File:KellerPlotT.png

Explicit plot of various iterations $t$ the Keller function


 * $ y=\mathrm{Keller}^t(x)=\mathrm{Shoka}\Big( t + \mathrm{ArcShoka}(x)\Big)$

To plot this graphic, the iterations of the Keller function are implemented through the Shoka function and the ArcShoka function.

C++ generator of curves]]
// File ado.cin shold 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.)


 * 1) include"ado.cin"

z_type Shoka(z_type z)  { return z + log(exp(-z)+(M_E-1.)); } z_type ArcShoka(z_type z){ return z + log((1.-exp(-z))/(M_E-1.)) ;}

main{ int j,k,m,n; DB x,y, a; FILE *o;o=fopen("KellerPlot.eps","w");ado(o,408,412); fprintf(o,"4 4 translate\n 100 100 scale 2 setlinecap 1 setlinejoin\n"); for(m=0;m<5;m++){ M(m,0)L(m,4)} for(n=0;n<5;n++){ M(0,n)L(4,n)} M(0,0)L(4,4) fprintf(o,".01 W 0 0 0 RGB S\n"); DO(n,134){x=.005+.01*n;y=Re(Shoka(3.+ArcShoka(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 0 .5 RGB S\n"); DO(n,216){x=.005+.01*n;y=Re(Shoka(2.+ArcShoka(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 0 .5 RGB S\n"); DO(n,154){x=.005+.02*n;y=Re(Shoka(1.+ArcShoka(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W 0 0 .5 RGB S\n"); DO(n,101){x=.005+.04*n;y=Re(Shoka(-1.+ArcShoka(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W .5 0 0 RGB S\n"); DO(n,101){x=.005+.04*n;y=Re(Shoka(-2.+ArcShoka(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W .5 0 0 RGB S\n"); DO(n,101){x=.005+.04*n;y=Re(Shoka(-3.+ArcShoka(x)));if(n==0)M(x,y)else L(x,y)} fprintf(o,".02 W .5 0 0 RGB S\n"); fprintf(o,"showpage\n%cTrailer",'%'); fclose(o); system("epstopdf KellerPlot.eps"); system(   "open KellerPlot.pdf"); //these 2 commands may be specific for macintosh getchar; system("killall Preview");// if run at another operational sysetm, may need to modify }
 * 1) define M(x,y) fprintf(o,"%6.3f %6.3f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%6.3f %6.3f L\n",0.+x,0.+y);

Latex generator of labels
% File KellerPlot.pdf should be generated with the code above in order to compile the Latex document below. %

\documentclass[12pt]{article} % \usepackage{geometry} % \usepackage{graphicx} % \usepackage{rotating} % \paperwidth 419pt % \paperheight 426pt % \topmargin -103pt % \oddsidemargin -83pt % \textwidth 1200pt % \textheight 600pt % \pagestyle {empty} % \newcommand \sx {\scalebox} % \newcommand \rot {\begin{rotate}} % \newcommand \ero {\end{rotate}} % \newcommand \ing {\includegraphics} % \begin{document} % \sx{1}{ \begin{picture}(810,410) % \put(1,9){\ing{KellerPlot}} % \put(-12,401){\sx{2.8}{$y$}} % \put(-12,303){\sx{2.8}{$3$}} % \put(-12,203){\sx{2.8}{$2$}} % \put(-12,103){\sx{2.8}{$1$}} % \put(0,-9){\sx{2.5}{$0$}} % \put(100,-9){\sx{2.5}{$1$}} % \put(200,-9){\sx{2.5}{$2$}} % \put(300,-9){\sx{2.5}{$3$}} % \put(392,-7){\sx{2.6}{$x$}} % %\put(560,214){\rot{37}\sx{4}{$y=\mathrm{Tania}(x)$}\ero} % \put( 88,354){\rot{53}\sx{2.8}{$t\!=\!3$}\ero} % \put(160,354){\rot{50}\sx{2.8}{$t\!=\!2$}\ero} % \put(246,354){\rot{48}\sx{2.8}{$t\!=\!1$}\ero} % \put(336,350){\rot{45}\sx{2.8}{$t\!=\!0$}\ero} % \put(340,218){\rot{44}\sx{2.8}{$t\!=\!-1$}\ero} % \put(344,136){\rot{41}\sx{2.7}{$t\!=\!-2$}\ero} % \put(338, 68){\rot{34}\sx{2.7}{$t\!=\!-3$}\ero} % \end{picture} % } % \end{document} %

Copyleft status
Copyleft 2012 by Dmitrii Kouznetsov. The image and the generators above may be used for free; attribute the source.