File:EncPlot100.png

Explicit plot of function Enc:

\( \displaystyle y = \mathrm{Enc}(n,x) \) \( \displaystyle = \left(\frac{\mathrm e^x\!-\!1}{x}\right)^{\!n}-1 \)

versus \(x\) for \(n=1\), \(n=2\), \(n=3\) and \(n=4\).

For comparison, the thin line shows \( y=\mathrm e^x \).

Requirements

For generation of the image, file ado.cin should be loaded. Then, one by one, the two files below can be compiled.

EncP.cc

#include <math.h>
#include <stdio.h>
#include <stdlib.h>
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
#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 "ado.cin"

int main(){ int j,k,m,n; DB f,x,y, p,q, t,r; //z_type z,c,d;
FILE *o;o=fopen("encP.eps","w");ado(o,808,1308);
#define M(x,y) fprintf(o,"%6.3f %6.3f M\n", 0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.3f %6.3f L\n", 0.+x,0.+y);
fprintf(o,"404 104 translate\n 100 100 scale 2 setlinecap\n");
for(m=-4;m<5;m++){M(m,-1)L(m,12)}
for(n=-1;n<13;n++){M( -4,n)L(4,n)}
fprintf(o,".008 W 0 0 0 RGB S\n");

for(n=0;n<161;n++){x=-4.05+.1*n; y=exp(x);                    ;if(n==0)M(x,y) else L(x,y) if(y>12.) break;} fprintf(o,".007 W 0 0 0 RGB S\n");
for(n=0;n<161;n++){x=-4.05+.1*n; f=(exp(x)-1.)/x; y=f-1.      ;if(n==0)M(x,y) else L(x,y) if(y>12.) break;} fprintf(o,".02 W 0 0 0 RGB S\n");
for(n=0;n<161;n++){x=-4.05+.1*n; f=(exp(x)-1.)/x; y=f*f-1.    ;if(n==0)M(x,y) else L(x,y) if(y>12.) break;} fprintf(o,".02 W .9 0 0 RGB S\n");
for(n=0;n<161;n++){x=-4.05+.1*n; f=(exp(x)-1.)/x; y=f*f*f-1.  ;if(n==0)M(x,y) else L(x,y) if(y>12.) break;} fprintf(o,".02 W 0 .8 0 RGB S\n");
for(n=0;n<261;n++){x=-4.05+.1*n; f=(exp(x)-1.)/x; y=f*f*f*f-1.;if(n==0)M(x,y) else L(x,y) if(y>12.) break;} fprintf(o,".02 W 0 0 .9 RGB S\n");

fprintf(o,"showpage\n%c%cTrailer",'%','%'); fclose(o);
      system("epstopdf encP.eps");
      system(    "xpdf encP.pdf");
}

EncPlot.tex

\documentclass[12pt]{article}
\usepackage{geometry}
\paperwidth 810pt
\paperheight 1210pt
\textheight 2000pt
\usepackage{graphicx}
\newcommand \sx {\scalebox}
\newcommand \ing {\includegraphics}
\usepackage{rotating}
\newcommand \rot {\begin{rotate}}
\newcommand \ero {\end{rotate}}
\topmargin -108pt
\oddsidemargin -72pt
\parindent 0pt
\begin{document}
\begin{picture}(602,1210)
\put(0,0){\ing{encP.pdf}}
\put(380,1187){\sx{4.1}{$y$}}
%\put(358,1192){\sx{4}{$11$}}
\put(358,1092){\sx{4}{$10$}}
\put(380,992){\sx{4}{$9$}}
\put(380,892){\sx{4}{$8$}}
\put(380,792){\sx{4}{$7$}}
\put(380,692){\sx{4}{$6$}}
\put(380,592){\sx{4}{$5$}}
\put(380,492){\sx{4}{$4$}}
\put(380,392){\sx{4}{$3$}}
\put(380,292){\sx{4}{$2$}}
\put(380,192){\sx{4}{$1$}}
%\put(368,92){\sx{4}{$0$}}
\put(62,66){\sx{4}{$-3$}}
\put(163,66){\sx{4}{$-2$}}
\put(264,66){\sx{4}{$-1$}}
\put(395,66){\sx{4}{$0$}}
\put(494,68){\sx{4}{$1$}}
\put(595,68){\sx{4}{$2$}}
\put(696,66){\sx{4}{$3$}}
\put(782,68){\sx{4.1}{$x$}}

\put(527,1108){\rot{87}\sx{4}{$n\!=\!4$}\ero}
\put(562,1108){\rot{86}\sx{4}{$n\!=\!3$}\ero}
\put(624,1108){\rot{85}\sx{4}{$n\!=\!2$}\ero}
\put(640,880){\rot{82}\sx{3.8}{$y\!=\!\exp(x)$}\ero}
\put(793,1108){\rot{83}\sx{4}{$n\!=\!1$}\ero}

\put(140,122){\rot{1}\sx{3.8}{$y\!=\!\mathrm e^x$}\ero}

\end{picture}
\end{document}

Generation

make EncP
./EncP
pdflatex EncPlot.tex
convert -density 100 EncPlot.pdf PNG8:EncPlot100.png

References

Abelfunction Ackermann ArkKneser