Difference between revisions of "File:TraplotT.jpg"

Fig.20.1 from page 268 of book «Superfunctions»^[1], 2020.

The image is used also as Рис.20.1 at page 277 of the Russian version «Суперфункции» ^[2], 2014.

The figure shows the explicit plot of the Trappmann function

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

For comparison, the exponential is shown,

\(y=\exp(x)\)

This function is interesting as a transfer function, because it has no fixed point. No superfunction was believed to be possible to construct for such a transfer function. However to year 2013, the superfunction and the abel function for this transfer function were reported. One of its superfunctions, namely, SuTra, is entire function with logarithmic asymptotic ^[3]. These results are used in the books mentioned above.

C++ generator of curves

/* File ado.cin, should be loaded 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 std::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 "doya.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);
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");

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}

References

Keywords

«[[]]», «Elementary function», «Fixed point», «Henryk Trappmann», «Superfunction», «Superfunctions», «SuTra», «Transfer function», «Trappmann function»,

«Суперфункции»