File:TraItu3T.jpg

Plot of the Trappmann function,

$y\!=\!\mathrm{tra}(x)\!=\!x+\mathrm e^x$, thick blue line;

and various iterates of the Trappmann function,

$y\!=\! \mathrm{tra}^c(x)~$ , thin lines, for $~c= -3,-2,-1,-0.8,-0.6,-0.4,-0.2,0,0.2,0.4,0.6,0.8,1,2,3~$.

C++ generator of curves

// Files ado.cin, Tania.cin, LambertW.cin, SuZex.cin, AuZex.cin //should be loaded to the working directory 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.)

#include "Tania.cin" // need for LambertW
#include "LambertW.cin" // need for AuZex
#include "SuZex.cin"
#include "AuZex.cin"

z_type tra(z_type z){ return exp(z)+z;}
z_type F(z_type z){ return log(suzex(z));}
z_type G(z_type z){ return auzex(exp(z));}

#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);

main(){ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d;  FILE *o;o=fopen("TraItu3.eps","w");  ado(o,604,604);
fprintf(o,"302 302 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
for(n=-3;n<4;n++) {M(-3,n)L(3,n)}
for(m=-3;m<4;m++) {M(m,-3)L(m,3)}
// M(M_E,0)L(M_E,1) M(0,M_E)L(1,M_E)  
fprintf(o,".004 W S\n");
// DO(m,700){x=.01 +.02*m; y=Re(LambertW(LambertW(x)));if(m==0) M(x,y) else L(x,y) if(x>12.03||y>12.03) break;} fprintf(o,".033 W 1 0 0 RGB S\n");
// DO(m,700){x=.01 +.02*m; y=Re(LambertW(x));if(m==0) M(x,y) else L(x,y) if(x>12.03||y>12.03) break;} fprintf(o,".04 W 1 .5 0 RGB S\n");
// M(0,0) L(12.03,12.03) fprintf(o,".04 W 0 1 0 RGB S\n");
DO(m,700){x=-3.02 +.02*m; y=Re(tra(x));          if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;} fprintf(o,".05 W 0 0 1 RGB S\n");
DO(m,700){x=-3.02 +.02*m; y=Re(tra(tra(x)));     if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;} fprintf(o,".012 W 0 0 1 RGB S\n");
DO(m,700){x=-3.02 +.02*m; y=Re(tra(tra(tra(x))));if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;} fprintf(o,".012 W 0 0 1 RGB S\n");
DO(m,700){y=-3.02+.02*m; x=Re(tra(y));          if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;} fprintf(o,".014 W 1 .4 0 RGB S\n");
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(y)));     if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;} fprintf(o,".014 W 1 0 0 RGB S\n");
DO(m,700){y=-3.02+.02*m; x=Re(tra(tra(tra(y))));if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;} fprintf(o,".014 W 1 0 0 RGB S\n");
for(n=-10;n<11;n+=2){
  DO(m,700){x=-3.01 +.02*m; y=Re(G(x)); y=Re(F(.1*n+y)); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.03) break;}  
  fprintf(o,".01 W 0 0 0 RGB S\n");
                   }
fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%');  fclose(o);
      system("epstopdf TraItu3.eps"); 
      system(    "open TraItu3.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

Latex Generator of labels

% %<br> % file PowPlo.pdf should be generated with the code above in order to compile the Latex document below. %<br> % Copyleft 2012 by Dmitrii Kouznetsov <br> % \documentclass[12pt]{article} % <br> \usepackage{geometry} % <br> \usepackage{graphicx} % <br> \usepackage{rotating} % <br> \paperwidth 606pt % <br> \paperheight 606pt % <br> \topmargin -105pt % <br> \oddsidemargin -73pt % <br> \textwidth 1100pt % <br> \textheight 1100pt % <br> \pagestyle {empty} % <br> \newcommand \sx {\scalebox} % <br> \newcommand \rot {\begin{rotate}} % <br> \newcommand \ero {\end{rotate}} % <br> \newcommand \ing {\includegraphics} % <br> \parindent 0pt% <br> \pagestyle{empty} % <br> \begin{document} % <br> \begin{picture}(602,602) % <br> %\put(10,10){\ing{PowPlo}} % <br> \put(0,0){\ing{TraItu3}} % <br> \put(311,590){\sx{2.5}{$y$}} % <br> \put(311,495){\sx{2.4}{$3$}} % <br> \put(311,395){\sx{2.4}{$1$}} % <br> \put(311,295){\sx{2.4}{$0$}} % <br> \put(307,194){\sx{2.4}{$-1$}} % <br> \put(307,093.4){\sx{2.4}{$-2$}} % <br> % <br> \put(083,308){\sx{2.4}{$-2$}} % <br> \put(183,308){\sx{2.4}{$-1$}} % <br> \put(297,308){\sx{2.4}{$0$}} % <br> \put(397,308){\sx{2.4}{$1$}} % <br> \put(497,308){\sx{2.4}{$2$}} % <br> \put(590,308){\sx{2.5}{$x$}} % <br> % <br> \put(228,532){\sx{2.4}{\rot{82}$c\!=\!3$\ero}} % <br> \put(278,532){\sx{2.4}{\rot{79}$c\!=\!2$\ero}} % <br> \put(354,532){\sx{2.4}{\rot{71}$c\!=\!1$\ero}} % <br> \put(264,350){\sx{3.1}{\rot{61}$y\!=\!x\!+\!\mathrm e^x$\ero}} % <br> %\put(427,528){\sx{2.3}{\rot{62}$c\!=\!0.6$\ero}} % <br> \put(462,534){\sx{2.3}{\rot{56}$c\!=\!0.4$\ero}} % <br> \put(506,536){\sx{2.3}{\rot{51}$c\!=\!0.2$\ero}} % <br> % <br> \put(528,531){\sx{2.4}{\rot{45}$c\!=\!0$\ero}} % <br> % <br> \put(520,475){\sx{2.4}{\rot{37}$c\!=\!-0.2$\ero}} % <br> \put(517,434){\sx{2.4}{\rot{31}$c\!=\!-0.4$\ero}} % <br> \put(527,338){\sx{2.4}{\rot{16}$c\!=\!-1$\ero}} % <br> \put(524,260){\sx{2.4}{\rot{8}$c\!=\!-2$\ero}} % <br> \put(523,212){\sx{2.4}{\rot{5}$c\!=\!-3$\ero}} % <br> \end{picture} % <br> \end{document} % <br> %