\!=\!2$ \ero } }
\put(94,39.6){\sx{.5}{\rot{5} $b\!=\! \rme$ \ero } }
\put(106,32){\sx{.5}{\rot{3} $b\!=\!10$ \ero } }
\put(96,22.4){\sx{.5}{\rot{0.} $b\!\rightarrow\!\infty$ \ero } }
\end{picture}}

\end{document}

References

Keywords

«Abel equation», «Abelfunction», «ArcTetration», «ate», «Superfunctions», «Tetration»,

«Суперфункции»,