File:Tetreal10bx10d.png

Plot of Tetration: $y=\mathrm{tet}_b(x)$ versus $x$ for
 * $b \rightarrow 1$,
 * $b = 1.1$,
 * $b = 1.2$,
 * $b = 1.3$,
 * $b = 1.4$,
 * $b = \sqrt{2}\approx 1.41,$
 * $b = \exp(1/ \mathrm e) \approx 1.44,$
 * $b = 1.5$,
 * $b = 1.6$,
 * $b = 1.7$,
 * $b = 1.8$,
 * $b = 1.9$,
 * $b = 2$,
 * $b = \mathrm e \approx 2.71$,
 * $b = 10$,
 * $b \rightarrow \infty$.

The fit is used, that provides several decimal digits, that seems to be sufficient to provide the "camera-ready copy"; the errors of the fitting are not seen even at the zooming-in.

This image is used as figure 17.2 of book Superfunctions .

Copyleft 2010 by Dmitrii Kouznetsov: Please attribute the source at the reuse: it helps to trace mistakes, if any

C++ generator of curves
// Files GLxw2048.inc, f2048ten.inc, f4ten.cin, fit1.cin should be loaded to the working directory in order to compile the C++ code below:

using namespace std; typedef complex z_type;
 * 1) include 
 * 2) include 
 * 3) include 
 * 4) define DB double
 * 5) define DO(x,y) for(x=0;x<y;x++)
 * 1) include
 * 1) define Re(x) x.real
 * 2) define Im(x) x.imag
 * 3) define I z_type(0.,1.)

//b=10
 * 1) include "f4ten.cin"


 * 1) 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"


 * 1) define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
 * 3) 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("tet10bx.eps","w");ado(o,134,124); 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,10)}} for(n= -2;n<11;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(0,10.1)            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"); fprintf(o,"1 setlinejoin 1 setlinecap\n");

//DO(m,400){x=-1.99+.008*m;y=Re(FIT1(log(10.),x)); if(y>10.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".03 W .5 0 .5 RGB S\n"); DO(m,400){x=-1.80+.02*m; y=Re(FIT1(log(2.),x)); if(y>10.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".03 W .5 0 .5 RGB S\n"); DO(m,400){x=-1.78+.02*m; y=Re(FIT1(log(1.9),x)); if(y>10.33) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n"); DO(m,400){x=-1.77+.02*m; y=Re(FIT1(log(1.8),x)); if(y>10.33) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n"); DO(m,400){x=-1.74+.02*m; y=Re(FIT1(log(1.7),x)); if(y>10.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n"); DO(m,400){x=-1.72+.03*m; y=Re(FIT1(log(1.6),x)); if(y>10.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n"); DO(m,400){x=-1.68+.04*m; y=Re(FIT1(log(1.5),x)); if(y>10.3) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n"); DO(m,400){x=-1.64+.04*m; y=Re(FIT1(log(1.4),x)); if(x>11.1) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n"); DO(m,400){x=-1.58+.04*m; y=Re(FIT1(log(1.3),x)); if(x>11.1) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n"); DO(m,400){x=-1.52+.04*m; y=Re(FIT1(log(1.2),x)); if(x>11.1) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n"); DO(m,400){x=-1.42+.04*m; y=Re(FIT1(log(1.1),x)); if(x>11.1) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 .5 0 RGB S\n"); // DO(m,400){x=-1.65+.04*m; y=Re(FIT1(1./M_E,x)); if(x>11.1) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".03 W 0 0 .7 RGB S\n"); DO(m,400){x=-1.64+.04*m; y=Re(FIT1(log(sqrt(2.)),x)); if(x>11.1) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".03 W .8 0 0 RGB S\n"); DO(m,400){x=-1.873+.01*m; y=Re(FIT1(1.,x)); if(y>11) break; if(m==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 0 0 RGB S\n"); // DO(m,401){x=-1.987+.01*m;y=Re(F4TEN(x));if(m==0)M(x,y)else L(x,y)}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"); fprintf(o,"showpage\n%cTrailer",'%'); fclose(o); system("epstopdf tet10bx.eps"); system(   "open tet10bx.pdf"); //mac //    system(    "xpdf tet10bx.pdf"); // linux getchar; system("killall Preview");// mac }

Latex generator of labels
% File tet10bx.pdf sould be generated with code above in order to compile the Latex document below % % \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}% \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 528pt% \paperheight 488pt% \topmargin -102pt% \oddsidemargin -78pt% \textwidth 610pt% \textheight 570pt% % \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% \sx{4}% {\normalsize% \begin{picture}(155,122)% \put(0,0){\ing{tet10bx}}% %\put(-2,125){\sx{.7}{$y\!=\!{\rm tet}_{b}(x)$ ~as solution of~ $F(z\!+\!1)=\exp_b(F(z))$ ~,~ $F(0)\!=\!1$}}% %\put( 0,125){\sx{.55}{$y\!=\!F(x)$ ~as solution of~ $F(z\!+\!1)\!=\!\exp_b(F(z))$ ,~ $F(0)\!=\!1$}}% \put( 17,120){\sx{.6}{$y$}}% \put( 17,100){\sx{.6}{$8$}}% \put( 17,80){\sx{.6}{$6$}}% \put( 17,60){\sx{.6}{$4$}}% \put( 18,48){\sx{.6}{e}}% \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( 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(2,90){\sx{.9}{$y\!=\!e\!\big(F_{1}(x\!+\!\rmi o)\big)$}}% %\put(183,105){\sx{1.}{$y\!=\!F_{3}(x)$}}% %\put(193,52){\sx{1.}{$y\!=\!\rme$}}% \put(26.6,86){\sx{.5}{\rot{90} $b\!\rightarrow\!\infty$ \ero } }% \put(31,106){\sx{.5}{\rot{88} $b\!=\!10$ \ero } }% %\put(39.2,106){\sx{.5}{\rot{86} $b\!=\! 3$ \ero } }% \put(42,87){\sx{.5}{\rot{87} $b\!=\! \rme$ \ero } }% \put(48.4,106){\sx{.5}{\rot{86} $b\!=\!2$ \ero } }% \put(53.3,106){\sx{.45}{\rot{85} $b\!=\!1.9$ \ero } }% \put(60.3,106){\sx{.45}{\rot{85} $b\!=\!1.8$ \ero } }% \put(68.4,106){\sx{.45}{\rot{84} $b\!=\!1.7$ \ero } }% \put(83,106){\sx{.45}{\rot{83} $b\!=\!1.6$ \ero } }% \put(128,106){\sx{.47}{\rot{80} $b\!=\!1.5$ \ero } }% %\put(55,88){\sx{1.3}{$y\!=\!\tet_b(x)$ } }% %\put(124,106){\sx{.6}{$b\!=\!1.5$}}% \put(109,46){\sx{.4}{\rot{2}$b\!=\!\exp(1/\rme)$\ero}}% \put(120,41.8){\sx{.34}{$b\!=\!\sqrt{2}$}}% \put(120,38){\sx{.36}{$b\!=\!1.4$}}% \put(120,33.6){\sx{.36}{$b\!=\!1.2$}}% %\put(115,29.6){\sx{.36}{$b\!=\!1.1$}}% \put(123,29){\sx{.36}{$b\!\rightarrow\!1$}}% \end{picture}}% \end{document}