File:ZexPlot.png

Explicit plot of the ArcLambertW function, \[ y=\mathrm{ArcLambertW}(x)=x\cdot \exp(x) \]

This plot is used as Fig.11.1 at page 135 of book «Superfunctions», 2020 ^[1][2]
in order to show the Transfer function considered in Chapter 11 as an example for the exotic iteration.

Generators

C++ generator of the curve

//File ado.cin

void ado(FILE *O, int X, int Y)
{       fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%');
       fprintf(O,"%c%cBoundingBox: 0 0 %d %d\n",'%','%',X,Y);
       fprintf(O,"/M {moveto} bind def\n");
       fprintf(O,"/L {lineto} bind def\n");
       fprintf(O,"/S {stroke} bind def\n");
       fprintf(O,"/s {show newpath} bind def\n");
       fprintf(O,"/C {closepath} bind def\n");
       fprintf(O,"/F {fill} bind def\n");
       fprintf(O,"/o {.1 0 360 arc C S} bind def\n");
       fprintf(O,"/times-Roman findfont 20 scalefont setfont\n");
       fprintf(O,"/W {setlinewidth} bind def\n");
       fprintf(O,"/RGB {setrgbcolor} bind def\n");}

should be loaded (if not yet loaded) to the current 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"ado.cin"
 #define M(x,y) fprintf(o,"%5.3f %5.3f M\n",0.+x,0.+y);
 #define L(x,y) fprintf(o,"%5.3f %5.3f L\n",0.+x,0.+y);
 
 main(){ int j,k,m,n; DB x,y, a;
 FILE *o;o=fopen("arclambertw.eps","w");ado(o,410,408);
 fprintf(o,"304 104 translate\n 100 100 scale\n");
 for(m=-3;m<2;m++){ M(m,-1)L(m,3)}
 for(n=-1;n<4;n++){ M(-3,n)L(1,n)}
 fprintf(o,".01 W 0 0 0 RGB S\n");
 for(n=0;n<410;n+=2){x=-3.02+.01*n; y=x*exp(x); if(n==0)M(x,y) else L(x,y) }
 fprintf(o,".03 W 0 .8 0 RGB S\n");
 M( 1,    M_E) L(0,    M_E)
 M(-1,-1./M_E) L(0,-1./M_E)
 fprintf(o,".005 W 0 0 0 RGB S\n");
 fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
 system("epstopdf arclambertw.eps");
 system(    "open arclambertw.pdf"); //these 2 commands may be specific for macintosh
 getchar(); system("killall Preview");// if run at another operational sysetm, may need to modify
 }

Latex generator of the lables

% Copyleft 2011 by Dmitrii Kouznetsov%<br>
 \documentclass[12pt]{article} %<br>
 \usepackage{geometry} %<br>
 \usepackage{graphicx} %<br>
 \usepackage{rotating} %<br>
 \paperwidth 410pt %<br>
 \paperheight 404pt %<br>
 \topmargin -103pt %<br>
 \oddsidemargin -94pt %<br>
 \textwidth 1200pt %<br>
 \textheight 600pt %<br>
 \pagestyle {empty} %<br>
 \newcommand \sx {\scalebox} %<br>
 \newcommand \rot {\begin{rotate}} %<br>
 \newcommand \ero {\end{rotate}} %<br>
 \newcommand \ing {\includegraphics} %<br>
 \begin{document} %<br>
 { \begin{picture}(408,410) %<br>
 \put(1,9){\ing{arclambertw}} %<br>
 \put(288,398){\sx{2.9}{$y$}} %<br>
 \put(309,378){\sx{3.1}{$\mathrm e$}} %<br>
 \put(288,303){\sx{2.8}{$2$}} %<br>
 \put(288,203){\sx{2.8}{$1$}} %<br>
 \put(288,103){\sx{2.8}{$0$}} %<br>
 \put(309,70){\sx{2.6}{$-\!1/\mathrm e$}} %<br>
 \put( 86, 90){\sx{2.5}{$-\!2$}} %<br>
 \put(186,90){\sx{2.5}{$-\!1$}}  %<br>
 % \put(300,-9){\sx{2.5}{$0$}} %<br>
 \put(398,90){\sx{2.6}{$x$}} %<br>
 \end{picture} %<br>
 } %<br>
 \end{document} 

References

Keywords

«Exotic iteration», «[[]]», «Exotic iteration», «Fixed point», «LambertW», «Superfunctions», «Transfer function», «Zex»,

«Суперфункции»,