File:SuTraAsy2plotT.png

Function SuTra of real argument and its asympototics.

at large negative values of the input, \[ \mathrm{SuTra}(x) \sim -\ln(-x) \]

At large positieve values of the input, \[ \mathrm{SuTra}(x) \underset{\mathrm{ate},\ x\to +\infty}{\sim} \mathrm{tet}(x\!-\!x_{\mathrm {st}} ) \] The finger estimate suggests that \(\ x_{\mathrm {st}} \approx 0.7 \)

Function SuTra is specified as «Entire Function with Logarithmic Asymptotic».
It is described in book «Superfunctions»^[1], 2020 and also in Applied Mathematical Sciences ^[2], 2013.

The generator below is copilefld. Please attribute the source at the reuse. The attribution helps to trace (and to correct) mistakes if any.

C++

/* subroutines ado.cin, Tania.cin, LambertW.cin, SuZex.cin, fslog.cin should be loaded in order to compile the source 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 "fslog.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);
int 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,812,812);
//FILE *o;o=fopen("19.eps","w");  ado(o,712,612);
FILE *o;o=fopen("SuTraAsy2plot.eps","w");  ado(o,712,612);
 fprintf(o,"202 202 translate\n 100 100 scale\n");
 fprintf(o,"1 setlinejoin 2 setlinecap\n");

 DO(m,240){x=-2.02+.02*m; y=Re(sutra(x)); if(m==0) M(x,y) else L(x,y) if(x>5.03||y>4) break;} fprintf(o,".03 W 0 0 1 RGB S\n");
 DO(m,98){x=-2.02+.02*m; y=-log(-x);     if(m==0) M(x,y) else L(x,y) if(x>5.03||y>4) break;} fprintf(o,".01 W 0 0 0 RGB S\n"); 
// DO(m,74){x=-2.02+.1*m; y=Re(FSLOG(x));  if(m==0) M(x,y) else L(x,y) if(x>5.03||y>4) break;} fprintf(o,".02 W 1 0 0 RGB S\n"); 
 DO(m,65){x=-2.05+.1*m; y=Re(FSLOG(sutra(x))); if(m==0) M(x,y) else L(x,y) if(x>5.03||y>4) break;} fprintf(o,".05 W 0 1 0 RGB S\n"); 

M(-2,-2-.7)L(5,5-.7) fprintf(o,".006 W 0 0 0 RGB S\n");
M(-1,-1-1.1)L(5,5-1.1) fprintf(o,".006 W 0 0 0 RGB S\n");

 for(n=-2;n<5;n++) {M(-2,n)L(5,n)}
 for(m=-2;m<6;m++) {M(m,-2)L(m,4)}
 // 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");

fprintf(o,"showpage\n"); fprintf(o,"%c%cTrailer\n",'%','%');  fclose(o);
      system("epstopdf SuTraAsy2plot.eps"); 
      system(    "open SuTraAsy2plot.pdf"); //for macintosh
//      getchar(); system("killall Preview"); // For macintosh
return 0;
}
//

Latex

\documentclass[12pr]{article}
\paperwidth 740pt
\paperheight 640pt
\textwidth 800pt
\textheight 700pt
\topmargin -96pt
\oddsidemargin -66pt
\usepackage{graphicx}
\usepackage{rotating}
\newcommand \sx {\scalebox}
\newcommand \ing {\includegraphics}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\parindent 0pt
\begin{document}
\begin{picture}(720,620)
\put(20,20){\ing{SuTraAsy2plot}}
\put( 4,606){\sx{3.1}{\(y\)}}
\put(224,590.4){\sx{2.9}{\(y\!=\!1\!+\!\mathrm e\)}}
\put( 6,516){\sx{2.4}{\(3\)}}
\put( 6,416){\sx{2.4}{\(2\)}}
\put( 6,316){\sx{2.4}{\(1\)}}
\put( 6,216){\sx{2.4}{\(0\)}}
\put(-6,114){\sx{2.3}{\(-\!1\)}}
\put(-6,14){\sx{2.3}{\(-\!2\)}}
\put(0,0){\sx{2.4}{\(-2\)}}
\put(102,0){\sx{2.4}{\(-1\)}}
\put(218,0){\sx{2.4}{\(0\)}}
\put(318,0){\sx{2.4}{\(1\)}}
\put(418,0){\sx{2.4}{\(2\)}}
\put(518,0){\sx{2.4}{\(3\)}}
\put(618,0){\sx{2.4}{\(4\)}}
\put(710,1){\sx{2.7}{\(x\)}}
\put(155,276){\sx{2.6}{\rot{72}\(y=-\ln(-x)\)\ero}}
\put(232,240){\sx{2.6}{\rot{48}\(y=\mathrm{SuTra}(x)\)\ero}}
\put(255,175){\sx{2.6}{\rot{44}\(y=x-0.7\)\ero}}
\put(375,255){\sx{2.6}{\rot{44}\(y=x-1.1\)\ero}}

\put(0,48){\sx{2.9}{\rot{14}\(y\!=\!\mathrm{ate}\big(\mathrm{SuTra}(x)\big)\)\ero}}
\end{picture}
\end{document}

References

Keywords

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

«Arctetral asymptotic», «Arctetration», «Asymptotic», «Superfunction», «Superfunctions», «SuTra», «Trappmann function»,