File:Tet10bxr.jpg

Explicit plot of tetration for real values of base $b\!>\!1$.

$y\!=\! \mathrm{tet}_b(x)$ versus $x$ for various $b$

C++ generator of curves
Files ado.cin, fit1.cin, efjh.cin, f2048ten.inc, f4ten.cin should be loaded in order to compile the code below //using namespace std; typedef std::complex z_type; //b=10 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" 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("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 }
 * 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.)
 * 1) include "f4ten.cin"
 * 2) include "fit1.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);

Latex generator of curves
\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}