Difference between revisions of "File:SuTraPlo3T.jpg"

Fig.20.5 from page 276 of book of «Superfunctions»^[1], 2020.

The image is used also as Рис.20.5 at page 284 of the Russian version «Суперфункции» ^[2], 2014.

The figure shows the explicit plot of function SuTra and its asymptotic \( x \mapsto -\ln(-x)\).

The additional horizontal gridline shows \(1\!+\!\mathrm e=\mathrm{SuTra}(2)\)

SuTra is superfunction of the Trappmann function \(\mathrm{tra} = z \mapsto z+\mathrm e^z\).

Function SuTra satisfies the Transfer equation

\(\mathrm{tra}(\mathrm{SuTra}(z))=\mathrm{SuTra}(z+1) \)

Here, the Trappmann function \(\mathrm{tra}\) is treated as a Transfer function.

Until year 2013, function SuTra had been believed to be difficult to evaluate (if at al) as the Trappmann function has no fixed point.
However, after to guess the asymptotic behavior (thin curve in the explicit plot), the evaluation of SuTra becomes straightforward.
It is described at Applied Mathematical Sciences ^[3], 2013.

To year 2026, function \(z \mapsto -\mathrm{ZuZex}(-z)\)
seems to be the only reported entire function with logarithmic asymptotic.

C++ generator of curves

/* Files ado.cin, SuZex.cin, LambertW.cin and Tania.cin sould be loaded to the working directory in order to compile the C++ 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));}

 z_type sutra(z_type z){ if( Re(z)<2. || fabs(Im(z))>2. ) return log(suzex(z));
                                                          return tra(sutra(z-1.));}

 #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("SuTraPlo3.eps","w");  ado(o,604,812);
 fprintf(o,"302 202 translate\n 100 100 scale\n");
 fprintf(o,"1 setlinejoin 2 setlinecap\n");
 for(n=-2;n<7;n++) {M(-3,n)L(3,n)}
 for(m=-3;m<4;m++) {M(m,-2)L(m,6)}
 // M(M_E,0)L(M_E,1) M(0,M_E)L(1,M_E)  
 M(0,1.+M_E) L(2,1.+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=-5.02 +.02*m; y=Re(sutra(x));  if(m==0) M(x,y) else L(x,y) if(x>3.03||y>6.1) break;} fprintf(o,".03 W 0 0 1 RGB S\n");
 DO(m,240){x=-5.02 +.02*m; y=-log(-x);      if(m==0) M(x,y) else L(x,y) if(x>3.03||y>6.03) break;} fprintf(o,".01 W .5 0 0 RGB S\n"); 
 fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%');  fclose(o);
      system("epstopdf SuTraPlo3.eps"); 
      system(    "open SuTraPlo3.pdf"); //for macintosh
      getchar(); system("killall Preview"); // For macintosh
}

Latex generator of labels

% file SuTraPlo3.pdf should be generated with the code above in order to compile the Latex document below. % Copyleft 2012 by Dmitrii Kouznetsov

\documentclass[12pt]{article} 
\usepackage{geometry} 
\usepackage{graphicx}
\usepackage{rotating}
\paperwidth 608pt
\paperheight 816pt
\topmargin -98pt
\oddsidemargin -71pt
\textwidth 900pt
\textheight 900pt
\pagestyle {empty}
\newcommand \sx {\scalebox}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\newcommand \ing {\includegraphics}
\parindent 0pt
\pagestyle{empty}
\begin{document} 
\begin{picture}(602,805)
\put(310,794){\sx{4}{$y$}}
\put(220,566){\sx{3.8}{$1\!+\!\mathrm e$}}
\put(310,690){\sx{4}{$5$}}
\put(310,590){\sx{4}{$4$}}
\put(310,490){\sx{4}{$3$}}
\put(310,390){\sx{4}{$2$}}
\put(310,289){\sx{4}{$1$}}
\put(312,187){\sx{4}{$0$}}
\put(290, 87){\sx{4}{$-\!1$}}
%\put(325,098){\sx{4}{$0$}}

\put(068,206){\sx{4}{$-\!2$}}
\put(168,206){\sx{4}{$-\!1$}}
\put(291,206){\sx{4}{$0$}}
\put(392,206){\sx{4}{$1$}}
\put(492,206){\sx{4}{$2$}}
\put(578,206){\sx{4}{$x$}}
 
\put(462,308){\sx{4.2}{\rot{70}$y\!=\!\mathrm{SuTra}(x)$\ero}}
\put(44,314){\sx{3.8}{\rot{0}$y\!=\!-\!\ln(-\!x)$\ero}}
%\put(10,10){\ing{ExpQ2plot}}
%\put(10,10){\ing{SuZexPlot511}}
\put(0,0){\ing{SuTraplo3}}
%\put(0,0){\ing{SuTraplo3}}
\end{picture}
\end{document}

References

Keywords

«ado.cin», «SuZex.cin», «LambertW.cin», «Tania.cin»

«Applied Mathematical Sciences», «Lambert Academic Publishing», «Logarithm», «Superfunction», «Superfunctions», «SuTra», «SuZex», «Tania function», «Zex»,

«Суперфункции»,