File:Autraplot.jpg

Fig.20.9 from page 283 of book «Superfunctions»^[1], 2020.

This image is used also as Рис.20.9 at page 293 of the Russian version ^[2], 2014.

The figure shows the explicit plot of function AuTra and two its asymptotics.

AuTra is inverse function of function SuTra, \( \mathrm{AuTra} = \mathrm{SuTra}^{-1} \)

Function SuTra is Entire Function with Logarithmic Asymptotic.

Functions AuTra and SuTra are described also at the «Applied Mathematical Sciences» ^[3], 2013.

Function SuTra is superfunction of the Trappmann function \(z \mapsto z+\exp(z) \).

AuTra is the Abelfunction.

Warning! There is misprint in the book in figure 20.9:

The label of the bottom asymptotic should be \(y\!=\!x_0\!+\!x/2\!-\!\mathrm e^{-x}\!-\!\mathrm e^x/6 \)

It is corrected in this picture and the Latex code. (But the misprint remains in the printed version of the Book.)

The C++ code is initially correct.

C++ generator of curves

/* Files ado.cin, conto.cin, sutran.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 std::complex<double> z_type;
#define Re(x) x.real()
#define Im(x) x.imag()
#define I z_type(0.,1.)
z_type tra(z_type z){ return exp(z)+z;}
#include "ado.cin"
#include "arctran.cin"
#include "sutran.cin"
#include "autran.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("autraplo.eps","w");  ado(o,412,412);
fprintf(o,"104 204 translate\n 100 100 scale\n");
fprintf(o,"1 setlinejoin 2 setlinecap\n");
for(n=-2;n<3;n++) {M(-1,n)L(3,n)}
for(m=-1;m<4;m++) {M(m,-2)L(m,2)}
// 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=-2.01 +.02*m; y=Re(sutran(x)); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.01) break;} fprintf(o,".01 W 0 0 1 RGB S\n");
//DO(m,740){x=-3.1 +.02*m; y=Re(sutran(x)); if(m==0) M(y,x) else L(y,x) if(x>3.03||y>6.01) break;} fprintf(o,".03 W 0 1 1 RGB S\n");
//DO(m,740){x=-1.25 +.02*m; y=Re(autra(x)); if(m==0) M(x,y) else L(x,y) if(x>6.03||y>6.01) break;} fprintf(o,".03 W 1 .5 0 RGB S\n"); 
DO(m,740){x=-1.1 +.02*m; y=Re(autran(x)); if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.01) break;} fprintf(o,".03 W 1 0 0 RGB S\n"); 
DO(m,740){x=-1.1 +.02*m; y=.5*x-exp(-x)+1.1259817765745026; if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.01) break;} fprintf(o,".01 W 0 0 0 RGB S\n"); 
DO(m,740){x=-1.1 +.02*m; y=.5*x-exp(-x)+1.1259817765745026-exp(x)/6.; if(m==0) M(x,y) else L(x,y) if(x>3.03||y>3.01) break;} fprintf(o,".01 W 0 0 0 RGB S\n"); 
//DO(m,740){x=-1.25 +.02*m; y=.5*x-exp(-x)+1.1259817765745026-exp(x)/6.+exp(2*x)/16; if(m==0) M(x,y) else L(x,y) if(x>6.03||y>6.01) break;} fprintf(o,".01 W 0 0 0 RGB S\n"); 
fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%');  fclose(o);
    system("epstopdf autraplo.eps"); 
    system(    "open autraplo.pdf"); //for macintosh
    getchar(); system("killall Preview"); // For macintosh
}

Latex generator of labels

\documentclass[12pt]{article} 
\usepackage{geometry} 
\usepackage{graphicx}
\usepackage{rotating}
\paperwidth 406pt
\paperheight 412pt
\topmargin -98pt
\oddsidemargin -75.6pt
\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}(402,404)
\put(112,391){\sx{2.2}{$y$}}
\put(112,296){\sx{2.2}{$1$}}
\put(112,196){\sx{2.2}{$0$}}
\put(102, 96){\sx{2.2}{$-1$}}
\put(99,180){\sx{2.2}{$0$}}
\put(199,180){\sx{2.2}{$1$}}
\put(299,180){\sx{2.2}{$2$}}
\put(390,180){\sx{2.2}{$x$}}
\put(0,0){\ing{autraplo}}
\put(153,303){\sx{2}{\rot{35}$y\!=\!x_0\!+\!x/2\!-\!\mathrm e^{-x}$\ero}}
\put(267,325){\sx{2.2}{\rot{17}$y\!=\!\mathrm{AuTra}(x)$\ero}}
%\put(185,128){\sx{2.2}{$y\!=\!x/2\!-\!\mathrm e^x\!+\!1.1259817765745026-exp(x)/6.+exp(2*x)/16)$}}
\put(166,122){\sx{2}{$y\!=\!x_0\!+\!x/2\!-\!\mathrm e^{-x}\!-\!\mathrm e^x/6$}}
\end{picture}
\end{document}

References

Keywords

«ado.cin», «AuTra», «Abel function», «Abelfunction», «Approximation», «Asymptotic», «Entire Function with Logarithmic Asymptotic», «Implementation», «Special function», «SuTra», «Sutran.cin», «Superfunction», «Superfunctions», «Trappmann function»,

«Суперфункции»,