File:TraplotT.jpg

Explicit plot of the Trappmann function

$y=\mathrm{tra}(x)=\exp(x)\!+\!x$

For comparison, the exponent is shown,

$y=\exp(x)$

This function is interesting as a transfer function, because it has no fixed point. One of its superfuncitons, namely, SuTra, is entire function with logarithmic asymptotic .

The image is used as Figure 20.1 in book Superfunction (In Russian, 2014; the English version is in preparation, 2015) .

C++ generator of curves
Files ado.cin, should be loaded to the working directory in order to compile the code below //using namespace std; typedef std::complex z_type; //#include "Tania.cin" // need for LambertW /* 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));} int main{ int j,k,m,n; DB x,y, p,q, t; z_type z,c,d; FILE *o;o=fopen("traplot.eps","w");  ado(o,504,806); fprintf(o,"302 302 translate\n 100 100 scale\n"); fprintf(o,"1 setlinejoin 2 setlinecap\n"); for(n=-3;n<7;n++) {M(-3,n)L(2,n)} for(m=-3;m<3;m++) {M(m,-3)L(m,6)} M(0,M_E)L(1,M_E) M(0,M_E+1)L(1,M_E+1) 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,".04 W 0 0 1 RGB S\n"); DO(m,700){x=-3.02+.02*m; y=exp(x); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>10.03) break;} fprintf(o,".01 W 0 0 0 RGB S\n");
 * 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 "doya.cin" // need for LambertW
 * 2) include "LambertW.cin" // need for AuZex
 * 3) include "SuZex.cin"
 * 4) include "AuZex.cin"
 * 1) include "ado.cin"
 * 2) define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
 * 3) define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
 * 1) define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);

fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%'); fclose(o); system("epstopdf traplot.eps"); system(   "open traplot.pdf"); //for macintosh getchar; system("killall Preview"); // For macintosh }

Latex generator of labelw
\documentclass[12pt]{article} \usepackage{geometry} \usepackage{graphicx} \usepackage{rotating} \paperwidth 506pt \paperheight 808pt \topmargin -105pt \oddsidemargin -73pt \textwidth 1100pt \textheight 1100pt \pagestyle {empty} \newcommand \sx {\scalebox} \newcommand \rot {\begin{rotate}} \newcommand \ero {\end{rotate}} \newcommand \ing {\includegraphics} \parindent 0pt \pagestyle{empty} \begin{document} \begin{picture}(604,804) %\put(10,10){\ing{PowPlo}} \put(0,0){\ing{traplot}} \put(280,784){\sx{3.4}{$y$}} \put(280,693){\sx{3.4}{$4$}} \put(224,666){\sx{3.4}{$1\!+\!\rm e$}} \put(280,593){\sx{3.4}{$3$}} \put(280,566){\sx{3.4}{$\rm e$}} \put(280,493){\sx{3.4}{$2$}} \put(280,393){\sx{3.4}{$1$}} \put(280,294){\sx{3.4}{$0$}} \put(252,191){\sx{3.4}{$-1$}} \put(252,091){\sx{3.4}{$-2$}} \put(070,272){\sx{3.4}{$-2$}} \put(170,272){\sx{3.4}{$-1$}} \put(294,272){\sx{3.4}{$0$}} \put(394,272){\sx{3.4}{$1$}} %\put(497,308){\sx{2.4}{$2$}} \put(482,274){\sx{3.4}{$x$}}

\put(24,342){\sx{3.4}{$y\!=\!\exp(x)$}} \put(56,42){\sx{3.4}{$y\!=\!\mathrm{tra}(x)$}}

\end{picture} \end{document}